Luigi Orsina

Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data

Abstract We consider the differential problem (*) { A ( u ) = μ in Ω , u = 0 on ∂ Ω , where Ω is a bounded, open subset of R N , N ≥ 2, A is a monotone operator acting on W 0 1 , p ( Ω ) , p > 1, and μ is a Radon measure on Ω that does not charge the sets of zero p -capacity. We prove a decomposition theorem for these measures (more precisely, as the sum of a function in L 1 (Ω) and of a measure in W −1, p ′ (Ω)), and an existence and uniqueness result for the so-called entropy solutions of (*) .

Definition and existence of renormalized solutions of elliptic equations with general measure data

Abstract We introduce a new definition of solution for the nonlinear monotone elliptic problem-div(a(a;, ∇u)) = μ in Ω u = 0 on ∂Ω, where μ is a Radon measure with bounded variation on Ω. We prove the existence of such a solution, a stability result, and partial uniqueness results.

Existence of critical points for some noncoercive functionals

Abstract In this paper we study critical points problems for some integral functionals with principal part having degenerate coerciveness, whose model is J(v)= 1 2 ∫ Ω |∇v| 2 (b(x)+|v|) 2α − 1 m ∫ Ω |v| m , v∈H 0 1 (Ω), with 1 ∗ (1−α) . We will prove several existence and nonexistence results depending on different assumptions on both m and α .

-nilpotent -ideals in () having a fixed class of nilpotence: combinatorics and enumeration

We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q,t)-analogue of the Catalan number C n . These (q,t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

Enumeration of ad-nilpotent ideals of a Borel subalgebra in type A by class of nilpotence

Abstract We provide a formula affording the number of ad-nilpotent ideals of a Borel subalgebra of sl (n+1, C ) having a fixed class of nilpotence.

Existence results for quasilinear elliptic and parabolic problems with quadratic gradient terms and sources

Abstract In this paper we deal with elliptic and parabolic quasilinear problems with a singular quadratic lower order term depending on the gradient, i.e., with the equations and with Ω a bounded open set of , T > 0, M a bounded measurable uniformly elliptic matrix, B > 0, 0 < θ < 1 and 0 < r < 2 – θ. We will prove existence result for solutions under various assumptions on ƒ and the initial datum u 0 Note that the elliptic equation is strongly related with the Euler–Lagrange equation of some integral functionals.

Strong maximum principle for Schrödinger operators with singular potential

We prove that for every p>1 and for every potential V∈Lp, any nonnegative function satisfying −Δu+Vu≥0 in an open connected set of RN is either identically zero or its level set {u=0} has zero W2,p capacity. This gives an affirmative answer to an open problem of Benilan and Brezis concerning a bridge between Serrin–Stampacchias strong maximum principle for p>N2 and Anconas strong maximum principle for p=1. The proof is based on the construction of suitable test functions depending on the level set {u=0}, and on the existence of solutions of the Dirichlet problem for the Schrodinger operator with diffuse measure data.

EXISTENCE OF FINITE ENERGY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH L 1 -VALUED NONLINEARITIES

In this paper, we are going to study the following elliptic system: where Ω is a bounded open subset of ℝN, a(x, s) and b(x, s) are positive and coercive Caratheodory functions, and f ∈ LM(Ω). The main purpose of this paper is to prove existence and regularity results with an improved regularity of the function z in the class of Sobolev spaces, and the existence of solutions (u, z) both with finite energy.

A semilinear problem with a

We study a degenerate elliptic equation, proving the existence of a W^{1,1}_0 distributional solution.

W^{1,1}_0

AbstractWe study the following relaxed Dirichlet problem