Fully nonlinear elliptic PDEs in thin domains with oblique boundary condition

TL;DR

This paper investigates fully nonlinear elliptic PDEs in thin domains with oblique boundary conditions, revealing new phenomena and limit equations with additional terms not present in the Neumann case, extending previous work on classical Laplacian problems.

Contribution

It introduces analysis of fully nonlinear elliptic PDEs with oblique boundary conditions in thin domains, uncovering new limit behaviors and terms absent in prior Neumann boundary condition studies.

Findings

Limit equations contain new terms of second, first, and zeroth order.

Identifies phenomena unique to oblique boundary conditions.

Extends understanding of PDE behavior in thin domains.

Abstract

In this preprint we consider fully nonlinear equations in thin domains with oblique boundary condition, finding some new phenomena, in particular the limit equation contains "new terms" of the second, first and zeroth order which don't have an equivalent in the Neumann case treated in our previous work arXiv:2404.19577. The classical laplacian problem with Neumann boundary condition, goes back to the well known result of Hale and Raugel (1992).