Salvador Villegas

Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations

This paper is devoted to the study of

Lyapunov-type Inequalities and Applications to PDE

L_p

Antisymmetry of solutions for some weighted elliptic problems

Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant

Systems of Equations

p \geq 1

Partial Differential Equations

. We consider ordinary and elliptic problems. The results obtained in the linear case are combined with Schauder fixed point theorem to provide new results about the existence and uniqueness of solutions for resonant nonlinear problems. The proof uses in a fundamental way the nontrivial relation between the best Lyapunov constants and the minimum value of some especial minimization problems.

A Variational Characterization of the Best Lyapunov Constants

This work is devoted to the study of resonant nonlinear boundary problems with Neumann boundary conditions. First, we consider the linear case doing a careful analysis which involves Lyapunov-type inequalities with the Lp— norms of the coefficient function. After this end, combining these results with Schauder fixed point theorem, we obtain some new results about the existence and uniqueness of solutions for resonant nonlinear problems.

An applied mathematical excursion through Lyapunov inequalities, classical analysis and Differential Equations

ABSTRACT This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.

Lyapunov inequalities for partial differential equations

This chapter is devoted to the study of L p Lyapunov-type inequalities for linear systems of ordinary differential equations with different boundary conditions (which include the case of Neumann, Dirichlet, periodic, and antiperiodic boundary value problems) and for any constant p ≥ 1. Elliptic problems are also considered. As in the scalar case, the results obtained in the linear case are combined with Schauder fixed point theorem to provide several results about the existence and uniqueness of solutions for resonant nonlinear systems. In addition, we study the stable boundedness of linear periodic conservative systems. The proof uses in a fundamental way the nontrivial relation (proved in Chap. 2) between the best Lyapunov constants and the minimum value of some especial constrained or unconstrained minimization problems (depending on the considered problems are resonant or nonresonant, respectively).

Liapunov-type inequalities and Neumann boundary value problems at resonance

This chapter is devoted to the study of L p Lyapunov-type inequalities (\(1 \leq p \leq +\infty\)) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in R N . In the case of Dirichlet conditions, it is possible to obtain analogous results in an easier way. We also treat the case of higher eigenvalues in the radial case, by using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions. It is proved that the relation between the quantities p and N∕2 plays a crucial role just to have nontrivial Lyapunov inequalities. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.

Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at −∞ and superlinear at +∞

This chapter is devoted to the definition and main properties of the L p Lyapunov constant, \(1 \leq p \leq \infty,\) for scalar ordinary differential equations with different boundary conditions, in a given interval (0, L). It includes resonant problems at the first eigenvalue and nonresonant problems. A main point is the characterization of such a constant as a minimum of some especial minimization problem, defined in appropriate subsets X p of the Sobolev space H1(0, L). This variational characterization is a fundamental fact for several reasons: first, it allows to obtain an explicit expression for the L p Lyapunov constant as a function of p and L; second, it allows the extension of the results to systems of equations (Chap. 5) and to PDEs (Chap. 4). For resonant problems (Neumann or periodic boundary conditions), it is necessary to impose an additional restriction to the definition of the spaces \(X_{p},\ 1 \leq p \leq \infty,\) so that we will have constrained minimization problems. This is not necessary in the case of nonresonant problems (Dirichlet or antiperiodic boundary conditions) where we will find unconstrained minimization problems. For nonlinear equations, we combine the Schauder fixed point theorem with the obtained results for linear equations.