arXiv subjects

Giampiero Palatucci

Publications and source records attributed to Giampiero Palatucci.

22 records · Page 2Linked to original sources

arXiv2012

Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian

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 $s=1$.

math.AP↗

arXiv2011

Local and global minimizers for a variational energy involving a fractional norm

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.

math.AP↗

arXiv2011

Hitchhiker's guide to the fractional Sobolev spaces

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.

math.FA↗

arXiv2009

$Γ$-convergence of some super quadratic functionals with singular weights

We study the $Γ$-convergence of the following functional ($p>2$) $$ F_ε(u):=ε^{p-2}\int_Ω|Du|^p d(x,\partial Ω)^{a}dx+\frac{1}{ε^{\frac{p-2}{p-1}}}\int_ΩW(u) d(x,\partial Ω)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrtε}\int_{\partialΩ}V(Tu)d\mathcal{H}^2, $$ where $Ω$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $α, β$ and $α', β'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\partialΩ$, $d(x,\partial Ω)$ stands for the distance function to the boundary $\partial\Om$. We show that the singular limit of the energies $F_ε$ leads to a coupled problem of bulk and surface phase transitions.

math.AP↗