### Abstract

We consider the situation when an elliptic problem in a subdomain
*n*-dimensional bounded domain Ω is coupled via inhomogeneous canonical transmission
conditions to a parabolic problem in

**MSC: **
35B45, 35M12.

##### Keywords:

transmission problem; elliptic-parabolic equation; a priori estimates### 1 Introduction

In the present paper we prove a priori estimates in

Here
*m*,
*λ* is a complex parameter. We assume

where
*u* along the interface Γ. Generalizing (1.2), we will consider inhomogeneous transmission
conditions of the form

where

Here we have set

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

On the other hand, the classical parabolic (in the sense of parameter-elliptic) a priori estimate would read

Here

This can be seen as a mixture of elliptic and parabolic a priori estimates. Note
that we do not reach the full order 2*m* with respect to

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 [1], 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 [7] and the references therein. In particular, in the paper [4] a Calderón method of reduction to the boundary was applied to deal with the case
where *ω* vanishes on a set
*λ* is present in each subdomain) and

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 [10] 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 ([11]) 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.*[12]. 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 [13]). A detailed investigation of the nonlinear elliptic-parabolic lithium battery model
and solvability in

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

where we used the standard notation

On the boundary, we will consider parameter-dependent trace norms given by

By
*u*, and

Let
*∂*Ω of class
*∂*Ω in common. Denote by
*∂*Ω. We define
*m* of the following structure:

with

being defined on *∂*Ω. We will write for short

**Assumption 2.1**

(1) Smoothness assumptions on the coefficients. We assume

for the coefficients of the differential operators and

(2a) Ellipticity of

(2b) Ellipticity with parameter of the boundary value problem

for all
*i.e.* in coordinates resulting from the original ones by rotation and translation such
that the positive

admits a unique solution. Here,

(3) Assumptions on the data. We assume

(4) In addition, we assume proper ellipticity, *i.e.* the polynomials
*m* from conditions (2a) and (2b) have exactly *m* roots in each half-plane

where
*λ* also holds for

Under these assumptions, we consider the inhomogeneous transmission boundary value problem

Here we have set

**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

**Remark 2.3** Our main task will be to study the problem for constant coefficient operators

The reflection

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

with a constant

**Remark 2.5** The boundary conditions in (2.6) are called canonical transmission conditions. In
the case

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 [18], Section 6.2 and [19], 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

**Lemma 3.1***Suppose 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
*m*-dimensional space of stable solutions to

and let
*m*-dimensional space of stable solutions to

Let
*m*-dimensional space of solutions to the equation

and *B* is linearly independent: Suppose there are nontrivial

Then (3.2) would possess a solution which is bounded on the entire real line, which
contradicts the fact that the polynomial
*B* is a fundamental system to (3.2) and the determinant of the Wronskian matrix

Now suppose that

If we plug in this approach into the transmission conditions, we obtain the system
of linear equations to determine

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
*i.e.* we study

Here

with

Note that

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,

**Definition 3.2** The fundamental solution

is defined as the unique solution of the ODE system (in

Following an idea of Leonid Volevich [20], we represent the solutions in a specific way. For this, we consider the elliptic
boundary value problem

**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

We set

The advantage of the basic solutions
*ω*. Let

with unknown coefficients

with

Therefore,

**Remark 3.4** Due to the unique solvability of equations (3.5) and (3.6), we have for

where we used the abbreviations

(See also (3.10) below for an explicit representation of

We summarize the representation of the solution in form of solution operators:

**Lemma 3.5***Let*
*and let*
*be a solution of* (3.4). *Let*
*be an extension of**g**to the half*-*space*. *Then**u**has 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

(‘Volevich trick’) and noting that

Our proofs are based on the Fourier multiplier concept, see, *e.g.*, [18]. Here a function

**Definition 3.6** Let
*M* a Michlin function if
*q*,

**Remark 3.7** (a) Michlin’s theorem (see [17], Section 2.2.4) states that every Michlin function is an

(b) By the product rule one immediately sees that the product of Michlin functions is a Michlin function, too.

(c) Let
*q*. If the norm of the inverse matrix is bounded by a constant independent of
*q*, then also

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

**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 [21], Section 1, there exist polynomials (with respect to *τ*)

with
*m* roots of the polynomial
*m* roots of
*m*.

This leads to the following representation for the basic solutions

To prove part (a), we will show that for all

are Michlin functions. Setting

are Michlin functions. We will restrict ourselves to

For

with

Due to the properties of
*j* in its arguments. Therefore,

From the fact that

for

which shows (3.11). In the same way, for the proof of (3.12) we set

This finishes the proof of (3.11) and (3.12) for

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 3.9***The functions*

*are Michlin functions*.

*Proof* By Lemma 3.8(b), we have

with Michlin functions

By a homogeneity argument we see that

For this, we write

with

with

By (3.8), the matrices
*S* in the form

We set

and write *S* as

The matrices

holds for all
*q*, a Neumann series argument shows that the norm of

For
*S*, and by continuity the inverse matrix
*q*.

Therefore, we have seen that
*q*. By Remark 3.7(c),

The above proof also shows that the modification

### 4 Proof of the a priori estimate

In this section, we will investigate the mapping properties of the solution operators

**Lemma 4.1** (a) *Let*

*Then for all*
*and all*
*we have*

*The same holds when*
*and*
*are replaced by*
*and*
*respectively*.

(b) *Let*

*Then for all*
*and all*
*we have*

*Proof* (a) For fixed

We have to show that

Inserting this into the definition of the solution operator, we obtain

Therefore,

Here we used the fact that

induces a bounded operator in

This shows the first statement in (a). Obviously, the uniform estimate also holds
in the case when

The proof of (b) follows exactly in the same way with