Theory Of Sobolev Spaces Research Articles

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

In this paper, we study Landau damping in the weakly collisional limit of a Vlasov-Fokker-Planck equation with nonlinear collisions in the phase-space $$(x,v) \in {{\mathbb {T}}}_x^n \times {\mathbb {R}}^n_v$$ . The goal is four-fold: (A) to understand how collisions suppress plasma echoes and enable Landau damping in agreement with linearized theory in Sobolev spaces, (B) to understand how phase mixing accelerates collisional relaxation, (C) to understand better how the plasma returns to global equilibrium during Landau damping, and (D) to rule out that collision-driven nonlinear instabilities dominate. We give an estimate for the scaling law between Knudsen number and the maximal size of the perturbation necessary for linear theory to be accurate in Sobolev regularity. We conjecture this scaling to be sharp (up to logarithmic corrections) due to potential nonlinear echoes in the collisionless model.

Spherical square functions of Marcinkiewicz type with Riesz potentials

We prove a pointwise equivalence between a spherical square function composed with the Riesz potential and a Littlewood–Paley function arising from the Bochner–Riesz operators. Also, its application to the theory of Sobolev spaces will be given.

Sobolev Spaces, Schwartz Spaces, and a Definition of the Electromagnetic and Gravitational Coupling

The concept of "multiplicity of solutions" was developed in arXiv:1509.02603v2 which is based on the theory of energy operators in the Schwartz space S^-(R) and some subspaces called energy spaces first defined in arXiv:1208.3385 and arXiv:1308.0874. The main idea is to look for solutions of a given linear PDE in those subspaces. Here, this work extends previous developments in S^-(R^m) (m in Z^+) using the theory of Sobolev spaces, and in a special case the Hilbert spaces. Furthermore, we also define the concept of "Energy Parallax", which is the inclusion of additional solutions when varying the energy of a predefined system locally by taking into account additional smaller quantities. We show that it is equivalent to take into account solutions in other energy subspaces. To illustrate the theory, one of our examples is based on the variation of ElectroMagnetic (EM) energy density within the skin depth of a conductive material, leading to take into account derivatives of EM evanescent waves, particular solutions of the wave equation. The last example is the derivation of the Woodward effect with the variations of the EM energy density under strict assumptions in general relativity. It finally leads to a theoretical definition of an electromagnetic and gravitational (EMG) coupling.

Littlewood–Paley Equivalence and Homogeneous Fourier Multipliers

We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for H\"ormander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given.

Nontrivial solutions for semilinear elliptic systems via Orlicz–Sobolev theory

In this paper, the semilinear elliptic systems with Dirichlet boundary value are consideredWe extend the notion of subcritical growth from polynomial growth to N-function growth. Under N-function growth, nontrivial solutions are obtained via Orlicz–Sobolev space theory and variational method. It is also noteworthy that the nonlinear term g(v) does not have to satisfy the usual Ambrosetti–Rabinowitz condition. So, in some sense, we enrich recent results of D.G. de Figueiredo, J.M. doÓ and B. Ruf [de Figueiredo DG, doÓ JM, Ruf B, An Orlicz-space approach to superlinear elliptic systems, J. Funct. Anal. 2005;224:471–496].

Time-harmonic acoustic wave scattering in an ocean with depth-dependent sound speed

Time-harmonic acoustic wave propagation in an inhomogeneous ocean with depth-dependent sound speed can be modeled by the Helmholtz equation in an infinite, three-dimensional (3D) waveguide of finite height. Using variational theory in Sobolev spaces we prove well posedness of the corresponding scattering problem from a bounded inhomogeneity inside such an ocean. To this end, we introduce an exterior Dirichlet-to-Neumann operator for depth-dependent sound speed and prove boundedness, coercivity, and holomorphic dependence of this operator in function spaces adapted to our weak solution theory. Analytic Fredholm theory then yields existence and uniqueness of solution for the scattering problem for all but a countable sequence of frequencies. The latter result generalizes corresponding theory for waveguide scattering with constant sound speed and easily extends to various related scattering problems, e.g. to scattering from impenetrable obstacles.

Embedding theorems and a variational problem for functions on a metric measure space

We use a new method to prove the Sobolev embedding theorem for functions on a metric space and study other questions of the theory of Sobolev spaces on a metric space. We prove the existence and uniqueness of solution to a variational problem.

Hodge theorem for the natural projection of complex horizontal Laplacian on complex Finsler manifolds

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.

A Sobolev space theory for parabolic stochastic PDEs driven by Lévy processes on [formula omitted]-domains

In this paper we study parabolic stochastic partial differential equations (SPDEs) driven by Lévy processes defined on Rd, R+d and bounded C1-domains. The coefficients of the equations are random functions depending on time and space variables. Existence and uniqueness results are proved in (weighted) Sobolev spaces, and Lp-estimates and various properties of solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular it can be negative or fractional.

Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with anisotropic viscosity acting only in the horizontal direction, which arises in ocean dynamics models. Global well-posedness for this system was proven by Danchin and Paicu; however, an additional smoothness assumption on the initial density was needed to prove uniqueness. They stated that it is not clear whether uniqueness holds without this additional assumption. The present work resolves this question and we establish uniqueness without this additional assumption. Furthermore, the proof provided here is more elementary; we use only tools available in the standard theory of Sobolev spaces, and without resorting to para-product calculus. We use a new approach by defining an auxiliary “stream-function” associated with the density, analogous to the stream-function associated with the vorticity in 2D incompressible Euler equations, then we adapt some of the ideas of Yudovich for proving uniqueness for 2D Euler equations.

A Compact Embedding for Sequence Spaces

The Rellich-Kondrachov Theorem is a fundamental result in the theory of Sobolev spaces. We prove an analogue of this theorem in the context of sequence spaces, using elementary methods.

Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation

In this paper we study absolute minimizers and the Aronsson equation for a noncoercive Hamiltonian. We extend the definition of absolutely minimizing functions (in a viscosity sense) for the minimization of the $L^\infty$ norm of a Hamiltonian, within a class of locally Lipschitz continuous functions with respect to possibly noneuclidian metrics. The metric structure is naturally associated to the Hamiltonian and it is related to the a-priori regularity of the family of subsolutions of the Hamilton-Jacobi equation. A special but relevant case contained in our framework is that of Hamiltonians with a Carnot-Carathéodory metric structure determined by a family of vector fields (CC for short in the following), in particular the eikonal Hamiltonian and the corresponding anisotropic infinity-Laplace equation. In this case, the definition of absolute minimizer can be written in an almost classical way, by the theory of Sobolev spaces in a CC setting. In general open domains and with a prescribed continuous Dirichlet boundary condition, we prove the existence of an absolute minimizer which satisfies the Aronsson equation as a viscosity solution. The proof is based on Perron's method and relies on a-priori continuity estimates for absolute minimizers.

Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition

We study the regularity of a diusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with Coulomb interaction and reflection between nearest neighbors. As our main result we establish the strong Feller property for the process in both cases of repulsion and attraction. In particular the system can be started from any initial state, including multiple point configurations. Moreover we show that the process is a Euclidean semi-martingale if and only if the interaction is repulsive. Hence, contrary to classical results about reflecting Brownian motion in smooth domains, in the attractive regime a construction via a system of Skorokhod SDEs is impossible. Finally, we establish exponential heat kernel gradient estimates in the repulsive regime. The main proof for the attractive case is based on potential theory in Sobolev spaces with Muckenhoupt weights.

Optimal Constants in the Theory of Sobolev Spaces and PDEs

Recent research activities on sharp constants and optimal inequalities have shown their impact on a deeper understanding of geometric, analytical and other phenomena in the context of partial differential equations and mathematical physics. These intrinsic questions have applications not only to a-priori estimates or spectral theory but also to numerics, economics, optimization, etc.

A Hardy-type inequality and its applications

We prove a Hardy-type inequality that provides a lower bound for the integral ∫0∞|f(r)|prp−1dr, p > 1. In the scale of classical Hardy inequalities, this integral corresponds to the value of the exponential parameter for which neither direct nor inverse Hardy inequalities hold. However, the problem of estimating this integral and its multidimensional generalization from below arises in some practical questions. These are, for example, the question of solvability of elliptic equations in the scale of Sobolev spaces in the whole Euclidean space ℝn, some questions in the theory of Sobolev spaces, hydrodynamic problems, etc. These questions are studied in the present paper.

Limit formulae and jump relations of potential theory in Sobolev spaces

In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces on ∂Σ for integrable functions, see for example Fichera (Ann Math Pura ed Appl Serie IV, Tomo 27, 1948) or Freeden and Kersten (The geodetic boundary value problem using the known surface of the Earth. Veröffentlichung des Geodätischen Instituts der RWTH Aachen, 29, 1980). The achievement of this paper is the L 2(∂Σ) convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and ∂Σ is a two dimensional suitable smooth submanifold in \({\mathbb{R}^3}\), called regular C m,α-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in C m(∂Σ) which is proved with help of some results taken from Günter (Die Potentialtheorie und ihre Anwendungen auf Grundaufgaben der mathematischen Physik. Teubner, Leipzig, 1957). Additionally, we need a result about the limit formulae in L 2(∂Σ), which can be found in Kersten (Result Math 3:17–24, 1980), and a reduction result which we took from Müller (Math Ann 123:235–262, 1951). Moreover we prove the convergence in the Hölder spaces C m,β(∂Σ). Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize density results, see e.g. Freeden and Michel (Multiscale potential theory. Birkhäuser, Boston, 2004), from L 2(∂Σ) to H m,2(∂Σ). For it we prove the limit formula for U 1 in (H m,2(∂Σ))' also. The last section is dedicated to oblique limit formulae for the single layer potential as well as for its first order oblique derivative.

Sobolev space theory of SPDEs with continuous or measurable leading coefficients

We study stochastic partial differential equations with variable coefficients defined on R d , R + d and bounded C 1 domains. For equations with continuous leading coefficients we give existence and uniqueness results in L q ( L p ) -spaces, where it is allowed for the powers of summability with respect to space and time variables to be different. For equations with measurable leading coefficients we give unique solvability in L p -spaces.

Sobolev Space Theory of Parabolic Equations Degenerating on the Boundary of C 1 Domains

Second-order parabolic equations degenerating on the boundary of C 1 domains are considered. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are presented.

On increased stability in the continuation of the Helmholtz equation

In this paper, we give analytical and numerical evidence of increasing stability in the Cauchy problem for the Helmholtz equation in the whole domain when frequency is growing. This effect depends upon the convexity properties of the surface where the Cauchy data are given. Proofs use previously obtained estimates in subdomains and the theory of Sobolev spaces: traces, embedding and interpolation theorems. The theory is illustrated by three-dimensional numerical examples. The results show that even in an acoustical frequency range the increase of resolution with growing frequency is quite dramatic. On the other hand, the resolution of continuation outside a unit sphere is decreasing.

The global domain of attraction and the initial value problems of a kind of GKdV equations

The initial value problem of a kind of Generalized Korteweg–de Vries (GKdV) equations are considered by using Sobolev space theory and finite element method. The well-posedness of the solution of the GKdV equation is obtained. Simple cell mapping method is employed to obtain the global domain of attraction and the finite step domain of attraction of GKdV equations.