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2024 | OriginalPaper | Chapter

4. Obstacle Problems

Authors : Xavier Fernández-Real, Xavier Ros-Oton

Published in: Integro-Differential Elliptic Equations

Publisher: Springer Nature Switzerland

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Abstract

In this chapter, we study obstacle problems for nonlocal elliptic operators. In order to do so, we first prove the boundary Harnack inequality for such operators in Lipschitz domains, a crucial tool in the regularity theory for free boundary problems. After that, we establish the smoothness of free boundaries near regular points and prove the optimal regularity of solutions. We finish the chapter with a brief discussion of some further results and open problems.

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previous chapter Fully Nonlinear Equations

We use \(v(x) \le v_h(x)\) and \(v(x+h) \le v_{-h}(x+h) = v(x) +\sigma (|h|)\).

This is because, since \(\mathcal {L} u=f\) in \(\{u>0\}\) and \(\mathcal {L} u\geq f\) everywhere, we have \(\mathcal {L} \big ((u(x+h)-u(x)\big )\geq f(x+h)-f(x)\) in \(\{u>0\}\).

The set \(\Omega =\{u_\circ >0\}\) is the complement of a convex set, and in particular it satisfies the hypothesis of Theorem 4.4.5.

It is interesting to notice that this result fails when \(s=1\), i.e., in case \(\mathcal {L}=-\Delta \).

Choose \(\varepsilon \), \(R_\circ \), and \(\eta \) from Proposition 4.4.15, which can then be applied by taking \(\eta > 0\) smaller if necessary thanks to Proposition 4.4.14.

In case of the fractional Laplacian \((-\Delta )^s\), one can use the extension property in order to prove such a semiconvexity estimate; see [8, 34, 101].

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Title: Obstacle Problems
Authors: Xavier Fernández-Real
Xavier Ros-Oton
Publisher: Springer Nature Switzerland
Book: Integro-Differential Elliptic Equations
Print ISBN: 978-3-031-54241-1

Electronic ISBN: 978-3-031-54242-8

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-54242-8_4

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