Variational inequalities for the fractional Laplacian
In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $Ω\subset\mathbb R^n$, under mild assumptions on the data.
math.AP↗
arXiv subjects
Roberta Musina
Publications and source records attributed to Roberta Musina.
On fractional Laplacians -- 2
The present paper is the natural evolution of arXiv:1308.3606. For $s>-1$ we compare two natural types of fractional Laplacians $(-Δ)^s$, namely, the "Navier" and the "Dirichlet" ones. As a main tool, we give the "dual" Caffarelli--Silvestre and Stinga--Torrea variational characterizations of these operators for $s\in(-1,0)$.
math.AP↗
The Hénon-Lane-Emden system: a sharp nonexistence result
We deal with very weak positive supersolutions to the Hénon-Lane-Emden system on neighborhoods of the origin. In our main theorem we prove a sharp nonexistence result.
math.AP↗
On fractional Laplacians -- 3
We investigate the role of the noncompact group of dilations in $\mathbb R^n$ on the difference of the quadratic forms associated to the fractional Dirichlet and Navier Laplacians. Then we apply our results to study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving the Navier-Laplacian.
math.AP↗
On the Sobolev and Hardy constants for the fractional Navier Laplacian
We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.
math.AP↗
The non-anticoercive Hénon-Lane-Emden system
We use variational methods to study the existence of a principal eigenvalue for the non-anticoercive Hénon-Lane-Emden system on a bounded domain. Then we provide a detailed insight into the problem in the linear case.
math.AP↗
Optimal Rellich-Sobolev constants and their extremals
We prove that extremals for second order Rellich-Sobolev inequalities have constant sign. Then we show that the optimal constants in Rellich-Sobolev inequalities on a bounded domain Ω and under Navier boundary conditions do not depend on Ω
math.AP↗
Non-critical dimensions for critical problems involving fractional Laplacians
We study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving Dirichlet-Laplacian of arbitrary real order $m>0$.
math.AP↗
On fractional Laplacians
We compare two natural types of fractional Laplacians $(-Δ)^s$, "Navier" and "Dirichlet" ones. We show that for $0<s<1$ their difference is positive definite and positivity preserving. Then we prove the coincidence of the Sobolev constants for these two fractional Laplacians.
math.AP↗
Symmetry breaking of extremals for the Caffarelli-Kohn-Nirenberg inequalities in a non-Hilbertian setting
We provide an explicit necessary condition to have that no extremal for the best constant in the Caffarelli-Kohn-Nirenberg inequality is radially symmetric.
math.AP↗
Weighted Sobolev spaces of radially symmetric functions
We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.
math.AP↗
Radially symmetric solutions to the Hènon-Lane-Emden system on the critical hyperbola
We use variational methods to study the existence of nontrivial and radially symmetric solutions to the Hènon-Lane-Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and nonexistence results.
math.AP↗
A class of second order dilation invariant inequalities
We compute the best constants in some second order dilation invariant inequalities, with weights being powers of the distance from the origin.
math.FA↗
Entire solutions for a class of variational problems involving the biharmonic operator and Rellich potentials
We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to second-order interpolation inequalities with weights.
math.AP↗
Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator: existence of extremal functions, breaking positivity and breaking symmetry
We investigate Caffarelli-Kohn-Nirenberg type inequalities for the weighted biharmonic operator on cones, both under Navier and Dirichlet boundary conditions. Moreover, we study existence and qualitative properties of extremal functions. In particular, we show that in some cases extremal functions do change sign; when the domain is the whole space, we prove some breaking symmetry phenomena.
math.FA↗
Rellich inequalities with weights
Let $Ω$ be a cone in $\mathbb{R}^{n}$ with $n\ge 2$. For every fixed $α\in\mathbb{R}$ we find the best constant in the Rellich inequality $\int_Ω|x|^α|Δu|^{2}dx\ge C\int_Ω|x|^{α-4}|u|^{2}dx$ for $u\in C^{2}_{c}(\barΩ\setminus\{0\})$. We also estimate the best constant for the same inequality on $C^{2}_{c}(Ω)$. Moreover we show improved Rellich inequalities with remainder terms involving logarithmic weights on cone-like domains.
math.FA↗
Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
math.AP↗
Hardy-Poincare' inequalities with boundary singularities
Let $Ø$ be a bounded domain in $\R^N$ with $0\in\deØ$ and $N\ge 2$. In this paper we study the Hardy-Poincaré inequality for maps in $H^1_0(Ω)$. In particular we give sufficient and some necessary conditions so that the best constant is achieved.
math.AP↗