Several new 1D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular, 1D symmetry is proven under theâ€¦ (More)

We provide an abstract framework for a symmetry result arising in a conjecture of G. W. Gibbons and we apply it to the fractional Laplace operator, to the elliptic operators with constantâ€¦ (More)

We study bounded, monotone solutions of âˆ†u = W â€²(u) in the whole of R, where W is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, uâ€¦ (More)

Abstract. This paper is devoted to the study of rigidity properties for special solutions of nonlinear elliptic partial differential equations on smooth, boundaryless Riemannian manifolds. As far asâ€¦ (More)

We consider some elliptic PDEs with Dirichlet and Neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classification of the solution andâ€¦ (More)

We prove the one-dimensional symmetry of solutions to elliptic equations of the form âˆ’div(ea(|âˆ‡u|)âˆ‡u) = f(u)e, under suitable energy conditions. Our results hold without any restriction on theâ€¦ (More)