In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain Ω. 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 Ω, and A(x) loses ellipticity at ∂Ω. The boundary ∂Ω is assumed to be the (disjoint) union of a finite number p of submanifolds of dimension κi ∈ {0, . . ., n − 1} (i = 1, . . ., p). Under suitable assumptions on the behavior of Q(x) and A(x), which also depend on κi, we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on Q and A are sharp.
A LIOUVILLE THEOREM FOR ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH A POTENTIAL
Biagi S.;Punzo F.
2025-01-01
Abstract
In this paper we are concerned with elliptic equations in divergence form with a potential, posed in a bounded domain Ω. 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 Ω, and A(x) loses ellipticity at ∂Ω. The boundary ∂Ω is assumed to be the (disjoint) union of a finite number p of submanifolds of dimension κi ∈ {0, . . ., n − 1} (i = 1, . . ., p). Under suitable assumptions on the behavior of Q(x) and A(x), which also depend on κi, we prove the validity of a Liouville-type theorem. Finally, we show an example for which our hypotheses on Q and A are sharp.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
