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Alberto Farina

Publications and source records attributed to Alberto Farina.

52 records · Page 3Linked to original sources
arXiv2014

Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations

We consider the Wulff-type energy functional $$ \mathcal{W}_Ω(u) := \int_ΩB(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.

math.AP↗
arXiv2014

Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs

We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations.

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arXiv2014

1D symmetry for semilinear PDEs from the limit interface of the solution

We study bounded, monotone solutions of$Δu=W'(u)$ in the whole of$\R^n$, where$W$ is a double-well potential. We prove that under suitable assumptions on the limit interface and on the energy growth, $u$ is $1$D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties and the cases studied lie outside the range of $Γ$-convergence methods. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are $1$D, at least up to dimension 4.

math.AP↗
arXiv2014

Symmetry properties of some solutions to some semilinear elliptic equations

In this paper we prove some symmetry results for entire solutions to the semilinear equation $-Δu=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.

math.AP↗
arXiv2014

Qualitative properties and classification of nonnegative solutions to $-Δu=f(u)$ in unbounded domains when $f(0)<0$

We consider nonnegative solutions to $-Δu=f(u)$ in unbounded euclidean domains, where $f$ is merely locally Lipschitz continuous and satisfies $f(0)<0$. In the half-plane, and without any other assumption on $u$, we prove that $u$ is either one-dimensional and periodic or positive and strictly monotone increasing in the direction orthogonal to the boundary. Analogous results are obtained if the domain is a strip. As a consequence of our main results, we answer affirmatively to a conjecture and to an open question posed by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and monotonicity results in the higher-dimensional case.

math.AP↗
arXiv2014

Existence and stability properties of entire solutions to the polyharmonic equation $(-Δ)^m u=e^u$ for any $m\ge 1$

We study existence and stability properties of entire solutions of a polyharmonic equation with an exponential nonlinearity. We study existence of radial entire solutions and we provide some asymptotic estimates on their behavior at infinity. As a first result on stability we prove that stable solutions (not necessarily radial) in dimensions lower than the conformal one never exist. On the other hand, we prove that radial entire solutions which are stable outside a compact always exist both in high and low dimensions. In order to prove stability of solutions outside a compact set we prove some new Hardy-Rellich type inequalities in low dimensions.

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arXiv2013

Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems

We prove the symmetry of components and some Liouville-type theorems for, possibly sign changing, entire distributional solutions to a family of nonlinear elliptic systems encompassing models arising in Bose-Einstein condensation and in nonlinear optics. For these models we also provide precise classification results for non-negative solutions. The sharpness of our results is also discussed.

math.AP↗
arXiv2013

Some symmetry results for entire solutions of an elliptic system arising in phase separation

We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.

math.AP↗
arXiv2013

Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces

We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.

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arXiv2012

Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds

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.

math.AP↗
arXiv2008

Stable solutions of elliptic equations on Riemannian manifolds

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 as stable solutions are concerned, we derive a new weighted Poincaré inequality which allows to prove Liouville type results and the flatness of the level sets of the solution in dimension 2, under suitable geometric assumptions on the ambient manifold.

math.AP↗
arXiv2008

Stable solutions of $-Δu = f(u)$ in $\R^N$

The paper proves Liouville-type results for stable solutions of semilinear elliptic PDEs with convex nonlinearity, posed on the entire Euclidean space. Extensions to solutions which are stable outside a compact set are also presented.

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