Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

Yannick Sire; Susanna Terracini; Stefano Vita; Yannick Sire; Susanna Terracini; Stefano Vita

doi:10.3934/mine.2021005

Mathematics in Engineering

2021, Volume 3, Issue 1: 1-50. doi: 10.3934/mine.2021005

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Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions

1.
Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, U.S.A
2.
Dipartimento di Matematica G. Peano, Università degli Studi di Torino, Via Carlo Alberto 10, 20123 Torino, Italy
3.
Dipartimento di Matematica, Università degli Studi di Milano Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milano, Italy

Received: 19 March 2020 Accepted: 19 July 2020 Published: 31 July 2020

We consider a class of equations in divergence form with a singular/degenerate weight $ -\mathrm{div}(|y|^a A(x, y)\nabla u) = |y|^a f(x, y)+\textrm{div}(|y|^aF(x, y))\; . $ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0, \alpha}$ and $C^{1, \alpha}$ a priori bounds for approximating problems in the form $ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x, y)\nabla u) = (\varepsilon^2+y^2)^{a/2} f(x, y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x, y)) $ as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.
- degenerate and singular elliptic equations,
- Liouville type theorems,
- blow-up,
- fractional Laplacian,
- divergence form elliptic operator,
- Schauder estimates,
- boundary Harnack,
- Fermi coordinates
Citation: Yannick Sire, Susanna Terracini, Stefano Vita. Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions[J]. Mathematics in Engineering, 2021, 3(1): 1-50. doi: 10.3934/mine.2021005

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Abstract

We consider a class of equations in divergence form with a singular/degenerate weight $ -\mathrm{div}(|y|^a A(x, y)\nabla u) = |y|^a f(x, y)+\textrm{div}(|y|^aF(x, y))\; . $ Under suitable regularity assumptions for the matrix $A$, the forcing term $f$ and the field $F$, we prove Hölder continuity of solutions which are odd in $y\in\mathbb{R}$, and possibly of their derivatives. In addition, we show stability of the $C^{0, \alpha}$ and $C^{1, \alpha}$ a priori bounds for approximating problems in the form $ -\mathrm{div}((\varepsilon^2+y^2)^{a/2} A(x, y)\nabla u) = (\varepsilon^2+y^2)^{a/2} f(x, y)+\textrm{div}((\varepsilon^2+y^2)^{a/2}F(x, y)) $ as $\varepsilon\to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

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