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Lorenzo Brasco

Publications and source records attributed to Lorenzo Brasco.

At least 55 records · Page 3Linked to original sources
arXiv2018

A note on homogeneous Sobolev spaces of fractional order

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\uı norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.

math.FA↗
arXiv2018

On fractional Hardy inequalities in convex sets

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\uı spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<\infty$ and $0<s<1$, with a constant which is stable as $s$ goes to $1$.

math.AP↗
arXiv2018

On the Lipschitz character of orthotropic $p-$harmonic functions

We prove that local weak solutions of the orthotropic $p-$harmonic equation are locally Lipschitz, for every $p\ge 2$ and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.

math.AP↗
arXiv2017

Schrödinger operators with negative potentials and Lane-Emden densities

We consider the Schrödinger operator $-Δ+V$ for negative potentials $V$, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of $-Δ+V$ is positive, provided that $V$ is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation $-Δu=u^{q-1}$ (for some $1\le q< 2$). In this case, the ground state energy of $-Δ+V$ is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.

math.AP↗
arXiv2017

A pathological example in Nonlinear Spectral Theory

We construct an open set $Ω\subset\mathbb{R}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.

math.AP↗
arXiv2017

Pohozaev identity for the fractional $p-$Laplacian on $\mathbb{R}^N$

By virtue of a suitable approximation argument, we prove a Pohozaev identity for nonlinear nonlocal problems on $\mathbb{R}^N$ involving the fractional $p-$Laplacian operator. Furthermore we provide an application of the identity to show that some relevant levels of the energy functional associated with the problem coincide.

math.AP↗
arXiv2016

A sharp estimate a la calderon-zygmund for the p--laplacian

We consider local weak solutions of the Poisson equation for the p--Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate.

math.AP↗
arXiv2016

The second eigenvalue of the fractional $p-$Laplacian

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $Ω\subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $λ_2(Ω)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \[ \inf \{λ_2(Ω)\,:\,|Ω|=c\}. \] We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.

math.AP↗
arXiv2016

Higher Sobolev regularity for the fractional $p-$Laplace equation in the superquadratic case

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p$-Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.

math.AP↗
arXiv2015

Compact Sobolev embeddings and torsion functions

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and only if it is compact. The proofs crucially exploit a {\it torsional Hardy inequality} that we investigate in details.

math.AP↗
arXiv2015

Global Lipschitz continuity for minima of degenerate problems

We consider the problem of minimizing the Lagrangian $\int [F(\nabla u)+f\,u]$ among functions on $Ω\subset\mathbb{R}^N$ with given boundary datum $φ$. We prove Lipschitz regularity up to the boundary for solutions of this problem, provided $Ω$ is convex and $φ$ satisfies the bounded slope condition. The convex function $F$ is required to satisfy a qualified form of uniform convexity {\it only outside a ball} and no growth assumptions are made.

math.AP↗
arXiv2014

Lipschitz regularity for local minimizers of some widely degenerate problems

We consider local minimizers of the functional \[ \sum_{i=1}^N \int (|u_{x_i}|-δ_i)^p_+\, dx+\int f\, u\, dx, \] where $δ_1,\dots,δ_N\ge 0$ and $(\,\cdot\,)_+$ stands for the positive part. Under suitable assumptions on $f$, we prove that local minimizers are Lipschitz continuous functions if $N=2$ and $p\ge 2$, or if $N\ge 2$ and $p\ge 4$.

math.AP↗