Roberta Musina

On Fractional Laplacians

We compare two natural types of fractional Laplacians (− Δ) s , namely, the “Navier” and the “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.

Hardy-Poincaré inequalities with boundary singularities

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincare inequality and to the Hardy inequality for maps in H 0 1 (Ω), where Ω is a bounded domain in ℝ N , N ≥ 2, with 0 ∈ ∂ Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved.

EXISTENCE OF MINIMAL H-BUBBLES

Given a function H ∈ C1 (ℝ3) asymptotic to a constant at infinity, we investigate the existence of H-bubbles, i.e., nontrivial, conformal surfaces parametrized by the sphere, with mean curvature H. Under some global hypotheses we prove the existence of H-bubbles with minimal energy.

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.

Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials

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.

Existence of H-bubbles in a perturbative setting

Given a C 1 function H:R 3 ! R, we look for H-bubbles, i.e, surfaces in R 3 parametrized by the sphere S 2 with mean curvature H at every regular point. Here we study the case H(u) = H0(u) + H 1(u) where H0 is some “good” curvature (for which there exist H0-bubbles with minimal energy, uniformly bounded in L 1 ), is the smallness parameter, and H1 is any C 1 function.

Holes and obstacles

Abstract In this paper we investigate the effect of a partial obstacle on a semilinear elliptic B.V.P. which has, in general, no solution. We show that highly unstable solutions arise, a phenomena previously observed for the same equation in presence of holes in the domain.

Non-critical dimensions for critical problems involving fractional Laplacians

Let f be a meromorphic function with bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that f has infinitely many repelling periodic points for any minimal period n ≥ 1, using a much simpler argument than the more general results for arbitrary entire transcendental functions.We study the Brezis–Nirenberg effect in two families of non-compact boundary value problems involving Dirichlet–Laplacian of arbitrary real order m∈(0,n/2).

An Approach to the Thin Obstacle Problem foŕ Variational Functionals Depending on Vector Valued Functions

In this paper we discuss a class of variational problems for integral functionals depending on vector valued functions u:Ω→R N , Ω⊆R n , subject to an obstacle condition on the image

S 2 -type Parametric Surfaces with Prescribed Mean Curvature and Minimal Energy

Given a functionHE Cl(I3) asymptotic to a constant at infinity, we investigate the existence of nontrivial, conformal surfaces parametrized by the sphere, with mean curvatureHand minimal energy.