>>> English version below! <<<

⟶ Data: Giovedì 22 Febbraio, ore 15

⟶ Dove: Aula Seminari (Pisa) e Online su Google meet: Adriano Prade (seminarimap.wixsite.com)

⟶ Abstract: After the breakthrough paper by Caffarelli and Silvestre in 2007, the study of fractional Laplacian and more general nonlocal operators has gained increasing popularity, from both an analytical and a probabilistic point of view. The purpose of the seminar is to present such a class of operators, starting from basic notions and then focusing on the PDEs’ theory developing from them. First, the formula for the square root of the Laplacian (-∆)^½ is provided, together with a few immediate remarks and the motivation behind its name. Then, after introducing the definition of fractional Laplacian (-∆)^s, we give an overview of its main properties, highlighting some similarities and differences with the classical Laplacian (-∆). The second part of the talk is entirely devoted to nonlocal PDEs, reserving particular attention to some regularity issues. We begin by dealing with the possible notions of solutions and next various results available in the literature are outlined. Finally, after introducing the class of Reifenberg flat sets, we present the problem of boundary Hölder regularity of some nonlocal PDEs on this kind of sets, sketching some possible solution strategies if time permits.

Jeremy Mirmina

Filippo Paiano

⟶ Date: Thursday 22nd February, 15:00

⟶ Where: Aula Seminari (Pisa) and online on Google meet: Adriano Prade (seminarimap.wixsite.com)

⟶ Title: Fractional Laplacian operator and regularity of nonlocal PDEs

⟶ Abstract: After the breakthrough paper by Caffarelli and Silvestre in 2007, the study of fractional Laplacian and more general nonlocal operators has gained increasing popularity, from both an analytical and a probabilistic point of view. The purpose of the seminar is to present such a class of operators, starting from basic notions and then focusing on the PDEs’ theory developing from them. First, the formula for the square root of the Laplacian (-∆)^½ is provided, together with a few immediate remarks and the motivation behind its name. Then, after introducing the definition of fractional Laplacian (-∆)^s, we give an overview of its main properties, highlighting some similarities and differences with the classical Laplacian (-∆). The second part of the talk is entirely devoted to nonlocal PDEs, reserving particular attention to some regularity issues. We begin by dealing with the possible notions of solutions and next various results available in the literature are outlined. Finally, after introducing the class of Reifenberg flat sets, we present the problem of boundary Hölder regularity of some nonlocal PDEs on this kind of sets, sketching some possible solution strategies if time permits.

Jeremy Mirmina

Filippo Paiano

Luciano Sciaraffia