# Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C 0 , α coefficients

Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C 0 , α... 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 $$C^{1,\min \{\alpha ,1/2\}}$$ C 1 , min { α , 1 / 2 } regularity of solutions in the presence of $$C^{0,\alpha }$$ C 0 , α coefficients $$a^{ij}$$ a i j and $$C^{1,\alpha }$$ C 1 , α obstacles $$\phi$$ ϕ . Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $$C^{1,\gamma }$$ C 1 , γ manifold for some $$\gamma \in (0,1)$$ γ ∈ ( 0 , 1 ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C 0 , α coefficients

, Volume 56 (5) – Aug 23, 2017
41 pages

/lp/springer_journal/optimal-regularity-for-the-thin-obstacle-problem-with-c-0-alpha-c-0-KRSp2WVwQX
Publisher
Springer Journals
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1230-9
Publisher site
See Article on Publisher Site

### 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 $$C^{1,\min \{\alpha ,1/2\}}$$ C 1 , min { α , 1 / 2 } regularity of solutions in the presence of $$C^{0,\alpha }$$ C 0 , α coefficients $$a^{ij}$$ a i j and $$C^{1,\alpha }$$ C 1 , α obstacles $$\phi$$ ϕ . Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $$C^{1,\gamma }$$ C 1 , γ manifold for some $$\gamma \in (0,1)$$ γ ∈ ( 0 , 1 ) .

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Aug 23, 2017

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