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 3.1 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 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 3.5 Let , and let be a solution of (3.4). Let be an extension of g to the half-space. Then u has the form
and where the solution operators
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 .
By the product rule one immediately sees that the product of Michlin functions is a Michlin function, too.
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 .
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.
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. □