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On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points

Nguyen Manh Hung1, Nguyen Thanh Anh1* and Phung Kim Chuc2

Author affiliations

1 Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

2 Department of Mathematics, Can Tho University, Can Tho City, Vietnam

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Citation and License

Boundary Value Problems 2011, 2011:17  doi:10.1186/1687-2770-2011-17

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/17


Received:22 January 2011
Accepted:25 August 2011
Published:25 August 2011

© 2011 Hung et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we deal with the second initial boundary value problem for higher order hyperbolic systems in domains with conical points. We establish several results on the well-posedness and the regularity of solutions.

1 Introduction

Boundary value problems in nonsmooth domains have been studied in differential aspects. Up to now, elliptic boundary value problems in domains with point singularities have been thoroughly investigated (see, e.g, [1,2] and the extensive bibliography in this book). We are concerned with initial boundary value problems for hyperbolic equations and systems in domains with conical points. These problems with the Dirichlet boundary conditions were investigated in [3-5] in which the unique existence, the regularity and the asymptotic behaviour near the conical points of the solutions are established. The Neumann boundary problem for general second-order hyperbolic systems with the coefficients independent of time in domains with conical points was studied in [6]. In the present paper we consider the Cauchy-Neumann (the second initial) boundary value problem for higher-order strongly hyperbolic systems in domains with conical points.

Our paper is organized as follows. Section 2 is devoted to some notations and the formulation of the problem. In Section 3 we present the results on the unique existence and the regularity in time of the generalized solution. The global regularity of the solution is dealt with in Section 4.

2 Notations and the formulation of the problem

Let Ω be a bounded domain in ℝn, n ≥ 2, with the boundary ∂Ω. We suppose that ∂Ω is an infinitely differentiable surface everywhere except the origin, in a neighborhood of which Ω coincides with the cone K = {x : x/|x| ∈ G}, where G is a smooth domain on the unit sphere Sn-1. For each t, 0 < t ≤ ∞, denote Qt = Ω × (0, t), Ωt = Ω × {t}. Especially, we set Q = Q, Γ = ∂Ω\{0}, S = Γ × [0, +∞).

For each multi-index p = (p1,..., pn) ∈ ℕn, we use notations |p| = p1 + ... + pn, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M1">View MathML</a>. For a complex-valued vector function u = (u1 ,..., us) defined on Q, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M2">View MathML</a>.

Let us introduce the following functional spaces used in this paper. Let l denote a nonnegative integer.

Hl (Ω) - the usual Sobolev space of vector functions u defined in Ω with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M3">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M4">View MathML</a>- the space of traces of vector functions from Hl (Ω) on Γ with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M5">View MathML</a>

Hl,0 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M6">View MathML</a>

Especially, we set L2(Q, γ) = H0,0(Q, γ).

Hl,1 (Q, γ) (γ ∈ ℝ)- the weighted Sobolev space of vector functions u defined in Q with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M7">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M8">View MathML</a>- the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M9">View MathML</a> with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M10">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M11">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M12">View MathML</a> -the weighted Sobolev space of vector functions u defined in Ω with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M13">View MathML</a>

If l ≥ 1, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M14">View MathML</a> denote the spaces consisting of traces of functions from respective spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M15">View MathML</a> on the boundary Γ with the respective norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M16">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M17">View MathML</a> - the weighted Sobolev space of vector functions u defined in Q with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M18">View MathML</a>

From the definitions it follows the continuous imbeddings

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M19">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M20">View MathML</a>

(2.2)

for arbitrary nonnegative integers l, k and real number α. It is also well known (see [[2], Th. 7.1.1]) that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M21">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M22">View MathML</a> then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M23">View MathML</a>

(2.3)

with the norms being equivalent.

Now we introduce the differential operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M24">View MathML</a>

where apq = apq (x, t) are the s × s matrices with the bounded complex-valued components in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M26">View MathML</a> for all |p|, |q| ≤ m, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M27">View MathML</a> is the transposed conjugate matrix to apq. This means the differential operator L is formally self-adjoint. We assume further that there exists a positive constant μ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M28">View MathML</a>

(2.4)

for all ηp ∈ℂs, |p| = m, and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M29">View MathML</a>.

Let v be the unit exterior normal to S. It is well known that (see, e.g., [[7], Th. 9.47]) there are boundary operators Nj = Nj (x, t, D), j = 1, 2,..., m on S such that integration equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M30">View MathML</a>

(2.5)

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M31">View MathML</a> and for all t ∈ [0, ∞). The order of the operator Nj is 2m - j for j = 1, 2,..., m.

In this paper, we consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M32">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M33">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M34">View MathML</a>

(2.8)

A complex vector-valued function u Hm,1(Q, γ) is called a generalized solution of problem (2.6)-(2.8) if and only if u|t = 0 = 0 and the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M35">View MathML</a>

(2.9)

holds for all η(x, t) ∈ Hm,1(Q) satisfying η(x, t) = 0 for all t T for some positive real number T.

3 The unique solvability and the regularity in time

First, we introduce some notations which will be used in the proof of Theorems 3.3 and 3.4. For each vector function u,v defined in Ω and each nonnegative integer k,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M36">View MathML</a>

For vector functions u and v defined in Q and τ > 0, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M37">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M38">View MathML</a>

Especially, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M39">View MathML</a>

From the formally self-adjointness of the operator L, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M40">View MathML</a>

(3.1)

Next, we introduce the following Gronwall-Bellman and interpolation inequalities as two fundamental tools to establish the theorems on the unique existence and the regularity in time.

Lemma 3.1 ([8], Lemma 3.1) Assume u, α, β are real-valued continuous on an interval [a, b], β is nonnegative and integrable on [a, b], α is nondecreasing satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M41">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M42">View MathML</a>

(3.2)

From [[9], Th. 4.14], we have the following lemma.

Lemma 3.2. For each positive real number ε and each integer j, 0 < j < m, there exists a positive real number C = C (Ω, m, ε) which is dependent on only Ω, m and ε such that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M43">View MathML</a>

(3.3)

holds for all u Hm(Ω).

Now we state and prove the main theorems of this section.

Theorem 3.3. Let h be a nonnegative integer. Assume that all the coefficients apq together with their derivatives with respect to t are bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25">View MathML</a>. Then there exists a positive real number γ0 such that for each γ > γ0, if f L2(Q, σ) for some nonnegative real number σ, the problem (2.6)-(2.8) has a unique generalized solution u in the space Hm,1(Q, γ + σ) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M44">View MathML</a>

(3.4)

where C is a constant independent of u and f.

Proof. The uniqueness is proved by similar way as in [[4], Th. 3.2]. We omit the detail here. Now we prove the existence by Galerkin approximating method. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M45">View MathML</a> is an orthogonal basis of Hm(Ω) which is orthonormal in L2(Ω). Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M46">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M47">View MathML</a> are the solution of the system of the following ordinary differential equations of second order:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M48">View MathML</a>

(3.5)

with the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M49">View MathML</a>

(3.6)

Let us multiply (3.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M50">View MathML</a>, take the sum with respect to l from 1 to N, and integrate the obtained equality with respect to t from 0 to τ (0 < τ < ∞) to receive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M51">View MathML</a>

(3.7)

Now adding this equality to its complex conjugate, then using (3.1) and the integration by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M52">View MathML</a>

(3.8)

With noting that, for some positive real number ρ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M53">View MathML</a>

we can rewrite (3.8) as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M54">View MathML</a>

(3.9)

By (2.4), the left-hand side of (3.9) is greater than

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M55">View MathML</a>

We denote by I, II, III, IV the terms from the first, second, third, and forth, respectively, of the right-hand sides of (3.9). We will give estimations for these terms. Firstly, we separate I into two terms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M56">View MathML</a>

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M57">View MathML</a>

Then, by the Cauchy inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M58">View MathML</a>

By the Cauchy inequality and the interpolation inequality (3.3), for an arbitrary positive number ε1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M59">View MathML</a>

where C1 = C1(ε1) is a nonnegative constant independent of uN, f and τ. Now using again the Cauchy and interpolation inequalities, for an arbitrary positive number ε2 with ε2 < μ, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M60">View MathML</a>

where C2 = C2(ε2) is a nonnegative constant independent of uN, f and τ. For the terms III and IV, by the Cauchy inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M61">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M62">View MathML</a>

where ε3 > 0, arbitrary. Combining the above estimations we get from (3.9) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M63">View MathML</a>

(3.10)

Now fix ε1, ε2 and consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M64">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M65">View MathML</a>

We see that the function g has a unique minimum at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M66">View MathML</a>

We put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M67">View MathML</a>

(3.11)

Now we take real numbers γ, γ1 arbitrarily satisfying γ0 < γ1 < γ. Then there are positive real numbers ε1, ε2, (ε2 < μ), ρ (ρ > C2(ε1, ε2)) and ε3 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M68">View MathML</a>

(3.12)

From now to the end of the present proof, we fix such constants ε1, ε2, ε3 and ρ. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M69">View MathML</a> stand for the left-hand side of (3.10). It follows from (3.10) and (3.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M70">View MathML</a>

(3.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M71">View MathML</a>. By the Gronwall-Bellman inequality (3.2), we receive from (3.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M72">View MathML</a>

(3.14)

We see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M73">View MathML</a>

Hence, it follows from (3.14) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M74">View MathML</a>

(3.15)

Now multiplying both sides of this inequality by e-2(γ+σ)τ, then integrating them with respect to τ from 0 to ∞, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M75">View MathML</a>

(3.16)

It is clear that |||.|||Q,γ+σ is a norm in Hm,1(Q, γ + σ) which is equivalent to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M76">View MathML</a>. Thus, it follows from (3.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M77">View MathML</a>

(3.17)

From this inequality, by standard weakly convergent arguments (see, e.g., [[10], Ch. 7]), we can conclude that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M78">View MathML</a> possesses a subsequence convergent to a vector function u Hm,1(Q, γ + σ) which is a generalized solution of problem (2.6)-(2.8). Moreover, it follows from (3.17) that the inequality (3.4) holds.   □

Theorem 3.4. Let h be a nonnegative integer. Assume that all the coefficients apq together with their derivatives with respect to t up to the order h are bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M25">View MathML</a>. Let γ0 be the number as in Theorem 3.3 which was defined by formula (3.11). Let the vector function f satisfy the following conditions for some nonnegative real number σ

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M79">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M80">View MathML</a>.

Then for an arbitrary real number γ satisfying γ > γ0 the generalized solution u in the space Hm,1(Q, γ + σ) of the problem (3.6)- (3.7) has derivatives with respect to t up to the order h with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M81">View MathML</a> for k = 0, 1,..., h and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M82">View MathML</a>

(3.18)

where C is a constant independent of u and f.

Proof. From the assumptions on the regularities of the coefficients apq and of the function f it follows that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M83">View MathML</a> of the system (3.5), (3.6) has generalized derivatives with respect to t up to the order h + 2. Now take an arbitrary real number γ1 satisfying γ0 < γ1 < γ. We will prove by induction that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M84">View MathML</a>

(3.19)

and for k = 0,..., h, where the constant C is independent of N, f and τ. From (3.15) it follows that (3.19) holds for k = 0 since the norm |||·||| is equivalent to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M85">View MathML</a>. Assuming by induction that (3.19) holds for k = h - 1, we will show it to be true for k = h. To this end we differentiate h times both sides of (3.5) with respect to t to receive the following equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M86">View MathML</a>

(3.20)

From these equalities together with the initial (3.6) and the assumption (ii), we can show by induction on h that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M87">View MathML</a>

(3.21)

Now multiplying both sides of (3.20) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M88">View MathML</a>, then taking sum with respect to l from 1 to N, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M89">View MathML</a>

(3.22)

Adding the equality (3.22) to its complex conjugate, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M90">View MathML</a>

Integrating both sides of this equality with respect to t from 0 to a positive real τ with using the integration by parts and (3.21), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M91">View MathML</a>

(3.23)

This equality has the form (3.8) with uN replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M92">View MathML</a> and the last term of the righthand side of (3.8) replaced by the following expression

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M93">View MathML</a>

Since the coefficients apq together with their derivatives with respect to t up to the order h are bounded, by the Cauchy and interpolation inequalities and the induction assumption, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M94">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M99">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M100">View MathML</a>

Thus, repeating the arguments which were used to get (3.15) from (3.8), we can obtain (3.19) for k = h from (3.23).

Now we multiply both sides of (3.19) by e-2((k+1)γ+σ)τ, then integrate them with respect to τ from 0 to ∞ to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M101">View MathML</a>

(3.24)

From this inequality, by again standard weakly convergent arguments, we can conclude that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M102">View MathML</a> possesses a subsequence convergent to a vector function u(k) Hm,1(Q, (k +1)γ +σ), moreover, u(k) is the kth generalized derivative in t of the generalized solution u of problem (2.6)-(2.8). The estimation (3.18) follows from (3.24) by passing the weak convergences.   □

4 The global regularity

First, we introduce the operator pencil associated with the problem. See [11] for more detail. For convenience we rewrite the operators L(x, t, D), Nj (x, t, D) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M103">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M104">View MathML</a>

Let L0(x, t, D), N0j (x, t, D), be the principal homogeneous parts of L(x, t, D), Nj (x, t, D). It can be directly verified that the derivative Dα can be written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M105">View MathML</a>

where Pα, p (ω, ω) are differential operators of order ≤ |α| - p with smooth coefficients on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M106">View MathML</a>, r = |x|, ω is an arbitrary local coordinate system on Sn-1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M107">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M108">View MathML</a>. Thus we can write L0(0, t, D) and N0j (0, t, D) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M109">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M110">View MathML</a>

The operator pencil associated with the problem is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M111">View MathML</a>

For every fixed λ ∈ ℂ and t ∈ (0, ∞), the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216">View MathML</a> continuously maps

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M112">View MathML</a>

For some fixed t ∈ (0, ∞), a complex number λ0 is called an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216">View MathML</a> if there exists φ0 H2m(G) such that φ0 ≠ 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M217">View MathML</a>. It is well known that the spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M218">View MathML</a> for each t ∈ (0, ∞), is an enumerable set of eigenvalues (see [[2], Th. 5.2.1]).

Now let us give the main theorem of this section:

Theorem 4.1. Suppose that all the assumptions of Theorem 3.4 hold for a given positive integer h. Assume further that the strip

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M113">View MathML</a>

(4.1)

does not contain any eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216">View MathML</a> for all t ∈ (0, +∞) and for some real numbers ε and α satisfying 0 ≤ α m + ε, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M114">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M115">View MathML</a>and n is even, otherwise ε = 0. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M116">View MathML</a>for k = 0, 1,..., h - 1 and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M117">View MathML</a>

(4.2)

where C is a constant independent of u and f.

To prove Theorem 4.1 we need to establish some following lemmas.

Lemma 4.2. Let l be a nonnegative integer, t0 be a fixed number in [0, ∞), and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M118">View MathML</a> be a solution of the following elliptic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M119">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M120">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M121">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M122">View MathML</a>and the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M123">View MathML</a>

(4.5)

holds with the constant C independent of u, f, gj and t0.

Proof. Without generality we assume that the domain Ω coincides with the cone K in the unit ball. Set Ω0 = {x ∈ Ω: |x| ≥ 2-1},

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M124">View MathML</a>

and Γk = ∂Ω ∩ ∂Ωk, k = 0, 1 .... According to well known results on the regularity of solutions of elliptic boundary problems in smooth domains (see, e.g., [12]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M125">View MathML</a>

with the constant C independent of u, f, gj and t0. By making change of variable x = 2-k x' for a positive integer k, we get from (4.3), (4.4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M126">View MathML</a>

(4.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M127">View MathML</a>

(4.7)

Similarly as above, from (4.6), (4.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M128">View MathML</a>

(4.8)

with the constant C independent of u, f, gj, t0 and k. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M129">View MathML</a> be arbitrary extensions of gj to Ω, j = 1,..., m. Then we have from (4.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M130">View MathML</a>

(4.9)

Returning to variable x with noting that, in Ωk+2, 2-k-2 r ≤ 2-k-1, from (4.9) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M131">View MathML</a>

(4.10)

Taking sum both sides of these inequalities with respect to k from 1 to ∞, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M132">View MathML</a>

(4.11)

Here it is noted that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M133">View MathML</a> can be estimated by the right-hand side of (4.11). It follows from (4.11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M134">View MathML</a>

with the constant C independent of u, f, gj and t0.   □

Lemma 4.3. Let t0 be a fixed number in [0, ∞), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M135">View MathML</a> be a generalized solution of the following elliptic boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M136">View MathML</a>

(4.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M137">View MathML</a>

(4.13)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M139">View MathML</a>, j = 1,..., m, ε is defined as in Theorem 4.1. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M140">View MathML</a>. Moreover, the following inequalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M141">View MathML</a>

(4.14)

holds with the constant C independent of u, and f, gj, and t0.

Proof. Firstly, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M142">View MathML</a> for j = 1,..., m, by (2.3), it holds that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M143">View MathML</a>, and therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M144">View MathML</a>. Moreover, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M145">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M147">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M148">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M149">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M150">View MathML</a> by (2.3) and (2.2). Thus the assertion of the lemma follows from Lemma 4.2 with noting that the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M151">View MathML</a> is continuously imbedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M152">View MathML</a> according to (2.1).

Now consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M115">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M153">View MathML</a> be the greatest integer not exceeding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M154">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M155">View MathML</a>. According to [[2], Th. 7.1.1], the function u which belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M156">View MathML</a> has the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M157">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M158">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M159">View MathML</a>

with the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M160">View MathML</a>

(4.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M161">View MathML</a>

(4.16)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M162">View MathML</a>, and C is a constant independent of u and t0. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M163">View MathML</a>. From (4.16) we see easily that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M164">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M165">View MathML</a>

where C is a constant independent of u and t0. From this we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M166">View MathML</a>

Hence, it follows from (4.12) and (4.13) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M167">View MathML</a>

(4.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M168">View MathML</a>

(4.18)

Now we can apply Lemma 4.2 to conclude from (4.17) and (4.18) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M169">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M170">View MathML</a> with the estimate (4.14). The lemma is completely proved.

Proof of Theorem 4.1: First, we show by induction on h that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M171">View MathML</a>

(4.19)

According to Theorem 3.4 it holds that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M172">View MathML</a>, k h. In particular, utt L2(Q, 2γ + σ). Thus, from the equality (2.9) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M173">View MathML</a>

(4.20)

for all η Hm(Ω) and a.e. t ∈ (0, ∞). Since f (·, t) - utt(·, t) ∈ L2(Ω) for a.e. t ∈ (0, ∞), according to results for elliptic boundary value problem in domains with smooth boundaries, it follows from (4.20) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M174">View MathML</a> for a.e. t ∈ (0, ∞), moreover, the function u satisfies the following equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M175">View MathML</a>

for a.e. (x, t) ∈ Q and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M176">View MathML</a>

in the trace sense. Thus, the assertion (4.19) holds for h = 1, and by (2.5) we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M177">View MathML</a>

(4.21)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M178">View MathML</a> and a.e. t ∈ (0, ∞). Assume now (4.19) holds for h - 2. It follows from (4.20) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M179">View MathML</a>

(4.22)

for all η Hm(Ω), a.e. t ∈ (0, ∞). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M180">View MathML</a>, by the induction assumption, we have from (4.21) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M181">View MathML</a>

(4.23)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M182">View MathML</a> and a.e. t ∈ (0, ∞). Combining (4.22) and (4.23) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M183">View MathML</a>

(4.24)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M184">View MathML</a> and a.e. t ∈ (0, ∞), where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M185">View MathML</a>

Similarly as above, it follows from (4.24) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M186">View MathML</a> for a.e. t ∈ (0, ∞), and therefore, (4.19) holds for h - 1.

Now we prove the assertion of the theorem by induction on h. Let us consider first the case h = 1. We rewrite (2.6), (2.7) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M187">View MathML</a>

(4.25)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M188">View MathML</a>

(4.26)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M189">View MathML</a> for a.e. t ∈ [0, ∞), by Lemma 4.3, it follows from (4.25) and (4.26) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M190">View MathML</a> for a.e. t ∈ (0, ∞) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M191">View MathML</a>

where C is a constant independent of u, f1 and t. Since the trip

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M192">View MathML</a>

does not contain any eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216">View MathML</a> for all t ∈ [0, +∞), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M193">View MathML</a> by the definition of the number ε, we can apply Theorem 7.2.4 and the note below Theorem 7.3.5 of [2] to conclude from (4.25), (4.26) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M194">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M195">View MathML</a>

(4.27)

where C is a constant independent of u, f1 and t. Now multiplying both sides of (4.27) with e-2(2γ+σ)t, then integrating with respect to t from 0 to ∞ and using estimates from Theorem 3.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M196">View MathML</a>

(4.28)

where C is a constant independent of u and f. Hence, the theorem is valid for h = 1.

Assume that the theorem is true for some nonnegative h - 2. We will prove it for h - 1. Differentiating (h - 1) times both sides of (4.25), (4.26) with respect to t, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M197">View MathML</a>

(4.29)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M198">View MathML</a>

(4.30)

By the induction assumption, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M199">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M200">View MathML</a>

by the assumption of the theorem and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M201">View MathML</a>

by Theorem 3.4. Thus, for a.e. t ∈ (0, ∞), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M202">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M203">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M204">View MathML</a>

where C is the constant independent of u, f and t. Now we can repeat the arguments above to conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M205">View MathML</a> with the estimate (4.2) for k = h -1. The proof is completed.

5 An example

In this section we apply the previous results to the Cauchy-Neumann problem for the classical wave equation. We consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M206">View MathML</a>

(5.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M207">View MathML</a>

(5.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M208">View MathML</a>

(5.3)

where Δ is the Laplace operator.

For problem (5.1)-(5.3) it can be directly verified that the constants μ, μ1 and γ0 are now defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M209">View MathML</a>

The operator pencil associated with the problem (5.1)-(5.3) is now defined by (see [[13], Sec. 2.3])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M210">View MathML</a>

(5.4)

where δ is the Laplace-Beltrami operator on the unit sphere Sn-1. It is well known that (see also [[13], Sec. 2.3]) the trip

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M211">View MathML</a>

(5.5)

does not contains any eigenvalue of the operator pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M216">View MathML</a>. We see that if 0 ≤ α ≤ 1 and n > 4 - 2α, then the trip (4.1) with m = 1 is contained in the trip (5.5), since ε can be chosen as zero or an arbitrary small positive real number. Thus, we can apply Theorem 4.1 to receive the following result.

Theorem 5.1. Let h be a nonnegative integer and α be a real number, 0 ≤ α ≤ 1. Assume that the vector function f satisfy the following conditions for some nonnegative real number σ

(i)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M212">View MathML</a>

(ii)<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M213">View MathML</a>

Assume further that n > 4 - 2α. Then for an arbitrary positive real number γ the problem (5.1)-(5.3) has a unique generalized solution u in the space H1,1(Q, γ + σ) which has derivatives with respect to t up to the order h with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M214">View MathML</a>for k = 0, 1,..., h - 1, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/17/mathml/M215">View MathML</a>

where C is a constant independent of u and f.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

This work was supported by National Foundation for Science and Technology Development (NAFOSTED), Vietnam, under project no. 101.01.58.09.

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