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Initial boundary value problems for second order parabolic systems in cylinders with polyhedral base

Vu Trong Luong1* and Do Van Loi2

Author Affiliations

1 Department of Mathematics, Taybac University, Sonla city, Sonla, Vietnam

2 Department of Mathematics, Hongduc University, Thanhhoa city, Thanhhoa, Vietnam

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Boundary Value Problems 2011, 2011:56  doi:10.1186/1687-2770-2011-56

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/56


Received:18 September 2011
Accepted:24 December 2011
Published:24 December 2011

© 2011 Luong and Loi; 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

The purpose of this article is to establish the well posedness and the regularity of the solution of the initial boundary value problem with Dirichlet boundary conditions for second-order parabolic systems in cylinders with polyhedral base.

1 Introduction

Boundary value problems for partial differential equations and systems in nonsmooth domains have been attracted attentions of many mathematicians for more than last 50 years. We are concerned with initial boundary value problems (IBVP) for parabolic equations and systems in nonsmooth domains. These problems in cylinders with bases containing conical points have been investigated in [1,2] in which the regularity and the asymptotic behaviour near conical points of the solutions are established. Parabolic equations with discontinuous coefficients in Lipschitz domains have also been studied (see [3] and references therein).

In this study, we consider IBVP with Dirichlet boundary conditions for second-order parabolic systems in both cases of finite cylinders and infinite cylinders whose bases are polyhedral domains. Firstly, we prove the well posedness of this problem by Galerkin's approximating method. Next, by this method we obtain the regularity in time of the solution. Finally, we apply the results for elliptic boundary value problems in polyhedral domains given in [4,5] and former our results to deal with the global regularity of the solution.

Let Ω be an open polyhedral domain in ℝn (n = 2, 3), and 0 < T ≤ ∞. Set QT = Ω × (0, T), ST = ∂Ω × (0, T). For a vector-valued function u = (u1, u2, ..., us) and p = (p1, p2, ..., pn) ∈ ℕn we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M1">View MathML</a>.

Let m, k be non negative integers. We denote by Hm(Ω), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M2">View MathML</a> the usual Sobolev spaces as in [6]. By the notation (., .) we mean the inner product in L2(Ω).

We denote by Hm,k(QT, γ) (γ ∈ ℝ) the weighed Sobolev space of vector-valued functions u defined in QT with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M3">View MathML</a>

Let us note that if T < +∞, then Hm,k(QT, γ) ≡ Hm,k(QT).

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M4">View MathML</a> is the closure in Hm,k(QT, γ) of the set consisting of all vector-valued functions u C(QT) which vanish near ST.

Let ∂singΩ be the set of all singular points of ∂Ω, namely, the set of vertexes of Ω for the case n = 2 and the union of all edges of Ω for the case n = 3. Let ρ(x) be the distance from a point x ∈ Ω to the set ∂singΩ. For a ∈ ℝ, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M5">View MathML</a> the weighed Sobolev space of vector functions u defined on Ω with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M6">View MathML</a>

It is obvious from the definition that continuous imbeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M7">View MathML</a> hold for all a ≤ 1.

The weighed Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M8">View MathML</a> are defined as sets of all vector-valued functions defined in QT with respect to the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M9">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M10">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M11">View MathML</a>

be a second-order partial differential operator, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M12">View MathML</a>, and Aij, Bi, C are s × s matrices of bounded functions with complex values from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M13">View MathML</a> is the transposed conjugate matrix of Aji.

We assume that the operator L is uniformly strong elliptic, that is, there exists a constant C > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M14">View MathML</a>

(1)

for all ξ ∈ ℝn, η ∈ ℂs and a.e. (x, t) ∈ QT.

In this article, we study the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M15">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M16">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M17">View MathML</a>

(4)

where f(x, t) is given.

Let us introduce the following bilinear form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M18">View MathML</a>

Then the following Green's formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M19">View MathML</a>

is valid for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M20">View MathML</a> and a.e. t ∈ [0, T).

Definition 1.1. A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M21">View MathML</a>is called a generalized solution of problem (2) -(4) if and only if u|t = 0 = 0 and the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M22">View MathML</a>

(5)

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M23">View MathML</a>.

From (1) it follows that there exist constants µ0 > 0, λ0 ≥ 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M24">View MathML</a>

(6)

holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M25">View MathML</a> and t ∈ [0, T). By substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M26">View MathML</a> into (2), we can assume for convenience that λ0 in (6) is zero. Hence, throughout the present paper we also suppose that B(., .; t) satisfies the following inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M27">View MathML</a>

(7)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M25">View MathML</a> and t ∈ [0, T).

Now, let us present the main results of this article. Firstly, we give a theorem on well posedness of the problem:

Theorem 1.1. Let f L2(QT, γ0), γ0 > 0, and suppose that the coefficients of the operator L satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M28">View MathML</a>

Then for each γ > γ0, problem (2) -(4) has a unique generalized solution u in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M29">View MathML</a>and the following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M30">View MathML</a>

(8)

where C is a constant independent of u and f.

Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M33">View MathML</a>. Next, we give results on the smoothness of the solution:

Theorem 1.2. Let m ∈ ℕ*, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M34">View MathML</a>, σ = γ - γ0, γk = (2k + 1)γ0. Assume that the coefficients of L satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M35">View MathML</a>

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M36">View MathML</a>

Then there exists η > 0 such that u belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M37">View MathML</a>for any |a| < η, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M38">View MathML</a>

(9)

where C is a constant independent of u and f.

2 The proof of Theorem 1.1

Firstly, we will prove the existence by Galerkin's approximating method. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M39">View MathML</a> be an orthogonal basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M40">View MathML</a> which is orthonormal in L2(Ω). Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M41">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M42">View MathML</a>, is the solution of the following ordinary differential system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M43">View MathML</a>

(10)

with the initial conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M44">View MathML</a>

(11)

Let us multiply (10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M45">View MathML</a>, then take the sum with respect to k from 1 to N to arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M46">View MathML</a>

Now adding this equality to its complex conjugate, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M47">View MathML</a>

(12)

Utilizing (7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M48">View MathML</a>

By the Cauchy inequality, for an arbitrary positive number ε, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M49">View MathML</a>

where C = C(ε) is a constant independent of uN, f and t. Combining the estimates above, we get from (12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M50">View MathML</a>

(13)

for a.e. t ∈ [0, T). Now write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M51">View MathML</a>

Then (13) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M52">View MathML</a>

Thus the differential form of Gronwall-Belmann's inequality yields the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M53">View MathML</a>

(14)

We obtain from (14) the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M54">View MathML</a>

Now multiplying both sides of this inequality by e-γt, γ > γ0 + ε, then integrating them with respect to t from 0 to T, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M55">View MathML</a>

(15)

Multiplying both sides of (13) by e-γt, then integrating them with respect to t from 0 to τ, τ ∈ (0, T), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M56">View MathML</a>

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M57">View MathML</a>

We employ the inequalities above to find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M58">View MathML</a>

(16)

Since the right-hand side of (16) is independent of τ, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M59">View MathML</a>

(17)

where C is a constant independent of u, f and N.

Fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M60">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M61">View MathML</a> and write v = v1 + v2, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M62">View MathML</a> and (v2, ωk) = 0, k = 1, ..., N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M63">View MathML</a>. We have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M64">View MathML</a>. Utilizing (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M65">View MathML</a>

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M66">View MathML</a>, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M67">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M68">View MathML</a>

Since this inequality holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M69">View MathML</a>, the following inequality will be inferred

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M70">View MathML</a>

(18)

Multiplying (18) by e-γt, γ > γ0 + ε, then integrating them with respect to t from 0 to T, and by using (17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M71">View MathML</a>

(19)

Combining (17) and (19), we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M72">View MathML</a>

(20)

where C is a constant independent of f and N.

From the inequality (20), by standard weakly convergent arguments, we can conclude that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M73">View MathML</a> possesses a subsequence weakly converging to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M74">View MathML</a>, which is a generalized solution of problem (2) -(4). Moreover, it follows from (20) that estimate (8) holds.

Finally, we will prove the uniqueness of the generalized solution. It suffices to check that problem (2)-(4) has only one generalized solution u ≡ 0 if f ≡ 0. By setting v = u(., t) in identity (5) (for f ≡ 0) and adding it to its complex conjugate, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M75">View MathML</a>

From (7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M76">View MathML</a>

Since u|t = 0 = 0, it follows from this inequality that u ≡ 0 on QT. The proof is complete.

3 The proof of Theorem 1.2

Firstly, we establish the results on the smoothness of the solution with respect to time variable of the solution which claims that the smoothness depends on the smoothness of the coefficients and the right-hand side of the systems.

To simplify notation, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M77">View MathML</a>

Proposition 3.1. Let h ∈ ℕ*. Assume that there exists a positive constant µ such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M78">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M79">View MathML</a>.

Then for an arbitrary real number γ satisfying γ > γ0, the generalized solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M80">View MathML</a>of problem (2)-(4) has derivatives with respect to t up to order h with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M81">View MathML</a>, and the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M82">View MathML</a>

(21)

holds, where C is a constant independent of u and f.

Proof. From the assumptions on the coefficients of operator L and the function f, it implies that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M83">View MathML</a> of problem (10)-(11) has derivatives with respect to t up to order h + 1. We will prove by induction that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M84">View MathML</a>

(22)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M85">View MathML</a>

(23)

Firstly, we differentiate h times both sides of (10) with respect to t to find the following equality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M86">View MathML</a>

(24)

From the equalities above together with the initial condition (11) and assumption (ii), we can show by induction on h that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M87">View MathML</a>

(25)

Equality (24) is multiplied by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M88">View MathML</a> and sum k = 1, ..., N, so as to discover

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M89">View MathML</a>

Adding this equality to its complex conjugate, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M90">View MathML</a>

(26)

Next, we show that inequalities (22) and (23) hold for h = 0. According to (26) (with h = 0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M91">View MathML</a>

Then the equality is rewritten in the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M92">View MathML</a>

Integrating both sides of this equality with respect to t from 0 to τ, τ ∈ (0, T), employing Garding inequality (7) and Cauchy inequality, and by simple calculations, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M93">View MathML</a>

Thus Gronwall-Belmann's inequality yields the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M94">View MathML</a>

(27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M95">View MathML</a>. Multiplying both sides of (27) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M96">View MathML</a>, then integrating them with respect to t from 0 to T, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M97">View MathML</a>

(28)

From inequalities (27) and (28), it is obvious that (22) and (23) hold for h = 0.

Assume that inequalities (22) and (23) are valid for k = h - 1, we need to prove that they are true for k = h. With regard to equality (26), the second term in left-hand side of (26) is written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M98">View MathML</a>

Hence, from (26) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M99">View MathML</a>

(29)

Integrating both sides of (29) with respect to t from 0 to τ, 0 < τ < T, and using the integration by parts, we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M100">View MathML</a>

(30)

For convenience, we abbreviate by I, II, III, IV, V the terms from the first to the fifth, respectively, of the right-hand side of (30). By using assumption (i) and the Cauchy inequality, we obtain the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M101">View MathML</a>

Employing the estimates above, we get from (30) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M102">View MathML</a>

(31)

By using (7) again, we obtain from (31) the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M103">View MathML</a>

(32)

From (32) and the induction assumptions, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M104">View MathML</a>

(33)

where ε > 0 is chosen such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M105">View MathML</a>

By the Gronwall-Bellmann inequality, we receive from (33) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M106">View MathML</a>

(γh > γj, for j = 0, ..., h - 1). Now multiplying both sides of this inequality by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M107">View MathML</a>, then integrating them with respect to τ from 0 to T, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M108">View MathML</a>

(34)

It means that the estimates (22) and (23) hold for k = h.

By the similar arguments in the proof of Theorem 1.1, we obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M109">View MathML</a>

(35)

Then the combination between (34) and (35) produces the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M110">View MathML</a>

(36)

Accordingly, by again standard weakly convergent arguments, we can conclude that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M111">View MathML</a> possesses a subsequence weakly converging to a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M112">View MathML</a>. Moreover, u(k) is the kth generalized derivative in t of the generalized solution u of problem (2)-(4). Estimate (21) follows from (36) by passing the weak convergences.   □

Next, by changing problem (2) -(4) into the Dirichlet problem for second order elliptic depending on time parameter, we can apply the results for this problem in polyhedral domains (cf. [4,5]) and our previous ones to deal with the regularity with respect to both of time and spatial variables of the solution.

Proposition 3.2. Let the assumptions of Theorem 3.1 be satisfied for a given positive integer h. Then there exists η > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M113">View MathML</a>belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M114">View MathML</a>for any |a| < η, k = 0, ..., h and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M115">View MathML</a>

(37)

where C is a constant independent of u and f.

Proof. We prove the assertion of the theorem by an induction on h. First, we consider the case h = 0. Equalities (2), (3) can be rewritten in the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M116">View MathML</a>

(38)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M117">View MathML</a>

(39)

Since u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M118">View MathML</a>

it is clear that for a.e. t ∈ (0, T), u is the solution of the Dirichlet problem for system (38) with the right-hand side <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M119">View MathML</a> for all a ≤ 1. From Theorem 4.2 in [5] (or Theorem 1.1. in [4]), it implies that there exists η > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M120">View MathML</a> for any |a| ≤ η. Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M121">View MathML</a>

(40)

where C is a constant independent of u, f and t. Now multiplying both sides of (40) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M122">View MathML</a>, then integrating with respect to t from 0 to T and using estimates from Theorem 3.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M123">View MathML</a>

where C is a constant independent of u, f. Thus, the theorem is valid for h = 0. Suppose that the theorem is true for h - 1; we will prove that this also holds for h. By differentiating h times both sides of (38)-(39) with respect to t, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M124">View MathML</a>

(41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M125">View MathML</a>

(42)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M126">View MathML</a>

By the induction assumption, it implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M128">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M129">View MathML</a>

by Theorem 3.1. Hence, for a.e. t ∈ (0, T), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M130">View MathML</a> and the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M131">View MathML</a>

(43)

Applying Theorem 4.2 in [5] again, we conclude from (41)-(42) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M132">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M133">View MathML</a>

From the inequality above and (43), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M134">View MathML</a>

(44)

Multiplying both sides of (44) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M135">View MathML</a>, then integrating with respect to t from 0 to T and using Theorem 3.1 with a note that γk < γh for k = 0, 1, ..., h - 1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M136">View MathML</a>

where C is the constant independent of u and f. The proof is completed.   □

Proof of Theorem 1.2. We will prove the theorem by an induction on m. It is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M137">View MathML</a>

Hence, Proposition 3.2 implies that the theorem is valid for m = 0. Assume that the theorem is true for m - 1, we will prove that it also holds for m. It is only needed to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M138">View MathML</a>

(45)

Suppose that (45) is true for s = m, m - 1, ..., j + 1, return one more to (41) (h=j), and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M139">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M140">View MathML</a>

(46)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M141">View MathML</a>. By the inductive assumption with respect to s, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M142">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M143">View MathML</a>

Thus, the right-hand side of (46) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M144">View MathML</a>. Applying Theorem 4.2 in [5] again, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M145">View MathML</a> for a.e. t ∈ (0, T). It means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M146">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M147">View MathML</a>.

Furthermore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M148">View MathML</a>

(47)

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M149">View MathML</a>

It implies that (45) holds for s = j. The proof is complete for j = 0.

An example. In order to illustrate the results above, we show an example for the case L = -Δ, and Ω is a curvilinear polygonal domain in the plane.

Denote by A1, A2, ..., Ak the vertexes of Ω. Let αj be the opening of the angle at the vertex Aj. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M150">View MathML</a>

as the angle at vertex Aj with sides <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M151">View MathML</a>. Here r, θ are the polar coordinates of the point x = (x1, x2), noting that r(x) = ρ(x) is the distance from a point x Kj U to the set {A1, A2, .... Ak}, where U is a small neighbourhood of Aj.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M152">View MathML</a> be the eigenvalue of the pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M153">View MathML</a> (cf. [7]) arises from the Dirichlet problem for Laplace operator via the Mellin transformation r λ. Let η = min{ηj}. We consider the Cauchy-Dirichlet problem for the classical heat equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M154">View MathML</a>

(48)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M155">View MathML</a>

(49)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M156">View MathML</a>

(50)

where f : QT → ℂ is given.

Combining Theorem 1.2 and Theorem 4.4 in [5] we receive the following theorem.

Theorem 3.1. Let Ω ⊂ ℝ2 be a bounded curvilinear polygonal domain in the plane. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M157">View MathML</a>

Then the generalized solution u of problem (48)-(50) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M158">View MathML</a>for any |a| < η := min ηj, as above, and u satisfies the following estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M159">View MathML</a>

(51)

where C is a constant independent of u and f.

Remark: Let us notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M161">View MathML</a>, the weighed Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M162">View MathML</a> is defined in [[7], p. 191]. Applying Theorem 6.1.4 in [[7], p. 205] with l2 = 2, β2 = 1 - a, l1 = 1, β1 = 0, n = 2 and the strip 0 < Reλ < a < η does not contain any eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M153">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M163">View MathML</a>. It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/56/mathml/M164">View MathML</a>. Hence, the regularity of the solution of problem (48)-(50) is better than the regularity result, which can obtain from helps of Theorem 6.1.4 in [[7], p. 205].

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors typed, read, and approved the final manuscript.

Acknowledgements

This study was supported by the Vietnam's National Foundation for Science and Technology Development (NAFOSTED: 101.01.58.09).

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