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A two-phase problem with degenerate operator in Orlicz-Sobolev spaces

Pedro F. Silva Pontes, Minbo Yang

TL;DR

This work studies a two-phase variational problem governed by the Φ-Laplacian with $Φ(s)=e^{s^2}-1$, focusing on the existence, boundedness, and regularity of minimizers in Orlicz–Sobolev spaces. To overcome lack of $\Delta_2$, it introduces truncated functionals with $Φ_k(s)=\sum_{i=1}^k \frac{1}{i!}|s|^{2i}$ and passes to the limit, establishing the existence of an $L^ ablafty$-bounded minimizer $u_0$ that is locally Log-Lipschitz. The minimizer satisfies a two-phase PDE $-Δu_0-2Δ_∞u_0 = f_+(x) e^{-|∇u_0|^2}$ in ${>0}$ and $f_-(x) e^{-|∇u_0|^2}$ in $\{u_0\le0\}$ both in weak and viscosity senses, and, under a gamma-condition, the phase boundary has locally finite perimeter with $(N-1)$-dimensional rectifiability. The work blends Orlicz–Sobolev theory with nonlinear PDE analysis to extend two-phase results beyond uniform ellipticity assumptions, providing rigorous regularity and geometric properties of the free boundary relevant to applications in porous media, materials, and biological models.

Abstract

In this paper we are interested in the study of a two-phase problem equipped with the $Φ$-Laplacian operator $$ Δ_Φu \coloneqq \mbox{div} \left(φ(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right), $$ where $Φ(s)=e^{s^2}-1$ and $φ=Φ'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the phase change free boundaries of the minimizers possess a finite perimeter.

A two-phase problem with degenerate operator in Orlicz-Sobolev spaces

TL;DR

This work studies a two-phase variational problem governed by the Φ-Laplacian with , focusing on the existence, boundedness, and regularity of minimizers in Orlicz–Sobolev spaces. To overcome lack of , it introduces truncated functionals with and passes to the limit, establishing the existence of an -bounded minimizer that is locally Log-Lipschitz. The minimizer satisfies a two-phase PDE in and in both in weak and viscosity senses, and, under a gamma-condition, the phase boundary has locally finite perimeter with -dimensional rectifiability. The work blends Orlicz–Sobolev theory with nonlinear PDE analysis to extend two-phase results beyond uniform ellipticity assumptions, providing rigorous regularity and geometric properties of the free boundary relevant to applications in porous media, materials, and biological models.

Abstract

In this paper we are interested in the study of a two-phase problem equipped with the -Laplacian operator where and . We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the phase change free boundaries of the minimizers possess a finite perimeter.
Paper Structure (8 sections, 18 theorems, 160 equations)

This paper contains 8 sections, 18 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Orlicz spaces
  3. A special case
  4. The truncated problem
  5. Existence and regularity for minimizers of $\mathcal{J}$
  6. Existence and regularity
  7. The operator $\Delta u + 2 \Delta_\infty u$
  8. Finite perimeter of the free boundary

Key Result

Theorem 1.1

There exists a minimizer $u_0 \in W_\psi^{1,\Phi}(\Omega)$ of $\mathcal{J}_{\Phi,f,\gamma}$. Furthermore, $u_0 \in L^\infty(\Omega)$ and there exists $C>0$ which depends on $\|f\|_\infty,\mathcal{L}^N(\Omega),N,\sup_{\overline{\Omega}}\psi$, such that

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more