We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and ...

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous ...

This article is a survey on boundedness results for stable solutions to semilinear elliptic problems. For these solutions, we present the currently known L8 estimates that hold for all nonlinearities. Such estimates are ...

We study hypersurfaces of RN with constant nonlocal (or fractional) mean curvature. This is the equation associated with critical points of the fractional perimeter functional under a volume constraint. We establish the ...

In this Note, we consider the linear heat equation ut - Au= a(x)u in (0,T)xW,u=0 on (0,T)xaW, and u(0)=uº on W, where W C RN is a smooth bounded domain. We assume that a€L^loc(W) a >=0 and u>=. A simple condition on the ...

We study in this note the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with slowly decaying kernel, an important example being the fractional Laplacian. Contrary to what happens ...

We prove the interior C2,α regularity of solutions for some nonconvex fully nonlinear elliptic equations F(D2u, x) = f (x), x ∈ B1 ⊂ Rn. Our hypothesis is that, for every x ∈ B1, F(·,x) is the minimum of a concave operator ...

This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized ...