Weighted Sobolev Spaces Research Articles

Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems

A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed.

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: Application to the nonlinearly damped $p$-system

A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta–Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows us to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform, which we avoid here, developing the “physical space version” of the hyperbolic hypocoercivity approach introduced in Beauchard and Zuazua [Arch. Ration. Mech. Anal. 199 (2011), 177–227], to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows us to recover the natural heat-like time decay of solutions under sharp rank conditions, without employing Fourier analysis or L^{1} assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped p -system.

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

A characterization of the subspace of radially symmetric functions in Sobolev spaces

In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.

Global differentiable structures for the Fisher-Rao and Kantorovich-Wasserstein-Otto metrics

We develop a class of non-parametric, Banach-Sobolev manifolds of probability measures that, despite having comparatively weak topologies, support the Fisher-Rao and Kantorovich-Wasserstein-Otto (KWO) Riemannian metrics. The manifolds employ the Kaniadakis κ-deformed logarithms in their charts, and are isomorphic to the (whole) model spaces, W1,λ(μ). These are weighted Sobolev spaces with Lebesgue exponents as small as λ=4, making them suitable for approximations. The KWO metric is analysed through a bundle of Markov semigroups having generators with irregular first-order terms. Strong a-priori estimates are obtained for the associated parabolic equations. Together with a weak Poincaré/Lp-convergence theorem, these are used to establish the Hölder continuity of the “velocity field” representation of tangent vectors—a property inherited by the KWO metric. The manifolds provide a simple framework for the study of problems involving both metrics.

Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

Novel convergence to forced traveling waves for a nonlocal dispersal population model in a shifting environment

Some new types of the nonlinear stability of forced traveling waves for a nonlocal dispersal population model in a shifting environment are given in this paper. Firstly, when the speed of the shifting habitat edge c is larger than the spreading speed of the species c⁎, by applying the comparison principle, the Fourier transform and the L2-weighted energy estimate, we show that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of e−μt, for some μ>0. In order to extend the stability results to c=c⁎, we take another weight function and construct the different weighted Sobolev space. Then by using the Fourier transform and the L1-weighted energy estimate, we prove that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of t−12e−νt, for some ν>0, while all forced traveling waves with the speed c=c⁎ are algebraically stable in the form of t−12, which are quite different from the previous stability results.

Existence of entropy solutions to nonlinear degenerate weighted elliptic p(.)-Laplacian problem and L1-data

Our aim in this paper is to study the existence result of entropy solution for a specific type of nonlinear degenerate weighted elliptic p(.)-Laplacian problems with Dirichlet-type boundary condition and L1 data. In the framework of the theory of weighted Sobolev spaces with variable exponents, we use the regularization approach combined with a priori estimates.

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Sharp Higher Order Adams’ Inequality with Exact Growth Condition on Weighted Sobolev Spaces

Sharp Higher Order Adams’ Inequality with Exact Growth Condition on Weighted Sobolev Spaces

Stability of traveling wavefronts for advection–reaction–diffusion equation

In this paper, we study an advection–reaction–diffusion equation, where the nonlinear advection has neither monotonicity nor variational structure. For all wavefronts with the speed c>c0, where c0 is the minimal wave speed, we use the technical weighted energy method to prove that these wavefronts are exponentially stable, when the initial perturbations are small in a weighted Sobolev space.

The existence and nonlinear stability of stationary solutions to outflow problem for a reduced gravity two and a half layer model

The main concern of this paper is to study large-time behavior of solutions for an outflow problem to the reduced gravity two and a half layer model in a half space. We establish the existence of stationary solutions and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small. Moreover, we obtain the convergence rate of the solution toward the stationary solution in some weighted Sobolev spaces only for the supersonic case. The proof is based on the basic energy method. A key point for the proof of convergence rates is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity satisfy some additional conditions.

Ill-posedness for the Half wave Schrödinger equation

We study the Cauchy problem for the half wave Schrödinger equation introduced by Xu [9]. There are some well-posedness results for the equation, however there is no ill-posedness result. We focus on the scale critical space and obtain the ill-posedness in the super-critical or at the critical space under certain condition. The proofs in the super-critical space are based on the argument established by Christ, Colliander and Tao [4]. More precisely, we analyze dispersionless equation with smooth initial data, namely the Schwartz function and it is locally well-posed in some weighted Sobolev space. We construct the solution for the half wave Schrödinger equation by using the solution for the dispersionless equation and we can exploit the norm inflation or the decoherence properties. For the critical space, we use the standing wave solution, which was proved the existence by Bahri, Ibrahim and Kikuchi [1].

Anisotropic obstacle Neumann problems in weighted Sobolev spaces with Hardy potential and variable exponent

Anisotropic obstacle Neumann problems in weighted Sobolev spaces with Hardy potential and variable exponent

Pólya–Szegö type inequality and imbedding theorems for weighted Sobolev spaces

Pólya–Szegö type inequality and imbedding theorems for weighted Sobolev spaces

Spectral Jacobi approximations for Boussinesq systems

AbstractThis paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results.

Weighted Sobolev spaces and Morse estimates for quasilinear elliptic equations

We establish critical groups estimates for the weak solutions of −Δpu=f(x,u) in Ω and u=0 on ∂Ω via Morse index, where Ω is a bounded domain, f∈C1(Ω‾×R) and f(x,s)>0 for all x∈Ω‾, s>0 and f(x,s)=0 for all x∈Ω‾, s≤0. The proof relies on new uniform Sobolev inequalities for approximating problems. We also prove critical groups estimates when Ω is the ball or the annulus and f is a sign changing function.

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

Nonlinear degenerate Navier problem involving the weighted biharmonic operator with measure data in weighted Sobolev spaces

Soliton resolution for the complex short-pulse positive flow with weighted Sobolev initial data in the space-time soliton regions

The Riemann Hilbert approach and ∂‾ steepest descent method are used to study the Cauchy problem of a new complex short-pulse positive flow, from which the long-time asymptotics and soliton resolution conjecture for the complex short-pulse positive flow with initial conditions is obtained in weighted Sobolev spaces. Resorting to the spectral analysis of Lax pair, the introduced transformations of field variables and transformations of independent variables, we establish the basic Riemann-Hilbert problem, and convert the solution of the Cauchy problem of the complex short-pulse positive flow into the corresponding solution of the Riemann Hilbert problem. The jump matrix then expands and deforms continuously over the initial contour. We finally obtain the long-time asymptotics and soliton resolution for the complex short-pulse positive flow in the soliton region by using the ∂‾ steepest descent method. The results also show that N-soliton solutions of the complex short-pulse positive flow are asymptotically stable.

Sharp Sobolev and Adams–Trudinger–Moser embeddings on weighted Sobolev spaces and their applications

Abstract We derive sharp Sobolev embeddings on a class of Sobolev spaces with potential weights without assuming any boundary conditions. Moreover, we consider the Adams-type inequalities for the borderline Sobolev embedding into the exponential class with a sharp constant. As applications, we prove that the associated elliptic equations with nonlinearities in both forms of polynomial and exponential growths admit nontrivial solutions.