-estimates for a transmission problem of mixed elliptic-parabolic type
© Denk and Seger; licensee Springer. 2014
Received: 25 July 2013
Accepted: 18 December 2013
Published: 22 January 2014
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.
Keywordstransmission problem elliptic-parabolic equation a priori estimates
Here we have set for .
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.
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
By we denote the Fourier transform of u, and stands for the partial Fourier transform with respect to the first variables .
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.
Ellipticity of . For the principal symbol , we have (, ).
- (2b)Ellipticity with parameter of the boundary value problem . The principal symbol of satisfiesfor 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, .
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(2.2)
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 .
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)
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.
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 .
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 .
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 .
admits a unique solution.
From (3.3) it now follows that the coefficients exist and are uniquely determined, which proves the assertion. □
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.
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.
We summarize the representation of the solution in form of solution operators:
Here the basic solution is defined in Definition 3.3, and the coefficient matrix is defined in (3.7).
(‘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.
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
- (b)The functions
are Michlin functions.
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.
are Michlin functions. We will restrict ourselves to , the result for follows in the same way.
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.
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.
are Michlin functions.
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.
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.
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).
This and equations (4.4) and (4.5) yield the statements of the theorem. □
Now we can consider the problem , in the half-space. As mentioned in Remark 2.3, this finishes the proof of the main theorem.
Proof (i) We start the proof with some preliminary remarks. Let be the restriction operator from to . Then is a retraction from to for every , and there exists a co-retraction (independent of k), i.e. a total extension operator satisfying for all k (see , Theorem 5.21).
For every , the trace operator to the boundary is a bounded operator from to . This holds both with respect to the parameter-independent norms and the parameter-dependent norms . For the latter, we refer to , Proposition 2.2. There exists a parameter-dependent extension operator which satisfies and whose operator norm with respect to the parameter-dependent norms is bounded by a constant independent of q for all with (see, e.g., , Proposition 2.3). In particular, we will consider which is a parameter-independent continuous extension operator.
- (ii)Let be a solution of , , and let with . We define . Then and . For
- (iii)Similarly, we set . It is well known (or easily seen by Michlin’s theorem) that with and(4.9)
We define and . Then w is a solution of , . Applying the parameter-independent extension operator to every component of g, we define . An extension of is given by omitting the trace to the boundary. Note that .
- (v)The proof of (4.7) follows the same lines. However, here we start with the refined estimate (4.3). For the left-hand side of (4.7), we note that
So we see that all terms in (4.11) can be estimated by the right-hand side of (4.7), and the proof of (4.7) is finished. □
- (b)For and , we obtain in particular