The optimal Euclidean L-Sobolev logarithmic inequality

We prove an optimal logarithmic Sobolev inequality in . Explicit minimizers are given. This result is connected with best constants of a special class of Gagliardo-Nirenberg-type inequalities.     

Poincaré inequality for linear SPDE driven by Lévy Noise

In this paper, we prove the Poincaré inequality and the integration by parts formula for the invariant measure of the linear SPDE driven by Lévy Noise. The equation was researched in Dong and Xie [5], which has proved the existence and uniqueness of the weak solution and the ergodicity of the Markov semigroup associated with the solution.     

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.     

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

G-convergence,Dirichlet to Neumann maps and invisibility

We establish optimal conditions under which the G-convergence of linear elliptic operators implies the convergence of the corresponding Dirichlet to Neumann maps. As an application we show that the approximate cloaking isotropic materials from [19] are independent of the source.     

Eigenvalues for radially symmetric fully nonlinear singular or degenerate operators

In this paper we present a one dimensional and radial theory for the existence of eigenvalues and eigenfunctions for fully nonlinear elliptic (α+1)$(\alpha +1)$-homogeneous operators, α>−1$\alpha >-1$. A general theory for the first eigenvalue and eigenfunction exists in the frame of viscosity solutions, but in this particular case a simpler theory can be established, that extends, via degree theory, to obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeros.     

“Bubble-tower” radial solutions in the slightly supercritical Brezis-Nirenberg problem

In this paper, we consider the Brezis-Nirenberg problem in dimension N?4, in the supercritical case. We prove that if the exponent gets close to and if, simultaneously, the bifurcation parameter tends to zero at the appropriate rate, then there are radial solutions which behave like a superposition of bubbles, namely solutions of the form

     

Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces

The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which involves a nonlinearity fulfilling a nonstandard growth condition. Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three weak solutions in an appropriate Orlicz-Sobolev space.     

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H¹(R^N) and satisfies the geometric hypotheses of the Mountain Pass Theorem.     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

Existence,non-existence and regularity of radial ground states for p-Laplacian equations with singular weights

By the Mountain Pass Theorem and the constrained minimization method existence of positive or compactly supported radial ground states for quasilinear singular elliptic equations with weights are established. The paper also includes the discussion of regularity and the validity of useful qualitative properties of the solutions, which seems of independent interest. Finally, a Pohozaev type identity is produced to deduce some non-existence results.     

Qualitative properties for solutions of singular elliptic inequalities on complete manifolds

We establish several qualitative properties for solutions of singular quasilinear elliptic differential inequalities on complete Riemannian manifolds, such as the validity of the compact support principle, of the strong maximum principle, existence of solutions to exterior Dirichlet problems, existence of dead core solutions.     

Quasilinear elliptic problems under strong resonance conditions

In this paper we establish the existence and multiplicity of solutions for a quasilinear elliptic problem under strong resonance conditions at infinity. In order to control the resonance we consider a new hypothesis on the nonlinear term. In all results we use Variational Methods, Critical Groups and the Morse Theory.     

A Szeg? condition for a multidimensional Schrödinger operator

We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szeg?-type condition.     

Minimizing the probability of lifetime ruin under borrowing constraints

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as lifetime ruin. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset.We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton’s model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results.     

An asymmetric Neumann problem with weights

We prove the existence of a first nonprincipal eigenvalue for an asymmetric Neumann problem with weights involving the p-Laplacian (cf. (1.2) below). As an application we obtain a first nontrivial curve in the corresponding Fu?ik spectrum (cf. (1.4) below). The case where one of the weights has meanvalue zero requires some special attention in connexion with the (PS) condition and with the mountain pass geometry.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.