Elliptic and parabolic equations with rough boundary data in Sobolev spaces with degenerate weights

TL;DR

This paper studies elliptic and parabolic equations with highly singular boundary data in weighted Sobolev spaces that lack boundary traces, establishing conditions for well-posedness despite degeneracies and singularities.

Contribution

It introduces a framework for solving boundary value problems with rough data in degenerate weighted Sobolev spaces, extending classical theory to more singular settings.

Findings

Solutions exist under a natural structural condition on right-hand sides.

Boundary data in negative order Besov spaces can be handled in these weighted spaces.

Well-posedness is achieved despite the absence of boundary traces for solutions.

Abstract

We investigate the inhomogeneous boundary value problem for elliptic and parabolic equations in divergence form in the half space $\{x_d > 0\}$, where the coefficients are measurable, singular or degenerate, and depend only on $x_d$. The boundary data are considered in Besov spaces of distributions with negative orders of differentiability in the range $(-1,0]$. The solution spaces are weighted Sobolev spaces with power weights that decay rapidly near the boundary, and are outside the Muckenhoupt $A_p$ class. Sobolev spaces with such weights contain functions that are very singular near the boundary and do not possess a trace on the boundary. Consequently, solutions may not exist for arbitrarily prescribed boundary data and right-hand sides of the equations. We establish a natural structural condition on the right-hand sides of the equations under which the boundary value problem is well-posed.