An elliptic boundary problem acting on generalized Sobolev spaces

We consider an elliptic boundary problem over a bounded region $\Omega$ in $\mathbb{R}^n$ and acting on the generalized Sobolev space $W^{0,\chi}_p(\Omega)$ for $1 < p < \infty$. We note that similar problems for $\Omega$ either a bounded region in $\mathbb{R}^n$ or a closed manifold acting on $W^{0,\chi}_2(\Omega)$, called H\"{o}rmander space, have been the subject of investigation by various authors. Then in this paper we will, under the assumption of parameter-ellipticity, establish results pertaining to the existence and uniqueness of solutions of the boundary problem. Furthermore, under the further assumption that the boundary conditions are null, we will establish results pertaining to the spectral properties of the Banach space operator induced by the boundary problem, and in particular, to the angular and asymptotic distribution of its eigenvalues

A transmission problem for elliptic equations involving a parameter and a weight

In the course of developing a spectral theory for nonselfajoint elliptic problems involving an indefinite weight function, there arises a transmission problem which has not previously been dealt with. By reducing our problem to one for ordinary differential equations with the aid of the Fourier transfornation, we are able to resolve the problem and to establish a priori estimates for its solutions which we require for the further development of the theory

Estimates for solutions of a parameter-elliptic multi-order system of differential equations

This paper is concerned with a boundary value problem defined over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of differential equations under limited smoothness assumptions. In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter-elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter-elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2x2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions

Necessity of parameter-ellipticity for multi-order systems of differential equations

In this paper we investigate parameter-ellipticity conditions for multi-order systems of differential equations on a bounded domain. Under suitable assumptions on smoothness and on the order structure of the system, it is shown that parameter-dependent a priori-estimates imply the conditions of parameter-ellipticity, i.e., interior ellipticity, conditions of Shapiro-Lopatinskii type, and conditions of Vishik-Lyusternik type. The mixed-order systems considered here are of general form; in particular, it is not assumed that the diagonal operators are of the same order. This paper is a continuation of an article by the same authors where the sufficiency was shown, i.e., a priori-estimates for the solutions of parameter-elliptic multi-order systems were established

Parameter-dependent estimates for mixed-order boundary value problems

In this paper we prove parameter-dependent a priori estimates for mixed-order boundary value problems of rather general structure. In particular, the diagonal operators are not assumed to be of the same order. Our assumptions on the structure of the boundary value problem covers the case of Dirichlet type boundary conditions

Weakly smooth nonselfadjoint spectral elliptic boundary problems

The paper is devoted to general elliptic boundary problems (A, B1 , ..., Bm) with a differential operator A of order 2m and general boundary conditions, acting in a bounded domain G of the n-dimensional space. No self-adjointness is assumed. The main goal is to minimize, to some extent, the smoothness assumptions under which the known spectral results are true. The main results concern the asymptotics of the trace of the q-th power of the resolvent, where q>n/2m, in an angle of ellipticity with parameter. For example, for the Dirichlet problem these asymptotics are obtained in the case of bounded and measurable coefficients in A and continuous coefficients in the principal part of A, while the boundary is assumed to belong to the Hölder space C2m-1,1. The asymptotics of the moduli of the eigenvalues are investigated. The last section is devoted to indefinite spectral problems, with a real-valued multiplier changing the sign in front of the spectral parameter

An elliptic boundary problem for a system involving a discontinuous weight

In a recent paper, Agranovich, Denk and Faierman dealt with a priori estimates, completeness, Abel-Lidskii summability, and eigenvalue asymptotics for scalar elliptic boundary eigenvalue problems involving discontinuous weights. Here we extend these results to the matrix valued case with a diagonal discontinuous weight matrix. The given region is subdivided into subregions on which the weights are continuous. Whereas in the scalar case the usual ellipticity conditions suffice to obtain a priori estimates, a counterexample shows that here transmission conditions at the boundaries of the subregions are also needed