Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions
Yannick Sire, Susanna Terracini, Stefano Vita, 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, <sup>†</sup><b>This contribution is part of the Special Issue:</b> Contemporary PDEs between theory and modeling—Dedicated to Sandro Salsa, on the occasion of his 70th birthday, Guest Editor: Gianmaria Verzini, Link: <a href="www.aimspress.com/mine/article/5753/special-articles" target=_blank>www.aimspress.com/mine/article/5753/special-articles</a>
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.