Calculus of Variations and Geometric Measure Theory
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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.

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