arXiv subjects
Publications and source records attributed to Enrico Valdinoci.
We prove that, in every dimension, Lipschitz nonlocal minimal surfaces are smooth. Also, we extend to the nonlocal setting a famous theorem of De Giorgi stating that the validity of Bernstein's theorem in dimension $n+1$ is a consequence of the nonexistence of $n$-dimensional singular minimal cones in $\R^n$.
We consider a minimization problem that combines the Dirichlet energy with the nonlocal perimeter of a level set, namely $$ \int_\Om |\nabla u(x)|^2\,dx+\Per\Big(\{u > 0\},\Om \Big),$$ with $σ\in(0,1)$. We obtain regularity results for the minimizers and for their free boundaries $\p \{u>0\}$ using blow-up analysis. We will also give related results about density estimates, monotonicity formulas, Euler-Lagrange equations and extension problems.
The main goal of this work is to prove that every non-negative {\it strong solution} $u(x,t)$ to the problem $$ u_t+(-Δ)^{α/2}u=0 \ \quad\mbox{for } (x,t)\in\mathbb{R}^{n}\times(0,T), \quad 0<α<2, $$ can be written as $$u(x,t)=\int_{\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\, dy},$$ where $$P_{t}(x)=\frac{1}{t^{n/α}}P\left(\frac{x}{t^{1/α}}\right), $$ and $$ P(x):=\int_{\mathbb{R}^{n}}{e^{ix\cdotξ-|ξ|^α}dξ}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \cite{W0} to the nonlocal diffusion framework.
We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$. As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and $s$ is close to 1.
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.
We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $Ω$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $Ω$. Moreover, we construct examples of sets for which the limit does not exist.
We review some recent results on minimisers of a non-local perimeter functional, in connection with some phase coexistence models whose diffusion term is given by the fractional Laplacian.
We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$ \int F(\nabla u, u, x) dx $$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions.
We show that the only nonlocal $s$-minimal cones in $\R^2$ are the trivial ones for all $s \in (0,1)$. As a consequence we obtain that the singular set of a nonlocal minimal surface has at most $n-3$ Hausdorff dimension.
This paper deals with the following class of nonlocal Schrödinger equations $$ \displaystyle (-Δ)^s u + u = |u|^{p-1}u \ \ \text{in} \ \mathbb{R}^N, \quad \text{for} \ s\in (0,1). $$ We prove existence and symmetry results for the solutions $u$ in the fractional Sobolev space $H^s(\mathbb{R}^N)$. Our results are in clear accordance with those for the classical local counterpart, that is when $s=1$.
We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \|u\|^2_{H^s(Ω)}+\int_ΩW(u)\,dx, $$ where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\mathbb{R}^n$. The results collected here will also be useful for forthcoming papers, where the second and the third author will study the $Γ$-convergence and the density estimates for level sets of minimizers.
This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
We consider gradient descent equations for energy functionals of the type S(u) = 1/2 < u(x), A(x)u(x) >_{L^2} + \int_Ω V(x,u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration. We consider the steepest descent equation for S where the gradient is an element of the Sobolev space H^β, β\in (0,1), with a metric that depends on A and a positive number γ> \sup |V_{22}|. We prove a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator. We provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional
We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $ε\ge0$ and $\bar u_ε$ minimizes the functional $$ \int_Ω(ε+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $ψ$, then 0 \le ÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}\bar u_ε|^2)^{(p/2)-1} \nabla_{\H^n}\bar u_ε\Big) \qquad \le (÷_{\H^n}\, \Big((ε+|\nabla_{\H^n}ψ|^2)^{(p/2)-1} \nabla_{\H^n}ψ\Big))^+.
We discuss the $Γ$-convergence, under the appropriate scaling, of the energy functional $$ \|u\|_{H^s(Ω)}^2+\int_ΩW(u)dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. When $s\in [1/2,\,1)$, we show that the energy $Γ$-converges to the classical minimal surface functional -- while, when $s\in(0,\,1/2)$, it is easy to see that the functional $Γ$-converges to the nonlocal minimal surface functional.
We prove density estimates for level sets of minimizers of the energy $$\eps^{2s}\|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s \in (0,1)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. As a consequence we obtain, as $\eps \to 0$, the uniform convergence of the level sets of $u$ to either a $H^s$-nonlocal minimal surface if $s\in(0,\frac 1 2)$, or to a classical minimal surface if $s \in[\frac 1 2,1)$.
We consider the minimizers of the energy $$ \|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s \in (0,1/2)$, where $\|u\|_{H^s(Ω)}$ denotes the total contribution from $Ω$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer $u$ in a large ball $B_R$ occupy a volume comparable with the volume of $B_R$.
We consider a functional obtained by adding a trace term to the Allen-Cahn phase segregation model and we prove some density estimates for the level sets of the interfaces. We treat in a unified way also the cases of possible degeneracy and singularity of the ellipticity of the model and the quasiminimal case.