On the multiple membranes problem
Ovidiu Savin, Hui Yu
TL;DR
This paper proves the best possible regularity for solutions to the multiple membranes problem, analyzes blow-up behavior at high multiplicity free boundary points, and classifies blow-up profiles in two dimensions using a Weiss-type monotonicity formula.
Contribution
It introduces the optimal regularity results and a comprehensive classification of blow-up profiles for the multiple membranes problem in the plane.
Findings
Established optimal regularity for solutions.
Performed blow-up analysis at high multiplicity points.
Classified blow-up profiles in two dimensions.
Abstract
We establish the optimal regularity for solutions to the multiple membranes problem, and perform a blow-up analysis at points on the free boundary with the highest multiplicity. This leads to a complete classification of blow-up profiles in the plane. The main technical tool is a Weiss-type monotonicity formula.
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Taxonomy
TopicsNonlinear Partial Differential Equations · Point processes and geometric inequalities · Optimization and Variational Analysis