Regularity in the two-phase Bernoulli problem for the p-Laplace operator

IF 2.1 2区数学 Q1 MATHEMATICS Calculus of Variations and Partial Differential Equations Pub Date : 2024-07-16 DOI:10.1007/s00526-024-02789-3

Masoud Bayrami, Morteza Fotouhi

引用次数: 0

Abstract

We show that any minimizer of the well-known ACF functional (for the p-Laplacian) constitutes a viscosity solution. This allows us to establish a uniform flatness decay at the two-phase free boundary points to improve the flatness, which boils down to \(C^{1,\eta }\) regularity of the flat part of the free boundary. This result, in turn, is used to prove the Lipschitz regularity of minimizers by a dichotomy argument. It is noteworthy that the analysis of branch points is also included.

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p 拉普拉斯算子两相伯努利问题中的正则性

我们证明，众所周知的 ACF 函数（p-Laplacian）的任何最小值都构成了粘性解。这使得我们可以在两相自由边界点建立均匀的平坦性衰减，以提高平坦性，这归结为自由边界平坦部分的（C^{1,\eta }\ ）正则性。这一结果反过来又被用来通过二分法论证最小化的 Lipschitz 正则性。值得注意的是，分支点的分析也包括在内。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.