Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions. Issue 2 (1st February 2021)

• Record Type: Journal Article
• Authors: Sire, Yannick; Terracini, Susanna; Vita, Stefano
• Is Part Of: Communications in partial differential equations. Volume 46:Issue 2(2021)
• Page Start: 310
• Page End: 361
• Publication Date: 2021-02-01
• Subjects: Degenerate and singular elliptic equations -- blow-up -- fractional divergence form elliptic operator -- fractional Laplacian -- Liouville type theorems -- Schauder estimates; 35J70 -- 35J75 -- 35R11 -- 35B40 -- 35B44 -- 35B53; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2020.1840586
• Languages: English
• ISSNs: 0360-5302

Abstract

Abstract: We consider a class of equations in divergence form with a singular/degenerate weight − div ( | y | a A ( x, y ) ∇ u ) = | y | a f ( x, y ) or div ( | y | a F ( x, y ) ) . Under suitable regularity assumptions for the matrix A and f (resp. F ) we prove Hölder continuity of solutions which are even in y ∈ R, and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the C 0, α and C 1, α a priori bounds for approximating problems in the form − div ( ( ε 2 + y 2 ) a / 2 A ( x, y ) ∇ u ) = ( ε 2 + y 2 ) a / 2 f ( x, y ) or div ( ( ε 2 + y 2 ) a / 2 F ( x, y ) ) as ε → 0 . Finally, we derive C 0, α and C 1, α bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.