Existence and uniqueness for P-area minimizers in the Heisenberg group

In [3] we studied p-mean curvature and the associated p-minimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized p-area and associated ( p-) minimizers in general dimensions. We prove the existence and investigate the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C ²-smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set.     

The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α>0. The high-frequency (or: semi-classical) parameter is ?>0. We let ? and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption.Under these assumptions, we prove that the solution u^? radiates in the outgoing direction, uniformly in ?. In particular, the function u^?, when conveniently rescaled at the scale ? close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in ?) of the limiting absorption principle.Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) for very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above-mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in ?.     

A Gradient Flow of the One-Dimensional Alt-Caffarelli Functional

We construct a candidate of the gradient flow for a variational functional with a singular term in the one-dimensional case. Applied is the scheme using the theory of Γ-convergence for variational functionals, where both the Dirichlet integral and the singular term are approximated at the same time. An approximated solution is constructed on the space of piecewise constant functions, and the desired gradient flow is built as the limit of it. We establish some properties of the gradient flow as well as obtain PDE including a term of integration by Radon measure.      

Existence of Solutions to a Semilinear Elliptic System through Orlicz-Sobolev Spaces

Using Orlicz-Sobolev spaces and a variant of the Mountain-Pass Lemma of Ambrosetti-Rabinowitz we obtain existence of a (positive) solution to a semilinear system of elliptic equations. The admissible nonlinearities are such that the system is superlinear and subcritical. The Orlicz setting used here allows us to consider nonlinearities which are not (asymptotically) pure powers. Moreover, by an interpolation theorem of Boyd we find an elliptic regularity result in Orlicz-Sobolev spaces. A bootstrapping argument implies that the above mentioned solutions are classical.     

Reproductivity for a nematic liquid crystal model

In this paper we prove existence of weak solution with the reproductivity in time property, for a penalized PDE’s system related to a nematic liquid crystal model. This problem is relatively explict when time-independent Dirichlet boundary conditions are imposed for the orientation of crystal molecules. Nevertheless, for the time-dependent case, the treatment of the problem is completely different. The verification of a maximum principle for weak reproductive solutions is fundamental in the argument. Finally, the relation between reproductive and periodic in time (regular) solutions will be pointed out, differenting the 2D and 3D cases. Basically, in two-dimensional domains every reproductive solution is regular and time periodic, whereas the problem remains open for three-dimensional domains.     

Congruence classes of Frenet curves in complex quadrics

Congruent classes of Frenet curves of order 2 in the complex quadric are studied, obtaining that each congruence class is a level set of a family of certain smooth functions, that are generalizations of isoparametric functions on the unit sphere in the tangent space of the complex quadric.     

Null congruence spacetimes constructed from 3-dimensional Robertson-Walker spaces

Null congruence spacetimes are constructed from three-dimensional time-orientable Lorentzian manifolds by taking a particular ellipse in the lightcone above every point. Starting from a three-dimensional Robertson-Walker space, new null congruence spacetimes are obtained and several of their curvature properties are deduced. In particular, it is shown that the static Einstein universe is locally isometric to a certain null congruence spacetime. Furthermore, a method is given to construct trapped surfaces which admit an isometric spacelike circle action in null congruence spacetimes.     

Integral transform and convolution of analytic functionals on abstract Wiener space

Yeh defined a convolution of functionals on classical Wiener space and investigated the relationship between the Fourier-Wiener transforms of functionals in certain classes and the Fourier-Wiener transform of their convolution. Yoo extended Yeh's results to abstract Wiener space. In this paper, we introduce the intergal transform and convolution of analytic functionals on abstract Wiener space. And we establish the relationship between the integral transforms of exponential type of analytic functionals and the integral transform of theor convolution. Also we obtain Parseval's and Plancherel's relations for those functionals from this relationship. The main results of Yeh and Yoo then follow from our results as corollaries.     

Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks

The author is concerned with the existence and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks. Previous methods do not apply in solving these problems because there is no maximum principle or conservation laws available to the integral differential equations. He applies fixed point theorems to prove the existence of the traveling waves. Then, he makes use of linearization technique as well as eigenvalue functions to study the exponential stability of the waves.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

Multiple Solitary Waves for Non-Homogeneous Schrödinger–Maxwell Equations

The aim of this paper is investigating the existence of standing waves which are solutions of a nonlinear Schr?dinger equation coupled with Maxwell’s equations when a non-homogeneous term breaks the symmetry of the associated functional. Dedicated to the memory of Professor Aldo Cossu This work was supported by M.I.U.R. (research funds ex 40% and 60%).     

半线性波动方程的高维古沙问题

In this paper we study the Goursat problem for semilinear wave equations with zero boundary condition in which the boundary is the characteristic cone for wave operator. Our result states that the solution is Lipschitz and is smooth awayfrom the characteristic cone.     

Symbolic software for soliton theory

Four symbolic programs, in Macsyma or Mathematica language, are presented. The first program tests forthe existence of solitons for nonlinear PDEs. It explicitly constructs solitons using Hirota's bilinear method. In the second program, the Painlevé integrability test for ODEs and PDEs is implemented. The third program provides an algorithm to compute conserved densities for nonlinear evolution equations. The fourth software package aids in the computation of Lie symmetries of systems of differential and difference-differential equations. Several examples illustrate the capabilities of the software.Research supported in part by NSF under Grant CCR-9300978.