带右手边的两相 p(x)-Laplacian 问题的平面自由边界的正则性

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-05-13 DOI:10.1007/s00526-024-02741-5

Fausto Ferrari, Claudia Lederman

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引用次数: 0

摘要

我们考虑了具有非零右边的 p(x)-Laplacian 两相自由边界问题的粘性解。我们证明平面自由边界是（C^{1,\gamma }\ ）。我们没有假设解的 Lipschitz 连续性。对于 p(x)-Laplacian 的两相自由边界问题以及具有非零右边的奇异/退化算子的两相问题，这些正则性结果是文献中首次出现的。即使当 \(p(x)\equiv p\), 即 p-拉普拉卡矩时，这些结果也是新的。我们的结果仅适用于粘性解，这一点使其具有广泛的适用性。

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Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are \(C^{1,\gamma }\). No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when \(p(x)\equiv p\), i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

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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.