Non UBC
DSpace
Susanna Terracini
2019-11-06T09:41:33Z
2019-05-09T09:02
We consider an equation in divergence form with a singular/degenerate weight \[
-\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y)\; \quad\textrm{or}\; \textrm{div}(|y|^aF(x,y))\;,
\]
Under suitable regularity assumptions for the matrix $A$ and $f$ (resp. $F$) we prove H\"older continuity of solutions and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $C^{0,\alpha}$ and $C^{1,\alpha}$ a priori bounds for approximating problems in the form
\[
-\mathrm{div}((\varepsilon^2+y^2)^a A(x,y)\nabla u)=(\varepsilon^2+y^2)^a f(x,y)\; \quad\textrm{or}\; \textrm{div}((\varepsilon^2+y^2)^aF(x,y))
\]
as $\varepsilon\to 0$. Finally, we derive $C^{0,\alpha}$ and $C^{1,\alpha}$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.
https://circle.library.ubc.ca/rest/handle/2429/72198?expand=metadata
36.0 minutes
video/mp4
Author affiliation: Università di Torino
Banff (Alta.)
10.14288/1.0385097
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Faculty
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Partial Differential Equations, Differential Geometry
Liouville type theorems and local behaviour of solutions to degenerate or singular problems.
Moving Image
http://hdl.handle.net/2429/72198