Cristina Trombetti

Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term

Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.

Existence results for a class of nonlinear elliptic problems with p-growth in the gradient

Abstract We prove the existence of bounded solutions for a class of nonlinear elliptic problems whose model is in the form: (∗) − div (a(x,u,Du))=k 1 (|u|)|Du| p +k 2 (|u|)f,u∈W 0 1,p (Ω)∩L ∞ (Ω), where 〈a(x,η,ξ)ξ〉⩾b(|η|)|ξ|p, b is a continuous monotone decreasing function and k1 and k2 are continuous monotone increasing functions.

On the Lipschitz regularity for certain elliptic problems

Abstract We prove the Lipschitz regularity of solutions to a class of elliptic problems characterized by weak growth, differentiability and ellipticity assumptions.

Non-Uniformly Elliptic Equations with Natural Growth in the Gradient

We prove the existence of bounded solutions for a class of nonlinear elliptic problems of type−div(a(x,u,Du))=H(x,u,Du)+fu∈W1,p0(Ω)∩L∞(Ω)where 〈a(x,η,ξ)ξ〉≥b(|η|)|ξ|p, b is a continuous monotone decreasing function and |H(x,η,ξ)| ≤ k(η)|ξ|p, k is a continuous monotone increasing function.

A quantitative Pólya-Szegö principle

Abstract The radially decreasing symmetrization is well known not to increase Dirichlet type integrals of Sobolev functions. In the present paper, the deviation of a function from its symmetral is estimated in terms of the gap between their Dirichlet integrals.

Sharp Estimates for Eigenfunctions of a Neumann Problem

In this paper we provide some bounds for eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of ℝ n . To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.

Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems

In this paper we prove a sharp lower bound for the first non-trivial Neumann eigenvalue μ 1 ( Ω ) for the p -Laplace operator ( p > 1) in a Lipschitz bounded domain Ω in ℝ n . Our estimate does not require any convexity assumption on Ω and it involves the best isoperimetric constant relative to Ω . In a suitable class of convex planar domains, our bound turns out to be better than the one provided by the Payne—Weinberger inequality.

On Pólya's inequality for torsional rigidity and first Dirichlet eigenvalue

Let

The Neumann eigenvalue problem for the

\Omega