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

Abstract

We consider a class of equations in divergence form with a singular/degenerate weight $$-\mathrm{div}(|y{|}^{a}A(x,y)\nabla u)=|y{|}^{a}f(x,y)+\text{div}(|y{|}^{a}F(x,y)).$$ 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 $$-\mathrm{div}(({\epsilon }^2+y^2{)}^{a/2}A(x,y)\nabla u)=({\epsilon }^2+y^2{)}^{a/2}f(x,y)+\text{div}(({\epsilon }^2+y^2{)}^{a/2}F(x,y))$$ as $\epsilon \to 0$. Our method is based upon blow-up and appropriate Liouville type theorems.

arXiv comment48 pages

References in corpus (3)

• An extension problem related to the fractional Laplacian
• A new boundary Harnack principle (equations with right hand side)
• Fractional elliptic equations, Caccioppoli estimates and regularity

Cited by in corpus (10)

• Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions
• Higher order boundary Harnack principle via degenerate equations
• Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations
• Schauder estimates for parabolic equations with degenerate or singular weights
• Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
• Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem
• Higher order boundary Harnack principles in Dini type domains
• Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem
• Optimal H{ö}lder convergence of a class of singular steady states to the Bahouri-Chemin patch
• The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains