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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 to understand the structure and regularity of free boundaries. In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author.

We study the regularity up to the boundary of solutions to fractional heat equation in bounded $$C^{1,1}$$ C1,1 domains. More precisely, we consider solutions to $$\partial _t u + (-\Delta )^su=0 \hbox { in }\Omega ,\ t > 0$$ ∂tu+(-Δ)su=0inΩ,t>0 , with zero Dirichlet conditions in $$\mathbb {R}^n{\setminus } \Omega $$ Rn\Ω and with initial data $$u_0\in L^2(\Omega )$$ u0∈L2(Ω) . Using the results of the second author and Serra for the elliptic problem, we show that for all $$t>0$$ t>0 we have $$u(\cdot , t)\in C^s(\mathbb {R}^n)$$ u(·,t)∈Cs(Rn) and $$u(\cdot , t)/\delta ^s \in C^{s-\epsilon }(\overline{\Omega })$$ u(·,t)/δs∈Cs-ϵ(Ω¯) for any $$\epsilon > 0$$ ϵ>0 and $$\delta (x) = \hbox {dist}(x,\partial \Omega )$$ δ(x)=dist(x,∂Ω) . Our regularity results apply not only to the fractional Laplacian but also to more general integro-differential operators, namely those corresponding to stable Lévy processes. As a consequence of our results, we show that solutions to the fractional heat equation satisfy a Pohozaev-type identity for positive times.

Abstract: Let K be a quadratic field, and let ζ K its Dedekind zeta function. In this paper we introduce a factorization of ζ K into two functions, L 1 and L 2, defined as partial Euler products of ζ K , which lead to a factorization of Riemann’s ζ function into two functions, p 1 and p 2. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when Δ K  > 0 the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.