Terracini: Liouville type theorems and local behaviour of solutions to degenerate or singular problems
Terracini: In order to join the seminar, please fill in the [mandatory participation form|https://forms.gle/HMZA6Y8ov3fzLRc68] before October 20th. Further information and instructions will be sent afterwards to the online audience. The streaming is also available on the Youtube channel [https://youtu.be/D-GUe6MCwZc] We consider an equation in divergence form with a singular-degenerate weight \[ -\mathrm{div}(|y|^a A(x,y)\nabla u)=|y|^a f(x,y,u)\; \quad\textrm{or}\; \textrm{div}(|y|^aF(x,y,u))\;, \] 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. http://cvgmt.sns.it/seminar/754/
When
Wed Oct 21, 2020 4pm – 5pm Coordinated Universal Time
Where
Colloquium Dipartimento di Matematica di Pisa - online (map)
