Some monotonicity results for general systems of nonlinear elliptic PDEs

In this paper we show that minima and stable solutions of a general energy functional of the form $$\int_{\Omega} F(\nabla u,\nabla v,u,v,x)dx$$ enjoy some monotonicity properties, under an assumption on the growth at infinity of the energy. Our results are quite general, and comprise some rigidity results which are known in the literature

A density property for fractional weighted Sobolev spaces

In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support. The additional difficulty in this nonlocal setting is caused by the fact that the weights are not necessarily translation invariant

(Non)local and (non)linear free boundary problems

We discuss some recent developments in the theory of free boundary problems, as obtained in a series of papers in collaboration with L. Caffarelli, A. Karakhanyan and O. Savin. The main feature of these new free boundary problems is that they deeply take into account nonlinear energy superpositions and possibly nonlocal functionals. The nonlocal parameter interpolates between volume and perimeter functionals, and so it can be seen as a fractional counterpart of classical free boundary problems, in which the bulk energy presents nonlocal aspects. The nonlinear term in the energy superposition takes into account the possibility of modeling different regimes in terms of different energy levels and provides a lack of scale invariance, which in turn may cause a structural instability of minimizers that may vary from one scale to another

Geometric inequalities and symmetry results for elliptic systems

We obtain some Poincar\'{e} type formulas, that we use, together with the level set analysis, to detect the one-dimensional symmetry of monotone and stable solutions of possibly degenerate elliptic systems of the form {eqnarray*} {{array}{ll} div(a(|\nabla u|) \nabla u) = F_1(u, v), div(b(|\nabla v|) \nabla v) = F_2(u, v), {array}. {eqnarray*} where $F\in C^{1,1}_{loc}(\R^2)$. Our setting is very general, and it comprises, as a particular case, a conjecture of De Giorgi for phase separations in $\R^2$.Comment: Minor change

On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data

We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and we apply them to the study of an associated nonlocal problem. We also establish a counterexample in the corresponding framework for the fractional Laplacian

All functions are locally ss-harmonic up to a small error

We show that we can approximate every function $f\in C^{k}(\bar{B_1})$ with a $s$-harmonic function in $B_1$ that vanishes outside a compact set. That is, $s$-harmonic functions are dense in $C^{k}_{\rm{loc}}$. This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.Comment: To appear in J. Eur. Math. Soc. (JEMS