Boundary value problems with rough boundary data

TL;DR

The article addresses solving linear higher-order parameter-elliptic boundary value problems with rough boundary data by introducing Sobolev spaces of mixed smoothness $H_p^{s,\sigma}$ that admit generalized traces into Besov spaces of negative order. It develops the mapping properties, interpolation, and parameter-dependent frameworks necessary to handle both model and variable-coefficient problems in the half-space and in domains, using localization and perturbation techniques to obtain uniform solvability and a priori estimates. The authors then apply the theory to the linearized Cahn--Hilliard equation with dynamic boundary conditions, proving that the associated operator generates a holomorphic semigroup on $L^p$ product spaces and establishing higher-regularity results. Overall, the work provides a robust functional-analytic toolkit for boundary value problems with rough data, with potential implications for SPDEs with boundary noise and dynamical boundary phenomena.

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 $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$.

Figures (1)

• Figure 2.1: In a first step, we see that the operator $M_{a}$ is continuous on $H^{P}_{p}(\mathbb{R}^{n})$ for every vertex $P\in \mathcal{H}$ of the outer hexagon and therefore by interpolation continuous on $H^{P_{\theta}}_{p}(\mathbb{R}^{n})$ for every $P_{\theta}$ on its boundary. Finally, we interpolate between that boundary and the origin to get the continuity on $H^{s,\sigma}_{p}(\mathbb{R}^{n})$ for every $(s,\sigma)$ on the boundary of the dashed hexagon. In the origin, we have $\gamma=1$, on the boundary of the outer hexagon, we have $\gamma=0$.

Theorems & Definitions (70)

• Definition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7: Roitberg spaces
• Definition 2.8