In the framework of linear elliptic equations of second order, the unique continuation principle states that if a solution vanishes on an open subset of a domain, then it vanishes identically in the domain. The principle applies under very general assumptions on the data and has various applications — in particular it implies the strict monotonicity property of eigenvalues with respect to domain inclusion. The unique continuation principle admits a straightforward extension to semilinear equations with Lipschitz nonlinearities, but it fails in general in the case of sublinear equations. In the talk, I will discuss very recent positive results for a rather large class of sublinear equations and also for some problems with discontinuous nonlinearities.

This is joint work with Nicola Soave (Politecnico di Milano).