Distribution spaces associated with elliptic operators

TL;DR

This paper investigates complex distribution spaces linked to elliptic operators on Lipschitz domains, establishing their interpolation properties, density, and separability, and applies these findings to elliptic boundary value problems with rough data.

Contribution

It introduces the use of quasi-Banach distribution spaces in the context of elliptic operators and characterizes their interpolation and regularity properties, including for the first time in this setting.

Findings

Spaces $X(A,Y)$ are shown to be separable and dense in $C^{ ext{infty}}(ar{ ext{Omega}})$ under certain conditions.

Explicit descriptions of interpolation spaces between $X(A,Y)$ are provided.

Maximal regularity results are established for elliptic problems with boundary Gaussian white noise.

Abstract

We study complex distribution spaces given over a bounded Lipschitz domain $\Omega$ and associated with an elliptic differential operator $A$ with $C^{\infty}$-coefficients on $\overline{\Omega}$. If $X$ and $Y$ are quasi-Banach distribution spaces over $\Omega$, then the space $X(A,Y)$ under study consists of all distributions $u\in X$ such that $Au\in Y$ and is endowed with the graph quasi-norm. Assuming $X$ to be an arbitrary Besov space or Triebel--Lizorkin space over $\Omega$, we find sufficient conditions for $Y$ under which the interpolation between the spaces $X(A,Y)$ preserves their structure, these spaces are separable, and the set $C^{\infty}(\overline{\Omega})$ is dense in them. We then explicitly describe the spaces obtained by the real, complex, and $\pm$ interpolation between the spaces under study. We apply these spaces to general elliptic problems with rough boundary data by proving the Fredholm property for bounded operators induced by these problems and defined on certain spaces $X(A,Y)$. Specifically, we establish the maximal regularity of solutions to some elliptic problems with Gaussian white noise in boundary conditions. Quasi-Banach distribution spaces are involved in the concept of $X(A,Y)$ for the first time. Our results are new even for inner product Sobolev spaces of integer-valued order.