PhD research topic - Isabeau Birindelli
Partial differential equations
Fully nonlinear elliptic PDE, Elliptic/ Degenerate elliptic PDE, Maximum principle, Regularity, Qualitative properties, Principal eigenvalue, Non-local PDE
The maximum principle can be seen as the backbone of the theory of elliptic and parabolic problems. When few functional analytical tools are available, it is the key instrument in order to
- give a definition of the solution in very weak settings i.e. viscosity solutions
- prove the existence of solutions: “uniqueness implies existence” L. Nirenberg
- prove regularity of solutions via Harnack’s inequality and the A-B-P inequality
- prove qualitative properties such as symmetry of solutions.
Many of these classical topics have been extended in the last decades to non linear equations, to degenerate elliptic operators, to nonsmooth domains.
Spectral properties and the maximum principle
It is a well-known classical fact that in general the validity of the maximum principle is related to the spectral properties of the operator in particular the principal or first eigenvalue see. Precisely the positivity of the principal eigenvalue of the homogeneous Dirichlet problem for a second order elliptic linear operator in a bounded domain is equivalent to the validity of the Maximum Principle. Both the concept of principal eigenvalue and its relationship with the Maximum Principle has been extended to fullynonlinear homogenous operators.
But in the case of degenerate elliptic operators, the question has only be started to be understood (see e.g. [BCDPR] and [BGI], many questions remain open. In particular in the case of the truncated Laplacians (see [BGI]) that arise naturally in many geometrical contest there many questions that need to be answered. A possible interesting extension is the case of non-local truncated Laplacians.
[BNV] Berestycki, Nirenberg, Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47 (1994)
[BCDPR] Berestycki, Capuzzo Dolcetta, Porretta, Rossi, Maximum principle and generalized principal eigenvalue for degenerate elliptic operators. J. Math. Pures Appl. (9) 103 (2015)
[BGI] Birindelli, Galise, Ishii, A family of degenerate elliptic operators: maximum principle and its consequences. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018)
[IL] Ishii, Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990)