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

Sabri Bensid

TL;DR

The paper extends the study of elliptic free boundary problems to stratified Lie groups, focusing on a generalized Prandtl–Batchelor type problem with a singular term that makes the energy $E(u)$ nondifferentiable. It combines elliptic regularity with a mountain-pass variational approach in a non-Euclidean setting and develops a smoothing approximation $E_\varepsilon$ to overcome the lack of $C^1$ regularity. For large $\lambda$ and small $\beta$, the authors prove the existence of two distinct Lipschitz solutions $u_0$ and $u_1$ with $E(u_0)<-\mathcal{H}(\Omega)\le -\mathcal{H}(\{u=1\})<E(u_1)$ and with their free boundaries understood in the viscosity sense. The analysis yields rigorous convergence of approximate solutions and establishes the free boundary condition, thereby extending Prandtl–Batchelor type results to sub-Riemannian geometry and broadening the applicability of variational methods to nondifferentiable energies on 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

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

TL;DR

The paper extends the study of elliptic free boundary problems to stratified Lie groups, focusing on a generalized Prandtl–Batchelor type problem with a singular term that makes the energy nondifferentiable. It combines elliptic regularity with a mountain-pass variational approach in a non-Euclidean setting and develops a smoothing approximation to overcome the lack of regularity. For large and small , the authors prove the existence of two distinct Lipschitz solutions and with and with their free boundaries understood in the viscosity sense. The analysis yields rigorous convergence of approximate solutions and establishes the free boundary condition, thereby extending Prandtl–Batchelor type results to sub-Riemannian geometry and broadening the applicability of variational methods to nondifferentiable energies on 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
Paper Structure (5 sections, 8 theorems, 131 equations)

This paper contains 5 sections, 8 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: stratified Lie groups and Sobolev spaces
  3. Proof of Theorem 1.1
  4. Approximate solution
  5. Proof of Theorem 1.2

Key Result

Lemma 1.1

Repsov Let $u>0$ be a Lipschitz continuous function on the unit ball $B_1(0)\subset \mathbb{G}$ satisfying the distributional inequalities for constants $A>0,$$0<\varepsilon<1.$ Suppose further that $\mathfrak{F}$ is a continuous function such that $\mathfrak{F}(t)=o(t^2)$ near infinity. Then, there exist $C=C(N,A)>0$ and $\int_{B_1(0)}u^2 dx$ but not on $\varepsilon$ such that

Theorems & Definitions (12)

  • Lemma 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • Theorem 2.1
  • Lemma 4.1
  • ...and 2 more