Thursday, December 19, 2019 - 14:30
Analisi MaTÉmatica allo SBAI
Gianmaria Verzini (Politecnico di Milano)
Normalized solutions to semilinear elliptic equations and systems
Abstract: We study the existence of solutions having prescribed L^2 norm to some semilinear elliptic problems in bounded domains. These kind of solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. When the nonlinearity is critical or supercritical with respect to the Gagliardo-Nirenberg inequality, though not Sobolev subcritical, we show that solutions having Morse index bounded from above can exist only when the mass is sufficiently small. On the other hand, we provide sufficient conditions for the existence of such solutions, also in the Sobolev critical case. Based on joint works with Benedetta Noris, Dario Pierotti and Hugo Tavares
Marco Cirant (Università di Parma)
Gradient regularity of viscous Hamilton-Jacobi equations with rough data
Abstract: The talk will be devoted to the study of Lipschitz regularity of solutions to viscous Hamilton-Jacobi equations, with data in Lebesgue spaces. I will first present some results obtained in collaboration with A. Goffi (GSSI-Padova). These are based on adjoint methods, namely on the analysis of regularity to dual equations of Fokker-Planck type. Finally, I will discuss further perspectives regarding the so-called problem of "maximal regularity".