ON THE SINGULAR SET IN THE THIN OBSTACLE PROBLEM: HIGHER-ORDER BLOW-UPS AND THE VERY THIN OBSTACLE PROBLEM

We consider the singular set in the thin obstacle problem with weight vertical bar x(n +1)vertical bar(a) for a epsilon (-1, 1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical obstacle problem. As a result, under a superharmonicity condition on the obstacle, we prove that each stratum of the singular set is locally contained in a single C-2 manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a C-1,C-alpha manifold for some alpha > 0 if a < 0. In studying the top stratum, we discover a dichotomy, until now unseen, in this problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups at singular points in the top stratum are global, homogeneous solutions to a codimension-2 lower-dimensional obstacle problem (or fractional thin obstacle problem) when a < 0, whereas second blow-ups at singular points in the top stratum are global, homogeneous, and a-harmonic polynomials when a >= 0. To do so, we establish regularity results for this codimension-2 problem, which we call the very thin obstacle problem. Our methods extend to the majority of the singular set even when no sign assumption on the Laplacian of the obstacle is made. In this general case, we are able to prove that the singular set can be covered by countably many C-2 manifolds, up to a lower-dimensional subset.

Automated summary

In this study, we investigate the fine properties of the singular set in the thin obstacle problem with weight and establish a hierarchical structure of singular points under certain conditions. For the top stratum of singular points, we uncover a previously unseen dichotomy, where second blow-ups are either global, homogeneous solutions to a lower-dimensional obstacle problem when the weight is less than 0, or global, homogeneous, and a-harmonic polynomials when the weight is greater than or equal to 0.