Calculus of Variations and Geometric Measure Theory

Liouville type theorems and local behaviour of solutions to degenerate or singular problems

Susanna Terracini

created by gelli on 16 Oct 2020
modified on 20 Oct 2020

21 oct 2020 -- 18:00   [open in google calendar]

Colloquium Dipartimento di Matematica di Pisa - online

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We consider an equation in divergence form with a singular-degenerate weight \[ -\mathrm{div}(
^a A(x,y)\nabla u)=
^a f(x,y,u)\; \quad\textrm{or}\; \textrm{div}(
^aF(x,y,u))\;, \] We first study the regularity of the nodal sets of solutions in the linear case. Next, when the r.h.s. does not depend on $u$, 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.