Marcello Lucia

Strong comparison principle for solutions of quasilinear equations

Let Ω C R N , N > 1, be a bounded smooth connected open set and a: Q x R N → R N be a map satisfying the hypotheses (H1)-(H4) below. Let f 1 ,f 2 ∈ L 1 loc(Ω) with f 2 ≥ f 1 , f 1 ≢ f 2 in Ω and u 1 ,u 2 ∈ C 1,θ (Ω) with 0 ∈ (0, 1] be two weak solutions of (P i ) - div(a(x, ⊇u i )) = f i in Ω, i = 1, 2. Suppose that u 2 > u 1 in Ω. Then we show that u 2 > u 1 in Ω under the following assumptions: either u 2 > u 1 on ∂Ω, or u 1 = u 2 = 0 on ∂Ω and u 1 ≥ 0 in Ω. We also show a measure-theoretic version of the Strong Comparison Principle.

Gamma-convergence of the Allen-Cahn energy with an oscillating forcing term

We consider a standard functional in the mesoscopic theory of phase transitions, consisting of a gradient term with a double-well potential, and we add to it a bulk term modelling the interaction with a periodic mean zero external field. This field is amplified and dilated with a power of the transition layer thickness \eps leading to a nontrivial interaction of forcing and concentration when \eps→0. We show that the functionals Γ-converge after additive renormalization to an anisotropic surface energy, if the period of the oscillation is larger than the interface thickness. Difficulties arise from the fact that the functionals have non constant absolute minimizers and are not uniformly bounded from below.

A Mean Field Equation on a Torus: One-Dimensional Symmetry of Solutions

ABSTRACT We study the equation for u ∈ E, where E = {u ∈ H 1(Ωϵ): u is doubly periodic, ∈ t Ωϵ u = 0} and Ωϵ is a rectangle of ℝ2 with side lengths 1/ϵ and 1, 0 < ϵ ≤ 1. We establish that every solution depends only on the x-variable when λ ≤ λ*(ϵ), where λ*(ϵ) is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters ϵ and λ in which every solution is identically zero. This range is optimal for ϵ ≤ 1/2.

Uniqueness and symmetry of equilibria in a chemotaxis model

Abstract We consider in a disc of a class of parameter-dependent, nonlocal elliptic boundary value problems that describes the steady states of some chemotaxis systems. If the appearing parameter is less than an explicit critical value, we establish several uniqueness results for solutions that are invariant under a group of rotations. Furthermore, we discuss the associated consequences for the time asymptotic behavior of the solutions to the corresponding time dependent chemotaxis systems. Our results also provide optimal constants in some Moser–Trudinger type inequalities.

Exact multiplicity of nematic states for an Onsager model

Isotropic–nematic phase transition due to excluded volume effects is considered for the 2D Onsager model of rigid rod-like polymers with a general interaction kernel involving a finite number of Fourier modes. Nematic phases appear above a concentration threshold. We prove the existence of continuous branches of nematic states bifurcating from the isotropic phase at concentrations proportional to the Fourier coefficients of the interaction kernel. The bifurcation structure is shown to depend on the size of the spectral gaps of the interaction operator. Exact multiplicity of nematic phases is proved for a class of two-mode trigonometric kernels. Our arguments use simple bifurcation and variational tools applied to an equivalent finite dimensional problem that involves multi-variable Bessel functions.

Laplacian eigenvalues for mean zero functions with constant Dirichlet data

Abstract We investigate the eigenvalues of the Laplace operator in the space of functions of mean zero and having a constant (unprescribed) boundary value. The first eigenvalue of such problem lies between the first two eigenvalues of the Laplacian with homogeneous Dirichlet boundary conditions and satisfies an isoperimetric inequality: in the class of open bounded sets of prescribed measure, it becomes minimal for the union of two disjoint balls having the same radius. Existence of an optimal domain in the class of convex sets is also discussed.

Gradient theory of phase transitions with a rapidly oscillating forcing term

We consider the Gamma-limit of a family of functionals which model the interaction of a material that undergoes phase transition with a rapidly oscillating conservative vector field. These functionals consist of a gradient term, a double-well potential and a vector field. The scaling is such that all three terms scale in the same way and the frequency of the vector field is equal to the interface thickness. Difficulties arise from the fact that the two global minimizers of the functionals are nonconstant and converge only in the weak L-2-topology.

A class of degenerate elliptic eigenvalue problems

Abstract. We consider a general class of eigenvalue problems where the leading elliptic term corresponds to a convex homogeneous energy function that is not necessarily differentiable. We derive a strong maximum principle and show uniqueness of the first eigenfunction. Moreover we prove the existence of a sequence of eigensolutions by using a critical point theory in metric spaces. Our results extend the eigenvalue problem of the p-Laplace operator to a much more general setting.

GLOBAL BIFURCATION FOR SEMILINEAR ELLIPTIC PROBLEMS

We study the existence of a global branch of solutions for the semilinear elliptic problem −∆u = λ a(x)u + b(x)r(u) , u ∈ D0 (Ω). We work in a general domain Ω of Rn , with indefinite weights a, b belonging to some Lorentz spaces, and the function r is either asymptotically linear or superlinear at infinity. To derive our result we first prove existence, uniqueness and simplicity of a principal eigenvalue for linear problems with weight in Lorentz spaces.

Some elliptic semilinear indefinite problems on R

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem −Δu=λf(x,u), u∈D 1,2 (R N ). The function f is allowed to change sign and has an asymptotically linear or a superlinear behaviour.