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A survey on obstacle-type problems for fourth order elliptic operators

Donatella Danielli, Alaa Haj Ali

Abstract

In this article we give a brief overview of some known results in the theory of obstacle-type problems associated with a class of fourth-order elliptic operators, and we highlight our recent work with collaborators in this direction. Obstacle-type problems governed by operators of fourth order naturally arise in the linearized Kirchhoff-Love theory for plate bending phenomena. Moreover, as first observed by Yang in \cite{Y13}, boundary obstacle-type problems associated with the weighted bi-Laplace operator can be seen as extension problems, in the spirit of the one introduced by Caffarelli-Silvestre, for the fractional Laplacian $(-Δ)^s$ in the case $1<s<2$. In our recent work, we investigate some problems of this type, where we are concerned with the well-posedness of the problem, the regularity of solutions, and the structure of the free boundary. In our approach, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas.

A survey on obstacle-type problems for fourth order elliptic operators

Abstract

In this article we give a brief overview of some known results in the theory of obstacle-type problems associated with a class of fourth-order elliptic operators, and we highlight our recent work with collaborators in this direction. Obstacle-type problems governed by operators of fourth order naturally arise in the linearized Kirchhoff-Love theory for plate bending phenomena. Moreover, as first observed by Yang in \cite{Y13}, boundary obstacle-type problems associated with the weighted bi-Laplace operator can be seen as extension problems, in the spirit of the one introduced by Caffarelli-Silvestre, for the fractional Laplacian in the case . In our recent work, we investigate some problems of this type, where we are concerned with the well-posedness of the problem, the regularity of solutions, and the structure of the free boundary. In our approach, we combine classical techniques from potential theory and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas.
Paper Structure (18 sections, 23 theorems, 79 equations)

This paper contains 18 sections, 23 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Obstacle problems for plates
  3. Unilateral displacements
  4. Unilateral displacements of the points of $\Omega$
  5. Unilateral displacements of the points of $\partial \Omega$
  6. The classical obstacle problem for the bi-Laplacian
  7. Regularity of solutions
  8. Regularity of the free boundary
  9. The thin obstacle problem for the poly-Laplacian
  10. The fractional Laplacian of order $1<s<2$
  11. The extension problem for the higher order fractional Laplacian
  12. The obstacle problem for a higher order fractional Laplacian
  13. A two-phase boundary obstacle-type problem for the bi-Laplacian
  14. Regularity of the minimizer
  15. Regularity of the free boundary
  16. ...and 3 more sections

Key Result

Lemma 3.1

(CF79) There exists an upper semicontinuous function $w$ such that $w=\Delta u$ a.e. in $\Omega$. Moreover, for every $x^0 \in \Omega$, and for any sequence of balls $B_{\rho}(x^0)$

Theorems & Definitions (24)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 5.1
  • Theorem 6.1
  • Lemma 7.1
  • Lemma 7.2
  • Lemma 7.3
  • Theorem 7.4
  • Theorem 7.5
  • ...and 14 more