Descripción: In this work we extend an inequality of Nehari to the eigenvalues of weighted quasilinear problems involving the p-Laplacian when the weight is a monotonic function. We apply it to different eigenvalue problems. © 2010 Elsevier Ltd. All rights reserved.

Descripción: In this paper we study the optimization problem for the first eigenvalue of the p-Laplacian plus a potential V with respect to V, when the potential is restricted to a bounded, closed and convex Bet of Lq(Ω).

Descripción: In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. © 2008 Elsevier Masson SAS. All rights reserved.

Descripción: In this note we give some remarks and improvements on our recent paper [5] about an optimization problem for the p-Laplace operator that were motivated by some discussion that we had with Prof. Cianchi. © 2009 Elsevier Ltd. All rights reserved.

Descripción: We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems. © 2006 Elsevier Inc. All rights reserved.

Descripción: In this paper we study the existence of nontrivial solutions for the problem Δpu = up-2u in a bounded smooth domain Ω ⊂ ℝN, with a nonlinear boundary condition given by |∇u|p-2∂u/∂v = f(u) on the boundary of the domain. The proofs are based on variational and topological arguments. © 2001 Academic Press.

Descripción: We derive oscillation and nonoscillation criteria for the one-dimensional p-Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.

Descripción: In this paper we study the applicability of energy methods to obtain bounds for the asymptotic decay of solutions to nonlocal diffusion problems. With these energy methods we can deal with nonlocal problems that not necessarily involve a convolution, that is, of the form ut (x, t) = ∫Rd G (x - y) (u (y, t) - u (x, t)) d y. For example, we will consider equations like,ut (x, t) = under(∫, Rd) J (x, y) (u (y, t) - u (x, t)) d y + f (u) (x, t), and a nonlocal analogous to the p-Laplacian,ut (x, t) = under(∫, Rd) J (x, y) | u (y, t) - u (x, t) |p - 2 (u (y, t) - u (x, t)) d y . The energy method developed here allows us to obtain decay rates of the form,{norm of matrix} u (ṡ, t) {norm of matrix}Lq (Rd) ≤ C t- α, for some explicit exponent α that depends on the parameters, d, q and p, according to the problem under consideration. © 2009 Elsevier Masson SAS. All rights reserved.

Descripción: We study a nonlinear problem of pendulum-type for a p-Laplacian with nonlinear periodic-type boundary conditions. Using an extension of Mawhin's continuation theorem for nonlinear operators, we prove the existence of a solution under a Landesman-Lazer type condition. Moreover, using the method of upper and lower solutions, we generalize a celebrated result by Castro for the classical pendulum equation. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Descripción: We study the existence of weak solutions to the equation Δpu = |u|p-2u + f(x, u) with the nonlinear boundary condition |∇u|p-2∂u/∂v = λ|u|p-2u - h(x, u). We assume Landesman-Lazer type conditions and use variational arguments to prove the existence of solutions.

Descripción: In this paper we study the limit of Monge-Kantorovich mass transfer problems when the involved measures are supported in a small strip near the boundary of a bounded smooth domain, Ω. Given two absolutely continues measures (with respect to the surface measure) supported on the boundary ∂Ω, by performing a suitable extension of the measures to a strip of width ε near the boundary of the domain Ω we consider the mass transfer problem for the extensions. Then we study the limit as ε goes to zero of the Kantorovich potentials for the extensions and obtain that it coincides with a solution of the original mass transfer problem. Moreover we look for the possible approximations of these problems by solutions to equations involving the p-Laplacian for large values of p.

Descripción: In this paper we introduce the generalized eigenvalues of a quasilinear elliptic system of resonant type. We prove the existence of infinitely many continuous eigencurves, which are obtained by variational methods. For the one-dimensional problem, we obtain an hyperbolic type function defining a region which contains all the generalized eigenvalues (variational or not), and the proof is based on a suitable generalization of Lyapunov's inequality for systems of ordinary differential equations. We also obtain a family of curves bounding by above the kth variational eigencurve. © 2006 Elsevier Inc. All rights reserved.

Descripción: We study the dependence on the subset A ⊂ Ω of the Sobolev trace constant for functions defined in a bounded domain Ω that vanish in the subset A. First we find that there exists an optimal subset that makes the trace constant smaller among all the subsets with prescribed and positive Lebesgue measure. In the case that Ω is a ball we prove that there exists an optimal hole that is spherically symmetric. In the case p = 2 we prove that every optimal hole is spherically symmetric. Then, we study the behavior of the best constant when the hole is allowed to have zero Lebesgue measure. We show that this constant depends continuously on the subset and we discuss when it is equal to the Sobolev trace constant without the vanishing restriction. © 2005 Elsevier SAS. All rights reserved.

Descripción: In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω) Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω), α) Lq(P(Ω), β). For the special case p = q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case. © 2003 Elsevier Inc. All rights reserved.