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arXiv · 2311.10185

One-phase free boundary solutions of finite Morse index

José Basulto·Nikola Kamburov
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Abstract

We study global solutions to the classical one-phase free boundary problem that have finite Morse index relative to the Alt-Caffarelli functional. We show that such solutions are stable outside a compact set and characterize the index as the maximal number of linearly independent $L^2$ integrable eigenfunctions of the corresponding Robin eigenvalue problem, associated to negative eigenvalues. As an application, we obtain a complete classification of global solutions of finite Morse index in the plane. Our results are counterparts to the minimal surface theorems of Fischer-Colbrie and Gulliver.

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José Basulto, Nikola Kamburov. 2023-11-16. One-phase free boundary solutions of finite Morse index. https://arxiv.org/abs/2311.10185

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