Workshop: The nodal set of solutions of some sublinear equations, with or without homogeneity

Speaker

Nicola Soave

Abstract

We are concerned with the nodal set of solutions to sublinear equations of the form
􀀀u = +􀀀 u+p􀀀1 􀀀 􀀀 􀀀 􀀀q􀀀1 in B1
where + > 0, 􀀀  0, and 1  p  q < 2. The main feature of this equation is that the right-hand has sublinear (or even discontinuous) character, and can be both homogeneous (p = q) and inhomogeneous (p < q). In this talk we present results regarding the nodal set of the solutions. In particular, we focus on the following issues:(a) the validity of the unique continuation principle;
(b) the niteness of the vanishing order at every point and the characterization
of the order spectrum;
(c) the non-degeneracy of solutions;
(d) the partial regularity of the nodal set.

Particular emphasis will be given on the dierences between the homogeneous and the inhomogeneous cases.

The talk is based on joint works with Susanna Terracini, and Giorgio Tortone, and Tobias Weth.