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arXiv · 2303.15843

Isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces

Tomasz Adamowicz·Giona Veronelli
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Abstract

We investigate the logarithmic and power-type convexity of the length of the level curves for $a$-harmonic functions on smooth surfaces and related isoperimetric inequalities. In particular, our analysis covers the $p$-harmonic and the minimal surface equations. As an auxiliary result, we obtain higher Sobolev regularity properties of the solutions, including the $W^{2,2}$ regularity. The results are complemented by a number of estimates for the derivatives $L'$ and $L''$ of the length of the level curve function $L$, as well as by examples illustrating the presentation. Our work generalizes results due to Alessandrini, Longinetti, Talenti and Lewis in the Euclidean setting, as well as a recent article of ours devoted to the harmonic case on surfaces.

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Tomasz Adamowicz, Giona Veronelli. 2023-03-28. Isoperimetric inequalities and regularity of $A$-harmonic functions on surfaces. https://arxiv.org/abs/2303.15843

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