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The obstacle problem for a higher order fractional Laplacian

Donatella Danielli, Alaa Haj Ali, Arshak Petrosyan

Abstract

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-Δ)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper half-space $\mathbb{R}_+^{n+1}$ to obtain a thin obstacle problem for the weighted biLaplace operator $Δ^2_b U$, where $Δ_b U=y^{-b}\nabla \cdot (y^b \nabla U)$. Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and $C_{loc}^{1,1}(\R^n) \cap H^{1+s}(\R^n)$-regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the case $b=0$ to the general case $-1<b<1$.

The obstacle problem for a higher order fractional Laplacian

Abstract

In this paper, we consider the obstacle problem for the fractional Laplace operator in the Euclidian space in the case where . As first observed in \cite{Y}, the problem can be extended to the upper half-space to obtain a thin obstacle problem for the weighted biLaplace operator , where . Such a problem arises in connection with unilateral phenomena for elastic, homogenous, and isotropic flat plates. We establish the well-posedness and -regularity of the solution. By writing the solutions in terms of Riesz potentials of suitable local measures, we can base our proofs on tools from potential theory, such as a continuity principle and a maximum principle. Finally, we deduce the regularity of the extension problem to the higher dimensional upper half space. This gives an extension of Schild's work in \cite{Sc1} and \cite{Sc2} from the case to the general case .
Paper Structure (11 sections, 22 theorems, 116 equations)

This paper contains 11 sections, 22 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Fractional Sobolev spaces and the fractional Laplacian of order $1<s<2$
  3. Existence and Uniqueness of solutions
  4. The problem without boundary values - Problem (\ref{['prb1']})
  5. The extension problem
  6. Fundamental solution at the origin
  7. Poisson Formula
  8. The problem with boundary values - Problem (\ref{['prb2']})
  9. $C^{1,1}_{\rm loc}$-regularity of Problem (\ref{['prb2']})
  10. $H^{1+s}$-regularity of Problem (\ref{['prb2']})
  11. Regularity of the extension problem (\ref{['extension_prb']})

Key Result

Theorem 1.2

Y Let $1<s<2$ be a non-integer and let $b=3-2s$. Suppose $U$ is a function in $H^{2} (\mathbb{R}_{+}^{n+1},y^b)$ satisfying for some $u\in \dot{H}^{s}(\mathbb{R}^n)$. Here $\Delta_b$ is as in (b-Laplace) and the Dirichlet boundary condition is satisfied in the sense of traces. Then,

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 3.1: Existence and Uniqueness of Minimizers
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 5.1
  • Lemma 6.1
  • Lemma 6.2
  • Lemma 7.1
  • ...and 14 more