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

TL;DR

This survey reviews obstacle problems related to fourth-order elliptic operators, highlighting recent work on well-posedness, regularity, and free boundary structure, with applications in plate bending and fractional Laplacian extensions.

Contribution

It provides a comprehensive overview of known results and introduces recent advances in the analysis of obstacle problems for fourth-order elliptic operators.

Findings

Analysis of well-posedness and regularity of solutions

Characterization of free boundary structures

Connections to fractional Laplacian extension problems

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 $(-\Delta)^s$ 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.