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arXiv · 2003.06989

On the finite $(Q-1)$-Hausdorff measure of the free boundary in the subelliptic obstacle problem

Agnid Banerjee·Nicola Garofalo
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Abstract

In this note, we prove the finite ${(Q-1)}$-Hausdorff measure of the free boundary in the obstacle problem in a Carnot group $\mathbb{G}$. Here, $Q$ represents the homogeneous dimension of $\mathbb{G}$. Our main result, Theorem 1.1, constitutes the subelliptic counterpart of the Euclidean result due to Caffarelli, but the analysis is complicated by the lack of commutation of the left-invariant vector fields. This obstruction is compensated by the use of right-invariant derivatives, and by a delicate compactness argument inspired to Caffarelli's fundamental works.

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Agnid Banerjee, Nicola Garofalo. 2020-03-16. On the finite $(Q-1)$-Hausdorff measure of the free boundary in the subelliptic obstacle problem. https://arxiv.org/abs/2003.06989

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