TY - ELEC
AU - Susanna Terracini
PY - 2019
TI - Liouville type theorems and local behaviour of solutions to degenerate or singular problems.
LA - eng
M3 - Moving Image
AB - 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.
N2 - 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.
UR - https://open.library.ubc.ca/collections/48630/items/1.0385097
ER - End of Reference