Geometric Properties for Parabolic and Elliptic PDEs The study of qualitative aspects of PDEs has always attracted much attention from the early beginnings. More recently once basic issues about PDEs such as existence uniqueness and stability of solutions have been understood quite well research on topological and or geometric properties of their solutions has become more intense. The study of these issues is attracting the interest of an increasing number of researchers and is now a broad and well established research area with contributions that often come from experts from disparate areas of mathematics such as differential and convex geometry functional analysis calculus of variations mathematical physics to name a few. This volume collects a selection of original results and informative surveys by a group of international specialists in the field analyzes new trends and techniques and aims at promoting scientific collaboration and stimulating future developments and perspectives in this very active area of research. Goro Akagi Stability and instability of group invariant asymptotic profiles for fast diffusion equations Elvise Berchio A family of Hardy Rellich type inequalities involving the L2 norm of the Hessian matrices Massimiliano Bianchini and Paolo Salani Power concavity for solutions of nonlinear elliptic problems in convex domains Lorenzo Brasco and Rolando Magnanini The heart of a convex set Giulio Ciraolo A viscosity equation for minimizers of a class of very degenerate elliptic functionals Adele Ferone Katos inequality in the half space: an alternative proof and relative improvements Ilaria Fragalà Filippo Gazzola and Jimmy Lamboley Sharp bounds for the p torsion of convex planar domains Giovanni Franzina and Enrico Valdinoci Geometric analysis of fractional phase transition interfaces Antonio Greco Existence of solutions to some classical variational problems Norihisa Ikoma Existence of minimizers for some coupled nonlinear Schrödinger equations Kazuhiro Ishige and Yoshitsugu Kabeya Decay rate of Lq norms of critical Schrödinger heat semigroups Shuichi Jimbo Hadamard variation for electromagnetic frequencies Toru Kan Global structure of the solution set for a semilinear elliptic problem related to the Liouville equation on an annulus Anna Mercaldo A priori estimates and comparison principle for some nonlinear elliptic equations Takeyuki Nagasawa Existence and uniqueness of the n dimensional Helfrich flow Bernhard Ruf and Federica Sani Ground states for elliptic equations in R2 with exponential critical growth Shigeru Sakaguchi Stationary level surfaces and Liouville type theorems characterizing hyperplanes Futoshi Takahashi Nonexistence of multi bubble solutions for a higher order mean field on equation on convex domains.