The aim of these notes is to examine the propagation of isotropic Sobolev (micro)singularities for an evolution problem whose generator is given by the Weyl quantization of a complex-valued quadratic form on phase space.
We present the results contained in [6,7], concerning radial solutions of fully nonlinear, uniformly elliptic equations posed in punctured balls, in presence of radial singular quadratic potentials. We discuss both the principal eigenvalues problem and the case of equations having also absorbing superlinear zero order terms: for the former problem, we explicitly compute the principal eigenvalues, thus obtaining an extension to the fully nonlinear framework of the Hardy-Sobolev constant; for the latter case, we provide a complete classification of solutions based on their asymptotic behavior near the singularity.
The differentiation operator in a space of continuous or $L^p$ functions defined on an interval generates a translation semigroup. Here we illustrate the generalization of this result to general first-order operators, possibly singular.
We present some results on the regularity and the singularities of the distance function in domains with boundary.
Let $\mathcal{L}$ be the hypoelliptic Ornstein-Uhlenbeck operator associated with the pair of matrices (A,B). In 2004, Priola and Zabczyk proved the following Liouville-type theorem: every bounded entire solution of $\mathcal{L}u=0$ is constant if and only if (*) every eigenvalue of B has real part less than or equal to zero. This remarkable result raised the following problem, which is still not completely solved: if condition (*) holds, is it true that every non-negative entire solution of $\mathcal{L}u=0$ is constant? In this note, along with a review of the current state of research on this problem, we present some recent new results.
