Classical Sobolev Spaces Research Articles

Shell interactions for Dirac operators

The self-adjointness of H+V is studied, where H=−iα⋅∇+mβ is the free Dirac operator in R3 and V is a measure-valued potential. The potentials V under consideration are given by singular measures with respect to the Lebesgue measure, with special attention to surface measures of bounded regular domains. The existence of non-trivial eigenfunctions with zero eigenvalue naturally appears in our approach, which is based on well known estimates for the trace operator defined on classical Sobolev spaces and some algebraic identities of the Cauchy operator associated to H.

Reconstruction of Interfaces from the Elastic Farfield Measurements Using CGO Solutions

In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a computable dominant part and an $H^\alpha$-decaying remainder term with $\alpha <3$, where $H^{\alpha}$ is the classical Sobolev space. Second, based on these properties, we estimate the convex hull as well as nonconvex parts of the interface using the far fields of only one of the two reflected body waves (pressure waves or shear waves) as measurements. The results are given for both the impenetrable obstacles, with traction boundary conditions, and the penetrable obstacles. In the analysis, we require the surfaces of the obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lame coefficients to be measurable and bounded with the usual jump conditions across the interface.

On the relaxation of variational integrals in metric Sobolev spaces

Abstract We give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for “relaxed” variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger–Keith's differentiable structure.

Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0,L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev space H s (0,L) for s> � 3 , which provides a positive answer to one of the open questions of Colin and Ghidaglia (Adv.

An Asymptotic Preserving Scheme Based on a New Formulation for NLS in the Semiclassical Limit

We consider the semiclassical limit for the nonlinear Schrodinger equation. We introduce a phase/amplitude representation given by a system similar to the hydrodynamical formulation, whose novelty consists in including some asymptotically vanishing viscosity. We prove that the system is always locally well-posed in a class of Sobolev spaces, and globally well-posed for a fixed positive Planck constant in the one-dimensional case. We propose a second order numerical scheme which is asymptotic preserving. Before singularities appear in the limiting Euler equation, we recover the quadratic physical observables as well as the wave function with mesh size and time step independent of the Planck constant. This approach is also well suited to the linear Schrodinger equation.

On the ill-posedness of the Gardner equation

We present ill-posedness results for the initial value problem (IVP) for the Gardner equation. We measure the regularity of the Cauchy problem in the classical Sobolev spaces Hs, and show the critical Sobolev index under which the local well-posedness of the problem is not present, in the sense that the dependence of solutions upon initial data fails to be continuous.

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part I: Shape Differentiability of Pseudo-homogeneous Boundary Integral Operators

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.

Reproducing Kernels and Variable Bandwidth

We show that a modulation space of type () is a reproducing kernel Hilbert space (RKHS). In particular, we explore the special cases of variable bandwidth spaces Aceska and Feichtinger (2011) with a suitably chosen weight to provide strong enough decay in the frequency direction. The reproducing kernel property is valid even if () does not coincide with any of the classical Sobolev spaces because unbounded bandwidth (globally) is allowed. The reproducing kernel will be described explicitly.

Reproducing kernels of Sobolev spaces via a green kernel approach with differential operators and boundary operators

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L:?=?P ???T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional adjoint operator P ??? of P is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators P and B. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.

On interpolation of cocompact imbeddings

Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we prove that, under quite general conditions, cocompactness of imbeddings of Banach spaces persists under both real and complex interpolation. As an application, we obtain that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces are cocompact relative to lattice shifts. We deduce this by interpolating the known cocompact imbeddings for classical Sobolev spaces (“vanishing” lemmas of Lieb and Lions). We also apply cocompactness to prove compactness of imbeddings of some radial subspaces and to show the existence of minimizers in some isoperimetric problems. Our research complements a range of previous results, and recalls that there is a natural conceptual framework for unifying them.

Well-posedness and stability results for the Gardner equation

In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also cover the energy space H^1(R) where global well-posedness follows from the conservation laws of the system. Moreover, we construct solitons of the Gardner equation explicitly and prove that, under certain conditions, this family is orbitally stable in the energy space.

On the well-posedness of the quasi-geostrophic equation in the Triebel-Lizorkin-Lorentz spaces

In this paper, we establish the local well-posedness for the quasi-geostrophic equations and obtain a blow-up criterion of smooth solutions in the framework of Triebel-Lizorkin-Lorentz spaces by adapting a method in Chen-Miao-Zhang (Arch. Rational Mech. Anal. 195: 2010, 561–578). Our new function spaces contain the classical Triebel-Lizorkin spaces and Sobolev spaces, and thus the corresponding results generalize several known ones, for instance, Chae (Nonlinearity 16: 2003, 479–495) and Castro et al. (Nonlinearity 22: 2009, 1791–1815). The main ingredients of our proofs are Littlewood–Paley decomposition and the paradifferential calculus.

Some stability results under domain variation for Neumann problems in metric spaces

A famous result of Chenais (8) (1975) says that if ›n is a sequence of extension domains in R N that converges to › in the characteristic functions topology, then the weak solutions un for the problem (0.1) ( i¢un + un = f in ›n; @ un = 0 on @›n converge strongly to the solution u of the same problem in ›. It is also proved in (8) using the method of Calderon that an -cone condition is sucient to obtain uniform extension domains. In this paper we establish this result in a metric space framework, replacing the classical Sobolev space H 1 (›) by the Newtonian space N 1;2 (›). Moreover, using the latest results about extension domains contained in (2), and which rely on the techniques of Jones, we give weaker conditions on the domains for still getting stability of the Neumann problem. Finally we prove that the Neumann problem is stable for a sequence of quasiballs with uniform distortion constant that converge in a certain measure sense. The latter result gives a new existence theorem for some shape optimisation problems under quasiconformal variations.

Continuous dependence on the data for nonlinear elliptic equations via symmetrization

We prove the continuous dependence on the data of weak solutions to Dirichlet problem for nonlinear elliptic equations with a first order term and datum in dual spaces of classical Sobolev spaces. We deduce uniqueness results.

Solitary waves for a class of quasilinear Schrödinger equations in dimension two

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrodinger equations of the form −e2Δu + V(z)u − e2 [Δ(u2)]u = h(u) in the whole two-dimension space where e is a small positive parameter and V is a continuous potential uniformly positive. The main feature of this paper is that the nonlinear term h(u) is allowed to enjoy the critical exponential growth with respect to the Trudinger–Moser inequality and also the presence of the second order nonhomogeneous term [Δ(u2)]u which prevents us to work in a classical Sobolev space. Using a version of the Trudinger–Moser inequality, a penalization technique and mountain-pass arguments in a nonstandard Orlicz space we establish the existence of solutions that concentrate near a local minimum of V.

SEMI-CLASSICAL STATES FOR QUASILINEAR SCHRÖDINGER EQUATIONS ARISING IN PLASMA PHYSICS

In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrödinger equations -ε2Δu + V(x)u - ε2[Δ(u2)]u = f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger–Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -ε2[Δ(u2)]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter ε for which is motivated by mathematical models in physics. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

On a capacity for modular spaces

The purpose of this article is to define a capacity on certain topological measure spaces X with respect to certain function spaces V consisting of measurable functions. In this general theory we will not fix the space V but we emphasize that V can be the classical Sobolev space W 1 , p ( Ω ) , the classical Orlicz–Sobolev space W 1 , Φ ( Ω ) , the Hajłasz–Sobolev space M 1 , p ( Ω ) , the Musielak–Orlicz–Sobolev space (or generalized Orlicz–Sobolev space) and many other spaces. Of particular interest is the space V : = W ˜ 1 , p ( Ω ) given as the closure of W 1 , p ( Ω ) ∩ C c ( Ω ¯ ) in W 1 , p ( Ω ) . In this case every function u ∈ V (a priori defined only on Ω ) has a trace on the boundary ∂ Ω which is unique up to a Cap p , Ω -polar set.

Good Banach spaces for piecewise hyperbolic maps via interpolation

We introduce a weak transversality condition for piecewise C1+α and piecewise hyperbolic maps which admit a C1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.

Local Exact Controllability and Stabilizability of the Nonlinear Schrödinger Equation on a Bounded Interval

This paper studies the exact controllability and the stabilization of the cubic Schrödinger equation posed on a bounded interval. Both internal and boundary controls are considered, and the results are given first in a periodic setting, and next with Dirichlet (resp., Neumann) boundary conditions. It is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical Sobolev space $H^s$ for any $s\geq0$. It is also shown that the systems with an internal stabilization are locally exponentially stabilizable in $H^s$ for any $s\geq0$.

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω ⊂ R n with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if s > n 2 , or 0 ⩽ s < n 2 with 1 ⩽ n < 2 + 2 s , or s = 0 , 1 with n = 2 , then the systems are locally exactly controllable in the classical Sobolev space H s ( Ω ) around any smooth solution of the cubic Schrödinger equation.