Global solutions to the thin obstacle problem with superquadratic growth

2025/04/30 by Xavier Fernández‐Real, Hui Yu, Fernández-Real, Xavier +1
Computer Science · Mathematics · Engineering · #Contact Mechanics and Variational Inequalities #Nonlinear Partial Differential Equations #Structural Analysis and Optimization

paper · pdf · doi:10.48550/arxiv.2504.21492

Abstract

We study rigidity/flexibility properties of global solutions to the thin obstacle problem. For solutions with bounded positive sets, we give a classification in terms of their expansions at infinity. For solutions with bounded contact sets, we show that the contact sets are highly flexible and can approximate arbitrary compact sets. These phenomena have no counterparts in the classical obstacle problem.

Citations

• Solutions to the Thin Obstacle Problem with non-2D frequency
• Compact contact sets of sub-quadratic solutions to the thin obstacle problem
• Generic regularity of free boundaries for the thin obstacle problem
• Complete classification of global solutions to the obstacle problem
• Contact points with integer frequencies in the thin obstacle problem
• The Thin Obstacle Problem: A Survey
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• Characterizing compact coincidence sets in the obstacle problem -- a short proof
• On global solutions of the obstacle problem
• Direct epiperimetric inequalities for the thin obstacle problem and applications
• Obstacle problems and free boundaries: an overview
• Some New Monotonicity Formulas and the Singular Set in the Lower Dimensional Obstacle Problem
• The structure of the free boundary for lower dimensional obstacle problems
• Optimal Regularity of Lower-Dimensional Obstacle Problems

Related

• Compact contact sets of sub-quadratic solutions to the thin obstacle problem
• Solutions to the nonlinear obstacle problem with compact contact sets
• Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian
• A local boundedness result for a class of obstacle problems with non-standard growth conditions
• Complete classification of global solutions to the obstacle problem