Sharp regularity for singular obstacle problems

We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for $0<\gamma<1$ and $p\ge2$. At the free boundary \(\partial\{u>\var...

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Bibliographic Details
Published in	arXiv.org
Main Authors	Araújo, Damião J, Teymurazyan, Rafayel, Voskanyan, Vardan
Format	Paper
Language	English
Published	Ithaca Cornell University Library, arXiv.org 17.10.2022
Subjects	Barriers Euler-Lagrange equation Free boundaries Regularity Smoothness
Online Access	Get full text

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Summary:	We obtain sharp local $C^{1,\alpha}$ regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for $0<\gamma<1$ and $p\ge2$. At the free boundary $\partial\{u>\varphi\}$, we prove optimal $C^{1,\tau}$ regularity of solutions, with $\tau$ given explicitly in terms of $p$, $\gamma$ and smoothness of $\varphi$, which is new even in the linear setting.
ISSN:	2331-8422

Summary:	We obtain sharp local \(C^{1,\alpha}\) regularity of solutions for singular obstacle problems, Euler-Lagrange equation of which is given by $$ \Delta_p u=\gamma(u-\varphi)^{\gamma-1}\,\text{ in }\,\{u>\varphi\}, $$ for \(0<\gamma<1\) and \(p\ge2\). At the free boundary \(\partial\{u>\varphi\}\), we prove optimal \(C^{1,\tau}\) regularity of solutions, with \(\tau\) given explicitly in terms of \(p\), \(\gamma\) and smoothness of \(\varphi\), which is new even in the linear setting.
ISSN:	2331-8422