Bruno Pini Mathematical Analysis Seminar

A basis of resolutive sets for the heat equation: an elementary construction

2023-01-05T16:18:37+01:00

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.

New concentration phenomena for radial sign-changing solutions of fully nonlinear elliptic equations

2023-01-05T16:38:18+01:00

We present recent results about radial sign-changing solutions of a class of fully nonlinear elliptic Dirichlet problems posed in a ball, driven by the extremal Pucci's operators and provided with power zero order terms. We show that new critical exponents appear, related to the existence or nonexistence of sign-changing solutions and due to the fully nonlinear character of the considered problem. Furthermore, we analyze the new concentration phenomena occurring as the exponents approach the critical values.

The basic problem of the calculus of variations: Du Bois-Reymond equation, regularity of minimizers and of minimizing sequences

2023-01-05T17:04:09+01:00

We consider the basic problem of the Calculus of variations of minimizing an integral functional among the absolutely continuous functions that satisfy prescribed boundary conditions. We resume the state of the art and our recent contributions concerning the validity of the Du Bois-Reymond condition, and the Lipschitz regularity of the minimizers and of minimizing sequences (e.g., Lavrentiev phenomenon).

On weighted second order Adams inequalities with Navier boundary conditions

2023-01-05T17:42:02+01:00

We obtain some sharp weighted version of Adams' inequality on second order Sobolev spaces with Navier boundary conditions. The weights that we consider determine a supercritical exponential growth, except in the origin, and the corresponding inequalities hold on radial functions only. We also consider the problem of extremal functions, and we show that the sharp suprema are achieved, as in the unweighted classical case.

The sub-Finsler Bernstein problem in H1

2023-01-05T18:06:26+01:00

This is a note based on the paper [32] written in collaboration with M. Ritoré. The purpose of this note is to present and discuss the Bernstein type problems in the sub-Finsler Heisenberg group H¹. We give a general idea of the state of the art and we prove that a complete, stable, (X,Y)-Lipschitz surface is a vertical plane.

On the regularity of the solutions and of analytic vectors for “sums of squares”

2023-01-05T19:01:45+01:00

We present a brief survey on some recent results concerning the local and global regularity of the solutions for some classes/models of sums of squares of vector fields with real-valued real analytic coefficients of H"ormander type. Moreover we also illustrate a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of H"ormander type, thus providing a microlocal version, in the analytic category, of a result due to M. Derridj.