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)