What type of study is this?

This is a Quantitative Study study.

September 30, 2025

A Liouville theorem for elliptic equations in divergence form with a potential

Key Points

A Liouville-type theorem is established under specific conditions on potential and diffusion matrix in bounded domains.
The coefficients of the diffusion matrix may diverge at the boundary, complicating the behavior of elliptic equations.
An example illustrates the sharpness of the hypotheses on potential and diffusion matrix behavior.
The boundary consists of a finite number of submanifolds, each with varying dimensions affecting the elliptic properties.

Abstract

In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain. We allow the coefficients of the diffusion matrix A (x) and the potential Q (x) to diverge at the boundary; in addition, we permit that Q (x) vanishes inside, and A (x) loses ellipticity at. The boundary is assumed to be the (disjoint) union of a finite number p of submanifolds of dimension ᵢ \0, , n-1\\, (i=1, , p). Under suitable assumptions on the behavior of Q (x) and A (x), which also depend on ᵢ, we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on Q and A are sharp.

Perguntar à IA

Bookmark