The regularity of solutions and free boundary for asymptotically linear and superlinear obstacle problems

Cong Wang

doi:10.3934/cpaa.2025065

Article Contents

2025, Volume 24, Issue 10: 1924-1961. Doi: 10.3934/cpaa.2025065

This issue Previous Article On the advanced balanced pantograph equation Next Article On the equivalence of derivatives for maps between Carnot groups

The regularity of solutions and free boundary for asymptotically linear and superlinear obstacle problems

Cong Wang

School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China

Received: October 2024

Revised: April 2025

Early access: April 29, 2025

Published online: April 29, 2025

This work is supported by NSFC(12301258) and Sichuan Science and Technology Program (2023NSFSC1298), the Fundamental Research Funds for the Central Universities 2682025CX060.

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In this paper, we study obstacle problems with nonlinearities that are asymptotically linear, superlinear, and of Sobolev-critical growth. We first establish the existence of solutions and prove their optimal regularity. Moreover, by combining blow-up analysis, the Weiss-type monotonicity formula, and the Monneau-type monotonicity formula, we obtain fine geometric properties of the free boundary. Specifically, we prove $ C^{1,\alpha} $-regularity near regular contact points and characterize the structure of the singular set.

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Mathematics Subject Classification: 35R35; 35A15; 35B65.

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