Mostly Maximum Principle (17w5116)
Isabeau Birindelli (Sapienza Università di Roma)
Antonio Vitolo (Università di Salerno)
Italo Capuzzo Dolcetta (Sapienza Università di Roma)
The maximum principle arises in so many instances and problems that we cannot hope to cover them all.
We plan to consider in particular the following themes:
• Regularity of solutions for fully nonlinear operators be they elliptic, degenerate elliptic, local or non local.
Alexandrov Bakelman Pucci maximum principle and Harnack’s inequality are a first step toward regularity. Although well established in classical settings these estimates are still object of research. An important realted tool is the “improvement of flatness”, many of the mathematicians that have used, enhanced or even created these techniques will be at the meeting (e.g. Barles, Boccardo, Cabré, Imbert, Ishii, Nirenberg, Silvestre, Sire, Swiech, Vitolo, etc.)
• Hamilton-Jacobi equations In the last fifty year the advantage of using a viscosity solution approach to study Hamilton-Jacobi or Hamilton-Jacobi-Bellman equation combined with optimal control and non smooth analysis techniques has been quite evident. Many questions are still open and pioneers of the subject as well as very active young researcher will meet (e.g. Bardi, Barles, Capuzzo Dolcetta, Figalli, Ishii, Koike, Ley, Porretta, Souganidis, Souplet, etc.)
• Liouville type results It has been known since Liouville that bounded entire harmonic functions are constants, this results has known an incredible number of extensions that go under the name of Liouville type result. Beside the intrinsic importance of such a result, it has also many applications e.g. to prove a priori estimate for solutions in bounded domains (à la Gidas Spruck), on the other hand it allow to study homogenous growth near the boundary etc.. Key players in this field are e.g. Berestycki, Birindelli, Capuzzo Dolcetta, Nirenberg, Leoni, Polacik, Quaas, Sirakov, Souplet, etc.
• Qualitative properties such as symmetry, lack of symmetry, overdetermined problems Starting with the moving plane method and continuing with the sliding method and the reflection method, the maximum principle has extensibly been a key tool to prove symmetry of the solution of semi-linear and fully nonlinear equations, as well as studying existence or lack of existence for solutions od overdetermined problems. Again we have gathered some of the principal contributors to this field (Berestycki, Del Pino, Farina, Garcia Melian, Pacella, Sicbaldi, Valdinoci, etc.
• Comparison principle and spectral properties In their acclaimed paper Berestycki, Nirenberg and Varadhan have linked the notion of principal eigenvalue with that of the validity of the maximum principle. This somehow is the milestone that on one hand opens the way to the extension of the notion of eigenvalue to non linear and non variational operators and on the other leads to the research of the limits of applications of the maximum principle. In these two directions we would like to mention the important contributions of e.g. Berestycki, Birindelli, Capuzzo Dolcetta, Demengel, Ishii, Leoni, Patrizi, Quaas, Rossi, Sirakov…for the spectral part and for the limit of applications of the maximum principle see e.g. Ishii, Leoni, Payne, Sweers, Vitolo
• Parabolic equations asymptotic analysis, traveling fronts. As is well known, the principal eigenvalue is related with the decay estimates of the corresponding heat equation, but, also, with the speed of propagation of traveling fronts; these model are particularly important in models of population dynamics. (Berestycki, Polacik, Rossi, Souplet, etc.)
• Minimal surfaces. The classical PDE approach to geometric problems still produces many interesting new ideas and results. Recently the charaterization and classification of solutions to geometric equations such as Yamabe or Allen Cahn equation has scattered new theorems and tools, some of the main contributors in this area will be present at our meeting e.g. Cabré, Da Lio, Da Del Pino, Farina, Figalli, Riviere, Sicbaldi, Sire, Valdinoci, etc.