Alberto Farina

On the classification of solutions of

In this short paper we prove that, for 3 < N < 9, the problem -Δu = e u on the entire Euclidean space R N does not admit any solution stable outside a compact set of R N . This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

-\Delta u= e^u

Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation − Δu = f(u) on a Riemannian manifold with non-negative Ricci curvature, we are able to classify both the solution and the manifold. We also discuss the classification of monotone (with respect to the direction of some Killing vector field) solutions, in the spirit of a conjecture of De Giorgi, and the rigidity features for overdetermined elliptic problems on submanifolds with boundary.

on

Several new

\mathbb {R}^N

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: Stability outside a compact set and applications

D results for solutions of possibly singular or degenerate elliptic equations, inspired by a conjecture of De Giorgi, are provided. In particular,

Splitting Theorems, Symmetry Results and Overdetermined Problems for Riemannian Manifolds

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1 D symmetry for solutions of semilinear and quasilinear elliptic equations

D symmetry is proven under the assumption that either the profiles at infinity are

CHAPTER 2 – Liouville-Type Theorems for Elliptic Problems

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On partially and globally overdetermined problems of elliptic type

D, or that one level set is a complete graph, or that the solution is minimal or, more generally,

Finite-energy solutions, quantization effects and Liouville-type results for a variant of the Ginzburg-Landau systems in ℝK*

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