A remark on an obstacle problem with lower regularity

TL;DR

This paper introduces a new monotone quantity for the classical obstacle problem with a non-smooth obstacle, demonstrating that blow-up solutions are homogeneous functions of degree less than 2, thus advancing understanding of obstacle problems with lower regularity.

Contribution

It constructs a monotone quantity for obstacle problems with non-smooth obstacles and characterizes the homogeneity degree of blow-up solutions, extending previous regularity results.

Findings

Monotone quantity for obstacle problem with non-smooth obstacle

Blow-ups are homogeneous functions of degree less than 2

Advances understanding of obstacle problems with lower regularity

Abstract

We construct a monotone quantity for the classical obstacle problem with non-smooth obstacle, and show that the blow-ups are homogeneous functions of degree $\alpha<2$.