On a class of thin obstacle-type problems for the bi-Laplacian operator
Donatella Danielli, Giovanni Gravina
TL;DR
This work analyzes a class of thin obstacle-type problems for the bi-Laplacian by using the Caffarelli–Silvestre extension framework with a degenerate/singular weight $y^b$ to formulate a fourth-order extended problem. The authors establish sharp local regularity results for the solution $u$ and its extension $v=\Delta_b u$, derive Almgren’s frequency formula and associated blow-up analysis, and introduce Monneau- and Weiss-type monotonicity formulas to classify homogeneous blow-ups. This enables a precise stratification of the thin free boundary into regular and singular parts, with detailed structural results depending on the dimension and weight parameter $b$. The combination of variational existence, bootstrapped regularity, frequency analysis, and blow-up/classification techniques provides a robust framework for two-phase thin obstacle problems and their higher-order extensions, connecting local regularity with global free-boundary geometry. The results have potential implications for higher-order nonlocal models and fractional powers of the Laplacian via higher-order extensions.
Abstract
This paper investigates the regularity of solutions and structural properties of the free boundary for a class of fourth-order elliptic problems with Neumann-type boundary conditions. The singular and degenerate elliptic operators studied naturally emerge from the extension procedure for higher-order fractional powers of the Laplacian, while the choice of non-linearity considered encompasses two-phase boundary obstacle problems as a special case. After establishing local regularity properties of solutions, Almgren- and Monneau-type monotonicity formulas are derived and utilized to carry out a blow-up analysis and prove a stratification result for the free boundary.