Partial differential equations (PDEs):
Isoperimetric inqualities & Fundamental Gap conjecture;
Regularity and the qualitative properties of solutions of elliptic and parabolic PDEs;
Maximum Principles;
Domain perturbation;
Kurdyka-Łojasiewicz-Simon inequality and long-time asymptotic behaviour.
All PDEs I am studying usually involve the (local or fractional) p-Laplace operator, or doubly-nonlinear operators.
Nonlinear harmonic anylsis with application to PDEs:
Regularity of nonlinear semigroups, gradient flows (in metric spaces)
Fractional powers of (nonlinear) accretive operators;
Dirichlet-to-Neumann operators and nonlinear Riesz operators;
Interplay between the regularising effects of semigroups and functional inequalities (Sobolev, Talagrand, etc.);
