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

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Susanna Terracini
Università di Torino
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. \[ \] In order to join the seminar, please fill in the participation form before October 20th. Further information and instructions will be sent afterwards to the online audience. Note that it is possibile to follow the seminar streaming on the Youtube link