In this paper we present a survey concerning unconstrained free boundary problems of type  F1(D u,∇u, u, x) = 0 in B1 ∩ Ω, F2(D u,∇u, u, x) = 0 in B1 \ Ω, u ∈ S(B1), where B1 is the unit ball, Ω is an unknown open set, F1, F2 are elliptic operators (admitting regular solutions), and S is a functions space to be specified in each case. Our main objective is to discuss a unifying approach to th...

We prove that the crossed product C∗-algebra C∗ r (Γ, ∂Γ) of a free group Γ with its boundary ∂Γ naturally sits between the reduced group C∗algebra C∗ r Γ and its injective envelope I(C∗ r Γ). In other words, we have natural inclusion C∗ r Γ ⊂ C∗ r (Γ, ∂Γ) ⊂ I(C∗ r Γ) of C∗-algebras.

Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is ...

We demonstrate that a method of Colesanti and Salani, which compares solutions of elliptic differential equations to their quasiconcave envelopes, can be extended to derive convexity of free boundaries. As examples we present the so-called dam problem, a free boundary problem modelling pollution and a Bernoulli problem. Moreover we prove strict convexity of the wet region in the dam problem in ...