2D Thin obstacle problem with data at infinity

TL;DR

This paper studies the 2D thin obstacle problem with data at infinity, proving existence, uniqueness, and symmetry of solutions, which aids in classifying higher-dimensional solutions and generalizes previous work on homogeneous solutions.

Contribution

It establishes existence, uniqueness, and symmetry properties of solutions to the 2D thin obstacle problem with data at infinity, extending classification results to higher dimensions.

Findings

Existence and uniqueness of solutions in 2D with data at infinity.

Symmetric solutions are half-space solutions.

Results facilitate classification of higher-dimensional solutions.

Abstract

In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are needed when classifying the half-space $(2k-\frac{1}{2})$-homogeneous solutions to the thin obstacle problems in $\mathbb{R}^3$. It is a generalization of one part of Savin-Yu's work \cite{savin2021halfspace} on classifying the half-space $\frac{7}{2}$-homogeneous solutions.