Elliptic problems with rough boundary data in Nikolskii spaces

TL;DR

This paper studies elliptic boundary value problems with rough boundary data in Nikolskii spaces, establishing key properties like Fredholmness, regularity, and a priori estimates, with applications to stochastic boundary conditions.

Contribution

It introduces a framework for analyzing elliptic problems with low-regularity boundary data in Nikolskii spaces, including negative order, and applies results to stochastic boundary conditions.

Findings

Fredholm property established for the problem

Maximal regularity and a priori estimates proven

Application to elliptic problems with Gaussian white noise boundary conditions

Abstract

We investigate a general elliptic problem given in a bounded Euclidean domain with boundary data in Nikolskii spaces of low, specifically, negative order. The right-hand side of the elliptic differential equation is supposed to be an integrable function. We establish the Fredholm property of the problem, the maximal regularity and a priori estimate of its generalized solutions in the spaces indicated. We give an application of these results to some elliptic problems with boundary conditions induced by a Gaussian white noise.