Direct Epiperimetric Inequalities for the Thin Obstacle Problem and Applications

We introduce a new logarithmic epiperimetric inequality for the 2m‐Weiss energy in any dimension, and we recover with a simple direct approach the usual epiperimetric inequality for the 3/2‐Weiss energy. In particular, even in the latter case, unlike the classical statements, we do not assume any a priori closeness to a special class of homogeneous functions. In dimension 2, we also prove for the first time the classical epiperimetric inequality for the (2m − 1/2)‐Weiss energy, thus covering all the admissible energies.

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