A priori regularity estimates for equations degenerating on nodal sets

2024/04/10 by Terracini, Susanna, Tortone, Giorgio, Vita, Stefano · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.06980

Abstract

We prove a priori and a posteriori Hölder bounds and Schauder C^1,α estimates for continuous solutions of degenerate elliptic equations with variable coefficients of the form div(|u|^a A∇ w)=0\qquadin Ω⊂\mathbb R², a∈\mathbb R, where the weight u is itself a solution to an elliptic equation of the type div(A ∇ u) = 0, with A a Lipschitz-continuous, uniformly elliptic matrix. The function u is allowed to have a nontrivial, possibly singular nodal set. The estimates are uniform with respect to u within a class of normalized solutions having bounded Almgren frequency. In the special case a = 2, our results apply to the ratio of two solutions to the same elliptic equation sharing a common zero set. Precisely, we prove higher-order boundary Harnack principles on nodal domains, via the derived Schauder estimates for the associated degenerate equations. The results are based upon a fine blow-up argument, a Liouville theorem, and quasiconformal maps.

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• A priori Hölder estimates for equations degenerating on nodal sets

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