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Obstacle Problems

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Integro-Differential Elliptic Equations

Part of the book series: Progress in Mathematics ((PM,volume 350))

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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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Notes

  1. 1.

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

  2. 2.

    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 \(\{u>0\}\).

  3. 3.

    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.

  4. 4.

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

  5. 5.

    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.

  6. 6.

    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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Authors and Affiliations

  1. Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Xavier Fernández-Real

  2. Departament de Matemàtiques i Informàtica, ICREA and Universitat de Barcelona, Barcelona, Spain

    Xavier Ros-Oton

Authors
  1. Xavier Fernández-Real
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Fernández-Real, X., Ros-Oton, X. (2024). Obstacle Problems. In: Integro-Differential Elliptic Equations. Progress in Mathematics, vol 350. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-54242-8_4

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