Resolvent estimates for Douglis–Nirenberg systems

Abstract

We study mixed order parameter-elliptic boundary value problems with boundary conditions of a certain structure. For such operators, we prove resolvent estimates in L p based Sobolev spaces of suitable order and the analyticity of the semigroup. Finally, we present an application of this theory to studies of the particle transport in a semi-conductor.

Large data solutions to the viscous quantum hydrodynamic model with barrier potential

“…The matrix operator turns out to be a parameter‐elliptic differential operator of mixed order in the sense of Douglis–Nirenberg, and the idea emerges to tackle using semigroup methods. Indeed, the first author succeeded in proving that this matrix differential operator (augmented by appropriate boundary conditions) does generate an analytic semigroup in certain L p ‐based Sobolev spaces, and then the local well‐posedness of can be shown in a short and elegant way .…”

“…‡ If we introduce the vector U :D .n, J/ > of main unknowns, then we can bring (1.1) into the formThe matrix operator turns out to be a parameter-elliptic differential operator of mixed order in the sense of Douglis-Nirenberg, and the idea emerges to tackle (1.1) using semigroup methods. Indeed, the first author succeeded in proving that this matrix differential operator (augmented by appropriate boundary conditions) does generate an analytic semigroup in certain L p -based Sobolev spaces, and then the local well-posedness of (1.1) can be shown in a short and elegant way [11][12][13][14].The key novelty of this paper is to include the barrier potential V B into the considerations, with the goal of rigorously proven analytical results. We focus our attention to the one-dimensional domain .0, 1/ and the time-independent caseWe wrote so-called because, in our opinion, the distinction between pure and applied analysis is misleading, because they belong together.…”