Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L+μ⋅∇+ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1,1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.     

p-Adic Schrödinger-type operator with point interactions

A p-adic Schrödinger-type operator Dα+VY is studied. Dα (α>0) is the operator of fractional differentiation and (bij∈C) is a singular potential containing the Dirac delta functions δx concentrated on a set of points Y={x₁,…,xn} of the field of p-adic numbers Qp. It is shown that such a problem is well posed for α>1/2 and the singular perturbation VY is form-bounded for α>1. In the latter case, the spectral analysis of η-self-adjoint operator realizations of Dα+VY in L₂(Qp) is carried out.     

Adaptive finite element methods for elliptic equations with non-smooth coefficients

Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000     

Coupled systems of non-smooth differential equations

We study the geometric qualitative behavior of a class of discontinuous vector fields in four dimensions. Explicit existence conditions of one-parameter families of periodic orbits for models involving two coupled relay systems are given. We derive existence conditions of one-parameter families of periodic solutions of systems of two second order non-smooth differential equations. We also study the persistence of such periodic orbits in the case of analytic perturbations of our relay systems. These results can be seen as analogous to the Lyapunov Centre Theorem.     

Lower norm error estimates for approximate solutions of differential equations with non-smooth coefficients

Summary In this paper, we derive error estimates inL _p-norm, 1p, for the ₂-Finite Element approximation to solutions of boundary value problems, where the coefficients are functions of bounded variation. The ₂-Finite Element Method was introduced in [3] and was shown to be effective for problems with non-smooth coefficient.The results of this paper form a part of a Ph.D. thesis written at the University of Maryland under the direction of Professor J.E. Osborn     

Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential

We prove spacetime weighted-L² estimates for the Schrödinger and wave equation with an inverse-square potential. We then deduce Strichartz estimates for these equations.     

Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications

We prove smoothing estimates for Schrödinger equations it_∂?+x_∂(a(x)x_∂?)=0 with a(x)∈BV, real and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing a∈BV to be in a sense a minimal requirement. Finally, we provide an application to sharp well-posedness for a generalized Benjamin-Ono equation.     

On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations

iφt+Δφ−²|x|φ+μ|φ|^p−1φ+γ|φ|^q−1φ=0,     

Self-adjointness of Schrödinger-type operators with locally integrable potentials on manifolds of bounded geometry

We consider a Schrödinger-type differential expression , where ∇ is a C^∞-bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with metric g and positive C^∞-bounded measure dμ, and V∈L_loc¹(EndE) is a linear self-adjoint bundle map. We define the maximal operator H_V,max associated to H_V as an operator in L²(E) given by H_V,maxu=H_Vu for all , where ∇∗∇u in is understood in distributional sense. We give a sufficient condition for the self-adjointness of H_V,max. The proof adopts Kato's technique to our setting, but it requires a more general version of Kato's inequality for Bochner Laplacian operator as well as a result on the positivity of u∈L²(M) satisfying the equation (Δ_M+b)u=ν, where Δ_M is the scalar Laplacian on M, b>0 is a constant and ν?0 is a positive distribution on M. For local estimates, we use a family of cut-off functions constructed with the help of regularized distance on manifolds of bounded geometry.     

Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

Fundamental systems for differential equations with operator coefficients

Fundamental systems for linear differential equations with operator coefficients are investigated, the concept of a Wronskian for such equations is introduced, and a method for constructing a differential equation from its fundamental system is proposed.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 777–782, December, 1970.We must thank V. É. Lyants for his help and the referee for his valuable advice.     

Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials

Recently, two-component systems of nonlinear Schrödinger equations with trap potentials have been well-known to describe a binary mixture of Bose-Einstein condensates called a double condensate. In a double condensate, the locations of spikes can be influenced by the interspecies scattering length and trap potentials so the interaction of spikes becomes complicated, and the locations of spikes are difficult to be determined. Here we study spikes of a double condensate by analyzing least energy (ground state) solutions of two-component systems of nonlinear Schrödinger equations with trap potentials. Our mathematical arguments may prove how trap potentials and the interspecies scattering length affect the locations of spikes. We use Nehari's manifold to construct least energy solutions and derive their asymptotic behaviors by some techniques of singular perturbation problems.     

Boundary stabilization of wave equations with variable coefficients

The aim of this paper is to obtain the exponential energy decay of the solution of the wave equation with variable coefficients under suitable linear boundary feedback. Multiplier method and Riemannian geometry method are used.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

Local energy decay for linear wave equations with variable coefficients

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L²-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].     

On holomorphic families of Schrödinger-type operators with singular potentials on manifolds of bounded geometry

We consider a family of Schrödinger-type differential expressions L(κ)=D²+V+κV(1), where κ∈C, and D is the Dirac operator associated with a Clifford bundle (E,∇E) of bounded geometry over a manifold of bounded geometry (M,g) with metric g, and V and V(1) are self-adjoint locally integrable sections of EndE. We also consider the family I(κ)=^*(∇F)∇F+V+κV(1), where κ∈C, and ∇F is a Hermitian connection on a Hermitian vector bundle F of bonded geometry over a manifold of bounded geometry (M,g), and V and V(1) are self-adjoint locally integrable sections of EndF. We give sufficient conditions for L(κ) and I(κ) to have a realization in L²(E) and L²(F), respectively, as self-adjoint holomorphic families of type (B). In the proofs we use Kato's inequality for Bochner Laplacian operator and Weitzenböck formula.     

On energy stability for the coupled nonlinear wave and Schrödinger systems

In this paper, we show that regular solutions to the coupled nonlinear wave and Schrödinger systems with coercive polynomial nonlinearities are unique among distributional solutions enjoying the energy inequality. The arguments also yield the stabilities of classical solutions of the evolution systems in the energy norm.     

Ground states for Schrödinger-type equations with nonlocal nonlinearity

The problem of the existence of stable solitary wave solutions for nonlinear Schrödinger-type equations with a generalized cubic nonlinearity is considered. These types of equations have recently arisen in the context of optical communications as averaging approximations to nonlinear dispersive equations with widely separated time scales. In this paper, it is shown that under general conditions on the kernel of the nonlocal term, stable standing wave solutions exist for these equations.     

Boundary-value problems for systems of differential equations with operator coefficients

The unique strong solvability is proved for several boundary-value problems for the systems Au_xx–Bu_y=f, A₁u_xx–Bu_yy=f, where A, A₁, B are operators of parabolic and hyperbolic types.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 12, pp. 1635–1641, December, 1990.