We consider the situation when an elliptic problem in a subdomain of an n-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission conditions to a parabolic problem in . In particular, we can treat elliptic-parabolic equations in bounded domains with discontinuous coefficients. Using Fourier multiplier techniques, we prove an a priori estimate for strong solutions to the equations in -Sobolev spaces.
MSC: 35B45, 35M12.
Keywords:transmission problem; elliptic-parabolic equation; a priori estimates
In the present paper we prove a priori estimates in -Sobolev spaces for the solution of a transmission problem of elliptic-parabolic type with discontinuous coefficients. More precisely, we consider a bounded domain which is divided into two subdomains , separated by a closed contour and a boundary value problem of the form
Here is a differential operator of order 2m, is a column of boundary operators , and λ is a complex parameter. We assume and are looking for a solution . The top-order coefficients of the operator are assumed to be continuous up to the boundary in each subdomain but may have jumps across the interface Γ. The condition leads to the canonical transmission conditions along Γ, given by
where stands for the jump of the th normal derivative of u along the interface Γ. Generalizing (1.2), we will consider inhomogeneous transmission conditions of the form
Here we have set for .
The aim of the paper is to prove uniform a priori estimates for the solutions of (1.1) and (1.3) under suitable ellipticity and smoothness assumptions on A and C, see Section 2 below for the precise formulations. To give an idea of our results, let us for the moment assume that and in (1.1) and (1.3). In classical elliptic theory, in the case of an uncoupled system we would expect a uniform a priori estimate of the form
On the other hand, the classical parabolic (in the sense of parameter-elliptic) a priori estimate would read
Here is the typical parameter-dependent norm appearing in parabolic theory. Concerning the coupled system (1.1), (1.3), the question arises if we still have similar estimates for and . We will see below that this is true in some sense. More precisely, we will obtain
This can be seen as a mixture of elliptic and parabolic a priori estimates. Note that we do not reach the full order 2m with respect to in the first inequality and not the full power with respect to in the second inequality. The general result for and and the precise formulation are stated in Section 2 below.
Applications of problem (1.1), (1.3) (in its parabolic form, i.e., the parameter λ being replaced by the time derivative) can be found, e.g., in , including the heat equation in a domain with vanishing thermal capacity in some subdomain and a model of an electric field generated by a current in a partially non-conducting domain. On the other hand, the problem under consideration is closely related to spectral problems with indefinite weight functions of the form
Here ω is a weight function which may change sign and may vanish on a set of positive measure. Such spectral problems have been investigated, e.g., in a series of papers by Faierman (see [2-4]) and by Pyatkov [5,6], see also  and the references therein. In particular, in the paper  a Calderón method of reduction to the boundary was applied to deal with the case where ω vanishes on a set of positive measure. For this, unique solvability of the Dirichlet boundary value problem in had to be assumed. Transmission problems of purely parabolic type (where the parameter λ is present in each subdomain) and -a priori estimates for their solution were considered in  and . Transmission problems in were also studied, with the same methods as in the present note, by Shibata and Shimizu in .
A standard approach to treat transmission problems is to use (locally) a reflection technique in one subdomain resulting in a system of differential operators which are coupled by the transmission conditions. A general theory of parameter-dependent systems can be found in a series of papers by Volevich and his co-authors (see  and the references therein). Here the so-called Newton polygon method leads to uniform a priori estimates for the solution. However, in the present case the Newton polygon is of trapezoidal form and thus not regular. Therefore, the Newton polygon approach cannot be applied to the transmission problem (1.1). On the other side, the resulting system is not parameter-elliptic in the classical sense () and is not covered by the standard parameter-elliptic theory. We also note the connection to singularly perturbed problems where a similar Newton polygon structure appears, cf.. The analysis of the elliptic-parabolic system below also serves as a starting point for more general (and nonlinear) elliptic-parabolic systems as, for instance, appearing in lithium battery models (see ). A detailed investigation of the nonlinear elliptic-parabolic lithium battery model and solvability in -Sobolev spaces can be found in the second author’s thesis . In  and  mathematical models for lithium battery systems can be found which lead to inhomogeneous transmission conditions.
In Section 2 we will state the precise assumptions and the main result of the present paper. The boundary value problem is analyzed by a localization method and the investigation of the model problem in the half-space. An explicit description of the solution of the model problem (in terms of Fourier multipliers) and resulting estimates can be found in Section 3. Finally, the proof of the main a priori estimate is given in Section 4.
2 Statement of the problem and main result
Let , , , and be open. By and we denote the Lebesgue and Sobolev spaces on Ω with their standard norms. We will further make use of the seminorms
where we used the standard notation . For real non-integer let denote the Besov space on Ω with its standard norm. Besides the standard norms, for the treatment of parameter-elliptic problems the following parameter-dependent norms will be convenient: Let and let be a complex parameter, varying in a closed sector with vertex at 0 where . Then for and , we define
On the boundary, we will consider parameter-dependent trace norms given by
By we denote the Fourier transform of u, and stands for the partial Fourier transform with respect to the first variables .
Let be a bounded domain with boundary ∂Ω of class , and let Γ be a closed Jordan contour in Ω, having no points with ∂Ω in common. Denote by and the resulting subdomains such that , , and . Note that, due to our assumptions, there is no contact point of Γ and ∂Ω. We define and will consider the differential operators for and for . Slightly generalizing the form of equation (1.1), we consider differential operators of even order 2m of the following structure:
with and for some . Furthermore, let the boundary operators of order be of the form
being defined on ∂Ω. We will write for short when we refer to the boundary value problem (1.1).
(1) Smoothness assumptions on the coefficients. We assume
for the coefficients of the differential operators and for the coefficients of the boundary operators.
(2a) Ellipticity of . For the principal symbol , we have ( , ).
(2b) Ellipticity with parameter of the boundary value problem . The principal symbol of satisfies
for all and all , and the Shapiro-Lopatinskii condition is satisfied for at each point . If denotes the principal symbol of the boundary operator, this condition reads as follows: For let the boundary value problem be rewritten in local coordinates associated with , i.e. in coordinates resulting from the original ones by rotation and translation such that the positive -axis coincides with the direction of the inner normal vector. Then for all and , the ODE problem on the halfline
admits a unique solution. Here, .
(3) Assumptions on the data. We assume , , for , and for .
(4) In addition, we assume proper ellipticity, i.e. the polynomials and of order 2m from conditions (2a) and (2b) have exactly m roots in each half-plane for all and , respectively, and for all and . Proper ellipticity allows a decomposition of the form with
where denote the roots in ( ) and ( ), respectively. A similar decomposition with an additional dependence on λ also holds for . We remark that proper ellipticity holds automatically if .
Under these assumptions, we consider the inhomogeneous transmission boundary value problem
Here we have set where denotes the derivative in direction of the outer normal with respect to . Our main result is the following a priori estimate for solutions to (2.3). Here, a solution of (2.3) is defined as a pair belonging to the Sobolev space for which the system (2.3) is satisfied as equality of -functions.
Theorem 2.2 (A priori estimate for the transmission boundary value problem)
Let Assumption 2.1 be satisfied and let be a solution to the transmission problem (2.3). Then there exists such that for all with the following estimates hold:
Note that with respect to g, inequality (2.4) is of elliptic type and (2.5) is of parameter-elliptic type. Due to the fact that the boundary operators act on , we have parameter-elliptic norms with respect to in both inequalities.
Remark 2.3 Our main task will be to study the problem for constant coefficient operators and in the half-spaces without lower order terms. This simplification can be justified by performing a localization procedure, using a finite covering with appropriate open sets , a corresponding partition of unity and perturbation results. For a detailed explanation of the localization procedure, we refer to , pp.151-153, but here we briefly list the types of local problems one has to deal with. If , one faces a local elliptic ( ) or parameter-elliptic operator in the whole space. For these situations, the estimates for are well-known results, see , Theorem 5.3.2, for the elliptic and , Proposition 2.5, for the parameter-elliptic case. If , the local problem is a standard boundary value problem in the half-space and the desired estimate is contained in , Proposition 2.6. It remains to consider the case where intersects both and , and in the sequel we restrict our considerations to the corresponding local model problem. This reads
The reflection , will be useful to treat problem (2.6). Therefore, we will use the notation for the symbol of the reflected operator, which is parameter-elliptic in . We set and .
By this substitution, we may rewrite (2.6) as a system in the half-space :
Here we have set
Remark 2.4 We see that the determinant of the principal symbol vanishes at the points . Hence the standard theory for parameter-elliptic systems is not applicable in this case. Due to continuity and homogeneity of the principal symbols we have the estimate
with a constant . Operators whose principal symbols allow an estimate of the form (2.8) are also called N-elliptic with parameter. Here the ‘N’ stands for the Newton polygon which is related to the principal symbol. In the case of (2.8), the Newton polygon is not regular, and therefore this equation is not covered by the results on N-ellipticity as in .
Remark 2.5 The boundary conditions in (2.6) are called canonical transmission conditions. In the case , they are equivalent to the condition for
Note that in (2.6) the number of conditions equals the order of the operator, in contrast to boundary value problems. We will show in Lemma 3.1 below that the ODE system corresponding to the transmission problem (2.6) is uniquely solvable. This is an analogue of the Dirichlet boundary conditions which are absolutely elliptic, i.e., for every properly elliptic operator the Dirichlet boundary value problem satisfies the Shapiro-Lopatinskii condition.
3 Fundamental solutions and solution operators
To represent the solution in terms of fundamental solutions, we start with the observation that the ODE system obtained from (2.6) by partial Fourier transform is uniquely solvable. This is the analogue of the Shapiro-Lopatinskii condition for transmission problems. For detailed discussions of this condition for boundary value problems, we refer to , Section 6.2 and , Chapter 11. The assertion of the following lemma is formulated for our situation of one elliptic and one parameter-elliptic operator but of course it also holds in the cases when both operators are of the same type.
To simplify our notation, we define and consider the differential operator with .
Lemma 3.1Suppose the operators and are elliptic and parameter-elliptic in , respectively. Fix , , and let ( ). Then the ODE problem
admits a unique solution.
Proof In the sequel, we do not write down the dependence of the polynomials and their roots on explicitly and fix as well as . We decompose and as indicated in (2.2) into . Let denote the m-dimensional space of stable solutions to
and let denote the m-dimensional space of stable solutions to
Let and be a basis of and , respectively. Then is obviously a subset of the 2m-dimensional space of solutions to the equation
and B is linearly independent: Suppose there are nontrivial ( ) with
Then (3.2) would possess a solution which is bounded on the entire real line, which contradicts the fact that the polynomial has only roots with nonzero imaginary part. Hence B is a fundamental system to (3.2) and the determinant of the Wronskian matrix is nonzero:
Now suppose that is a solution to (3.1). Then there exist constants for , such that
If we plug in this approach into the transmission conditions, we obtain the system of linear equations to determine and :
From (3.3) it now follows that the coefficients exist and are uniquely determined, which proves the assertion. □
From now on, we restrict ourselves to the model problem (2.7) which is the only non-standard step in the proof of the main theorem, see Remark 2.3. We first consider the case in (2.7), i.e. we study
Here , ,
Note that and are also called generalized Dirichlet and Neumann conditions, respectively.
Due to Lemma 3.1, the ODE system corresponding to (3.4) is uniquely solvable. The main step in the proof of Theorem 2.2 will be to find a priori estimates for the fundamental solutions of this ODE system. In the following, stands for the ( )-dimensional unit matrix.
Definition 3.2 The fundamental solution
is defined as the unique solution of the ODE system (in )
Following an idea of Leonid Volevich , we represent the solutions in a specific way. For this, we consider the elliptic boundary value problem and the parameter-elliptic boundary value problem separately. It is well known that the (generalized) Dirichlet and Neumann boundary conditions are absolutely elliptic, hence the Shapiro-Lopatinskii condition holds for both subproblems. We will call the canonical basis for these boundary value problems the basic solutions and . More precisely, we define the following.
Definition 3.3 We define the basic solution
as the unique solution of the ODE system
Analogously, the basic solution
is defined as the unique solution of the ODE system
The advantage of the basic solutions , lies in the fact that classical (parameter-)elliptic estimates are easily available for them. We have to compare these solutions with the fundamental solution ω. Let . As the function is a solution of ( ), it can be written as a linear combination of the basic solutions. Therefore, we can write
with unknown coefficients . The analogous representation holds for . In matrix notation, we obtain
with . By the definition of the fundamental solution, we have
Remark 3.4 Due to the unique solvability of equations (3.5) and (3.6), we have for the following scaling properties for all :
where we used the abbreviations
(See also (3.10) below for an explicit representation of and .) We will apply this with for and for . Note that these scaling properties also yield the identities
We summarize the representation of the solution in form of solution operators:
Lemma 3.5Let , and let be a solution of (3.4). Let be an extension ofgto the half-space. Thenuhas the form
where and where the solution operators and are given by
Here the basic solution is defined in Definition 3.3, and the coefficient matrix is defined in (3.7).
Proof By definition of the fundamental solution, we have . Writing this in the form
(‘Volevich trick’) and noting that , we obtain the above representation. □
Our proofs are based on the Fourier multiplier concept, see, e.g., . Here a function is called an -Fourier multiplier if , (being defined on the Schwartz space ) extends to a continuous mapping . We will apply Michlin’s theorem to prove the Fourier multiplier property. For this, we introduce the notion of a Michlin function.
Definition 3.6 Let be a matrix-valued function. Then we call M a Michlin function if for all and if there exists a constant , independent of q, , and , such that
Remark 3.7 (a) Michlin’s theorem (see , Section 2.2.4) states that every Michlin function is an -Fourier multiplier for all .
(b) By the product rule one immediately sees that the product of Michlin functions is a Michlin function, too.
(c) Let be a Michlin function, and let be invertible for all and q. If the norm of the inverse matrix is bounded by a constant independent of and q, then also is a Michlin function. This follows iteratively noting that
Now we will show that the basic solution Y as well as the coefficient matrix Ψ satisfy uniform estimates. Here and in the following, C stands for a generic constant which may vary from inequality to inequality but is independent of the variables appearing in the inequality. We will scale the functions with and with
Lemma 3.8 (a) For all and all , the function
is a Michlin function with constant independent of .
(b) The functions
are Michlin functions.
Proof We use an explicit description of the basic solutions. According to , Section 1, there exist polynomials (with respect to τ) and such that
with being the Kronecker delta symbol. Here is a smooth closed contour in the upper half-plane , depending on and enclosing the m roots of the polynomial with positive imaginary part, while is a smooth closed contour in depending on and enclosing the m roots of in . Moreover, is positively homogeneous in its arguments of degree for while and are positively homogeneous in their arguments of degree m.
This leads to the following representation for the basic solutions and :
To prove part (a), we will show that for all
are Michlin functions. Setting and noting the definitions of and , this immediately implies (a). Similarly, to show (b) we have to prove that
are Michlin functions. We will restrict ourselves to , the result for follows in the same way.
For and , we substitute in the integral representation (3.10) and obtain
with . Note for the first equality that it is not necessary to differentiate the contour because it may be chosen locally independent of . In the last equality, we replaced the contour by a fixed contour which is possible by a compactness argument.
Due to the properties of and , the function is homogeneous of degree −j in its arguments. Therefore, is homogeneous of degree in its arguments, and we obtain
From the fact that may be chosen in and the elementary inequality ( ) we get
for . Inserting this and the homogeneity of into the above representation, we see
which shows (3.11). In the same way, for the proof of (3.12) we set in the above integral representation and obtain
This finishes the proof of (3.11) and (3.12) for . For , we use the substitution in the integral representation. As indicated above, (a) and (b) are immediate consequences of (3.11) and (3.12), respectively. □
The last lemma in connection with the following result is the essential step for the proof of the a priori estimates from the main theorem.
Lemma 3.9The functions
are Michlin functions.
Proof By Lemma 3.8(b), we have
with Michlin functions and . For we obtain
By a homogeneity argument we see that and are Michlin functions, and therefore the matrix on the right-hand side of (3.13) is a Michlin function. In order to apply Remark 3.7(c), we have to show that the norm of is uniformly bounded.
For this, we write in the form of a Schur complement: For an invertible block matrix, we have
with . Applied to the matrix , we obtain
By (3.8), the matrices and and, consequently, the matrix are homogeneous of degree 0 in their arguments. Thus we can write S in the form
We set and
and write S as
The matrices , , and are bounded for all and . By for all , we see that there exists a such that
holds for all and with . For these and q, a Neumann series argument shows that the norm of is bounded by 2.
For and with , the tuple belongs to the compact set . Now we use the fact that for all and , the matrix is invertible, and therefore the matrix on the right-hand side of (3.13) is invertible, too. This yields the invertibility of S, and by continuity the inverse matrix is bounded for these and q.
Therefore, we have seen that holds for all and . From the explicit description of in (3.14) and the uniform boundedness of the other coefficients in (3.14), we see that holds for all and q. By Remark 3.7(c), is a Michlin function.
The above proof also shows that the modification is a Michlin function. Note that remains unchanged and that we obtain an additional factor in the right upper corner which does not affect the boundedness. □
4 Proof of the a priori estimate
In this section, we will investigate the mapping properties of the solution operators , introduced in Lemma 3.5. As above, let . In the following, we will use the abbreviations and . Based on Lemma 3.8 and 3.9 and on the continuity of the Hilbert transform, it is not difficult to obtain the following result.
Lemma 4.1 (a) Let
Then for all and all we have
The same holds when and are replaced by and , respectively.
Then for all and all we have
Proof (a) For fixed , let and
We have to show that . For this, we write
Inserting this into the definition of the solution operator, we obtain
Here we used the fact that and are Michlin functions and therefore Fourier multipliers and that the (one-sided) Hilbert transform
induces a bounded operator in for every .
This shows the first statement in (a). Obviously, the uniform estimate also holds in the case when and are multiplied with the same factor, as this factor cancels out.
The proof of (b) follows exactly in the same way with being replaced by from Lemma 3.5. □
The next result shows the key estimate for the solution of (3.4).