# Susanna Terracini

## Università degli Studi di Torino

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

We consider an equation in divergence form with a singular/degenerate weight

$-\operatorname{div}(|y|^a A(x,y)∇ u)=|y|^a f(x,y,u)\; \quad\textrm{or}\; \operatorname{div}(|y|^aF(x,y,u))\;,$

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 Hölder continuity of solutions and possibly of their derivatives up to order two or more (Schauder estimates). In addition, we show stability of the $$C^{0,α}$$ and $$C^{1,α}$$ a priori bounds for approximating problems in the form

$-\operatorname{div}((ε^2+y^2)^a A(x,y)∇ u)=(ε^2+y^2)^a f(x,y)\; \quad\textrm{or}\; \operatorname{div}((ε^2+y^2)^aF(x,y))$

as $$ε→ 0$$. Finally, we derive $$C^{0,α}$$ and $$C^{1,α}$$ bounds for inhomogenous Neumann boundary problems as well. Our method is based upon blow-up and appropriate Liouville type theorems.

#### References

1. Yannick Sire, Susanna Terracini and Giorgio Tortone, On the nodal set of solutions to degenerate or singular elliptic equations with an application to $$s$$-harmonic functions, Jour. Math. Pures Appl., to appear (https://arxiv.org/abs/1808.01851)

2. Yannick Sire, Susanna Terracini and Stefano Vita, Liouville type theorems and local behaviour of solutions to degenerate or singular problems part I: even solutions. Preprint 2019 (https://arxiv.org/abs/1903.02143)

3. Yannick Sire, Susanna Terracini and Stefano Vita, Liouville type theorems and local behaviour of solutions to degenerate or singular problems part II: odd solutions. Preprint 2020 (https://arxiv.org/abs/2003.09023)