Optimal regularity for the thin obstacle problem with C0 , α coefficients

Angkana Rüland, Wenhui Shi

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Abstract

In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson (Invent Math 204(1):1–82, 2016. doi:10.1007/s00222-015-0608-6) and the epiperimetric inequality from Focardi and Spadaro (Adv Differ Equ 21(1–2):153–200, 2016), Garofalo, Petrosyan and Smit Vega Garcia (J Math Pures Appl 105(6):745–787, 2016. doi:10.1016/j.matpur.2015.11.013), we prove the optimal C1 , min { α , 1 / 2 } regularity of solutions in the presence of C0 , α coefficients ai j and C1 , α obstacles ϕ. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a C1 , γ manifold for some γ∈ (0 , 1).

Original languageEnglish
Article number129
Number of pages41
JournalCalculus of Variations and Partial Differential Equations
Volume56
Issue number5
DOIs
Publication statusPublished - 1 Oct 2017
Externally publishedYes

Keywords

  • 35R35

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