High-Order regularity for fully nonlinear elliptic transmission problems under weak convexity assumption

TL;DR

This paper develops Schauder regularity theory for fully nonlinear elliptic transmission problems, including cases with weak convexity assumptions and small ellipticity aperture, advancing understanding of solution regularity under less restrictive conditions.

Contribution

It introduces new regularity results for transmission problems with operators lacking concavity or convexity, especially under weak convexity and small ellipticity aperture conditions.

Findings

Established Schauder regularity for operators with small ellipticity aperture.

Proved regularity results for quasiconcave and quasiconvex operators.

Extended regularity theory to operators without standard convexity assumptions.

Abstract

This paper studies Schauder theory to transmission problems modelled by fully nonlinear uniformly elliptic equations of second order. We focus on operators F that fails to be concave or convex in the space of symmetric matrices. In a first scenario, it is considered that F enjoys a small ellipticity aperture. In our second case, we study regularity results where the convexity of the superlevel (or sublevel) sets is verified, implying that the operator F is quasiconcave (or quasiconvex).