arXiv subjects
Publications and source records attributed to Eugenio Vecchi.
We consider sequences of elliptic and parabolic operators in divergence form and depending on a family of vector fields. We show compactness results with respect to G-convergence, or H-convergence, by means of the compensated compactness theory, in a setting in which the existence of affine functions is not always guaranteed, due to the nature of the family of vector fields.
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by $p$-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
In this note we obtain an asymptotic estimate for growth behavior of variational eigenvalues of the $p-$fractional eigenvalue problem on a smooth bounded domain with Dirichlet boundary condition.
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.
Given a bounded open set $Ω\subseteq{\mathbb{R}}^n$, we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of $Ω$. We prove that the second eigenvalue $λ_2(Ω)$ is always strictly larger than the first eigenvalue $λ_1(B)$ of a ball $B$ with volume half of that of $Ω$. This bound is proven to be sharp, by comparing to the limit case in which $Ω$ consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.
In this paper we consider a Hénon-type equation driven by a nonlinear operator obtained as a combination of a local and nonlocal term. We prove existence and non-existence akin to the classical result by Ni, and a stability result as the fractional parameter $s \to 1$.
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
In this note we study the limit as $s\downarrow 0$ of fractional Orlicz-Sobolev seminorms in Carnot groups. This closes the study started in [10]
We prove Pohozaev-type identities for smooth solutions of Euler-Lagrange equations of second and fourth order that arise from functional depending on homogeneous Hörmander vector fields. We then exploit such integral identities to prove non-existence results for the associated boundary value problems.
In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form $-Δ+(-Δ)^s$, with $s\in(0,1)$. We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.
In this note we prove the validity of the Maz'ya-Shaposhnikova formula in magnetic fractional Orlicz-Sobolev spaces. This complements a previous study of the limit as $s \uparrow 1$ performed by the second author in [21].
In this article we define a class of fractional Orlicz-Sobolev spaces on Carnot groups and, in the spirit of the celebrated results of Bourgain-Brezis-Mironescu and of Maz'ya-Shaposhnikova, we study the asymptotic behavior of the Orlicz functionals when the fractional parameter goes to $1$ and $0$.
The aim of this note is to prove a representation theorem for left--invariant functionals in Carnot groups. As a direct consequence, we can also provide a $Γ$-convergence result for a smaller class of functionals.
We obtain symmetry results for solutions of an elliptic system of equation possessing a cooperative structure. The domain in which the problem is set may possess "holes" or "small vacancies" (measured in terms of capacity) along which the solution may diverge. The method of proof relies on the moving plane technique, which needs to be suitably adapted here to take care of the complications arising from the vacancies in the domain and the analytic structure of the elliptic system.
The composite plate problem is an eigenvalue optimization problem related to the fourth order operator $(-Δ)^2$. In this paper we continue the study started in [10], focusing on symmetry and rigidity issues in the case of the hinged composite plate problem, a specific situation that allows us to exploit classical techniques like the moving plane method.
We consider an elliptic polyharmonic problem of any order which takes place in a punctured bounded domain with Navier conditions. We prove that if the domain is convex in one direction and symmetric with respect to the reflections induced by the normal hyperpane to such a direction, then the solution is necessarily symmetric under this reflection and monotone in the corresponding direction.
In this paper we deal with the composite plate problem, namely the following optimization eigenvalue problem $$ \inf_{ρ\in \mathrm{P}} \inf_{u \in \mathcal{W}\setminus\{0\}} \frac{\int_Ω(Δu)^2}{\int_Ω ρu^2}, $$ where $\mathrm{P}$ is a class of admissible densities, $\mathcal{W}= H^{2}_{0}(Ω)$ for Dirichlet boundary conditions and $\mathcal W= H^2(Ω) \cap H^1_{0}(Ω)$ for Navier boundary conditions. The associated Euler-Lagrange equation is a fourth-order elliptic PDE governed by the biharmonic operator $Δ^2$. In the spirit of [10], we study qualitative properties of the optimal pairs $(u,ρ)$. In particular, we prove existence and regularity and we find the explicit expression of $ρ$. When $Ω$ is a ball, we can also prove uniqueness of the optimal pair, as well as positivity of $u$ and radial symmetry of both $u$ and $ρ$.
We prove a general magnetic Bourgain-Brezis-Mironescu formula. In particular, after developing a theory of magnetic bounded variation functions, we prove the validity of the formula in this class.