On a Generalized Prandtl-Batchelor Free Boundary Problem with a Singularity on a Stratified Lie Group

TL;DR

This paper studies a class of elliptic free boundary problems with singularities on stratified Lie groups, extending classical methods to non-Euclidean settings and addressing challenges posed by nondifferentiable energy functionals.

Contribution

It introduces a novel approach combining elliptic regularity and mountain pass techniques to analyze free boundary problems on stratified Lie groups with singularities.

Findings

Existence of solutions established using mountain pass theorem.

Regularity results for solutions in a non-Euclidean setting.

Extension of classical free boundary problem techniques to stratified Lie groups.

Abstract

We investigate a class of elliptic free boundary problems, including a generalized Prandtl Batchelor type problem with a singularity on a stratified Lie group. The associated energy functional is nondifferentiable, which precludes the direct application of standard variational techniques. Our analysis combines elliptic regularity theory with the mountain pass theorem in a non-Euclidean setting