Boundary value problems with rough boundary data

TL;DR

This paper develops a framework for solving higher-order elliptic boundary value problems with rough boundary data using Sobolev spaces of mixed smoothness, extending classical trace theory and applying it to the Cahn--Hilliard equation.

Contribution

It introduces a generalized boundary trace in Sobolev spaces of mixed smoothness for higher-order elliptic problems with non-classical boundary data.

Findings

Proves unique solvability for rough boundary data in half-space and smooth domains.

Establishes that the linearized Cahn--Hilliard operator generates a holomorphic semigroup.

Extends boundary value problem theory to include data in Besov spaces of negative order.

Abstract

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn--Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$.