Weighted Sobolev Spaces Research Articles

Weighted mixed-norm Lp estimates for equations in non-divergence form with singular coefficients: The Dirichlet problem

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the Ap-Muckenhoupt class.

The outflow problem for the radiative and reactive gas: existence, stability and convergence rate

In this paper, we show that the small-amplitude stationary solutions are time-asymptotically stable to the half-space problem for the one-dimensional radiative and reactive gas (pressure , internal energy ) provided that the initial perturbation is sufficiently small in some weighted Sobolev spaces. We also obtain the convergence rate of global solutions toward corresponding stationary solutions. The proofs are given by a weighted energy method. A key point is to capture the positivity of the energy dissipation functional and boundary terms with suitable space weight functions and using the precise estimates about stationary solutions.

The double phase Dirichlet problem when the lowest exponent is equal to 1

In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when the lowest exponent is equal to 1. Our solution is a function of bounded variation that simultaneously lies in a suitable weighted Sobolev space and is found as the limit of a sequence of solutions of intermediate double phase Dirichlet problems whose lowest exponent p goes to 1. As a result of that, our approach involves the study of some relevant properties of generalized Orlicz-Sobolev spaces.

Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model

We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator \(\Delta_p\) for \(1 < p < \infty\). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics. See also https://ejde.math.txstate.edu/special/02/t1/abstr.html

Analysis of a subdiffusion model with a variable-order fractional calibration term

A subdiffusion model with a variable-order fractional calibration term is investigated. The well-posedness of a corresponding ordinary differential equation is firstly proved and then the well-posedness of the proposed model is analyzed in weighted Sobolev spaces, in which the weight function is selected to accommodate the blow up of the derivative of the solutions at the initial time.

Degenerate linear parabolic equations in divergence form on the upper half space

We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.

From the Vlasov-Poisson-Boltzmann System to the Vlasov-Poisson-Landau System with Coulomb Potential

In the celebrated work [Y. Guo., J. Amer. Math. Soc., 25 (2012), pp. 759–812], Guo proved global well-posedness of the Vlasov–Poisson–Landau (VPL) system with Coulomb potential in algebraically weighted Sobolev spaces. Thanks to this breakthrough, there are rich developments on the VPL system and also on the Vlasov–Poisson–Boltzmann (VPB) system with cutoff or noncutoff Boltzmann kernels. In this work, we are interested in the connection between the two important physical systems where particles interact through the Coulomb force and are influenced by their self-consistent electrostatic field. The Coulomb potential yields the famous Rutherford scattering cross section which is too singular in the angular variable for the Boltzmann equation to be meaningful. With an angular cutoff and a proper scaling as in [R. Alexandre and C. Villani, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), pp. 61–95], we prove global well-posedness of VPB in algebraically weighted Sobolev spaces. To this end, we develop a new two-stage energy method and merely rely on the -norm (from the dissipation of the linearized Boltzmann operator) to control the nonlinear terms containing the electrostatic field. Consequently, the same well-posedness result also holds for the VPB system with all the other soft potentials, which removes the technical constraint in [R. Duan and S. Liu, Comm. Math. Phys., 324 (2013), pp. 1–45]. Moreover, when the cutoff parameter tends to zero, convergence of VPB to VPL is established with an explicit rate.

Weighted Jump and Variational Inequalities for Hypersingular Integrals with Rough Kernels

In this paper, we study the weighted jump function and variation of hypersingular integral operators with rough kernels which are defined as $${T_{\Omega,\alpha,\varepsilon}}f\left(x \right) = \int_{\left| y \right|>\varepsilon} {{{\Omega ({y^\prime})} \over {{{\left| y \right|}^{n + \alpha}}}}f(x - y)dy,} $$ where α ≥ 0, Ω is an integrable function on the unit sphere $${\mathbb S^{n - 1}}$$ satisfying certain cancellation conditions. More precisely, we show that for 1 < p < ∞, the jump function and variation of the family of truncated hypersingular integrals {TΩ,α, ε}ε>0 extends to bounded operators from the weighted Sobolev space L (w) to the weighted Lebesgue space Lp(w) with $$\Omega \,\,{L^q}({\mathbb S^{n - 1}})$$ where q > 1.

Homogenization results for the generator of multiscale Langevin dynamics in weighted Sobolev spaces

AbstractWe study the homogenization of the Poisson equation with a reaction term and of the eigenvalue problem associated to the generator of multiscale Langevin dynamics. Our analysis extends the theory of two-scale convergence to the case of weighted Sobolev spaces in unbounded domains. We provide convergence results for the solution of the multiscale problems above to their homogenized surrogate. A series of numerical examples corroborate our analysis.

A regularity result for the bound states of N-body Schrödinger operators: blow-ups and Lie manifolds

We prove regularity estimates in weighted Sobolev spaces for the L^2-eigenfunctions of Schrödinger-type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual N-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is , where is the usual Euclidean distance to the union of the set of collision planes {mathcal {F}}. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification {overline{X}} of the underlying space X and we first blow up the spheres {mathbb {S}}_Y subset {mathbb {S}}_X at infinity of the collision planes Y in {mathcal {F}} to obtain the Georgescu–Vasy compactification. Then, we blow up the collision planes {mathcal {F}}. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher-order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

Planar Stationary Solution to Outflow Problem for Compressible Heat Conducting Gas in \(\mathbb{R}^3_+\): Stability and Convergence Rate

We are concerned with the stability and convergence rate of planar stationary solution to the compressible heat conducting gas in half space under outflow condition. (1) It is shown that a corresponding planar stationary solution is time asymptotically stable in three cases (i.e., supersonic, subsonic, and transonic), respectively, provided the initial perturbation in a certain Sobolev space and the boundary strength are sufficiently small. (2) Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Precisely, we obtain an algebraic decay rate provided that an initial perturbation decays in a tangential direction with the algebraic rate for a supersonic flow. The corresponding algebraic convergence rate is also obtained for a transonic flow, which is worse than that for the supersonic flow due to a degenerate property of the transonic flow. Each proof is given by deriving a priori estimates of the perturbation from the stationary wave by using a time and space weighted energy method.

Maximal operators associated with Fourier multipliers and applications

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on Lp(Rd) for each p∈(1,∞). Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which are not necessarily radial. To do so, we make use of modified square function estimates and bilinear interpolation. For the bilinear interpolation we introduce a function space Σ2(B) where B denotes a Banach space of functions on Rd, which is a variant of weighted Sobolev spaces. In result, we obtain convergence results for fractional half-wave equations and surface averages as well as the Lp boundedness for the maximal operators.

On the Neumann problem for the nonstationary Stokes system in angles and cones

AbstractThe authors consider the Neumann problem for the nonstationary Stokes system in a two‐dimensional angle or a three‐dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.

A unified approach to degenerate problems in the half-space

We study elliptic and parabolic problems governed by the singular elliptic operatorsL=yα1Δx+yα2(Dyy+cyDy−by2),α1,α2∈R in the half-space R+N+1={(x,y):x∈RN,y>0}. We prove elliptic and parabolic Lp-estimates and solvability for the associated problems. In the language of semigroup theory, we prove that L generates an analytic semigroup, characterize its domain as a weighted Sobolev space and show that it has maximal regularity.

Strong and weak associativity of weighted Sobolev spaces of the first order

A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered. Bibliography: 49 titles.

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Rothe Time-Discretization Method for Nolinear Parabolic Problems in Weighted Sobolev Space with Variable Exponents

Линейный финитный функционал в весовом пространстве Соболева

In this paper, a representation of a linear functional in the weighted Sobolev space is obtained. The space is normed without use of pseudodifferential operators. The norm contains partial derivatives of all intermediate orders of the test function. The Sobolev space is considered to be of non-Hilbert type. First, we deduce the representation of linear functional via a boundary element of the test space. The boundary element corresponds to the given functional. This way, referring to Clarkson’s inequalities, we prove the uniqueness of the boundary element. Then, to obtain a condition for the boundary element, we differentiate the function built based on the norm. The result leads to a representation of an arbitrary linear functional via the boundary element. When considering the boundary element as unknown, the representation performs as a nonlinear differential equation. Second, we consider a finite linear functional. The extreme function of such a functional was built in our earlier papers. The extreme function is expressed via convolution of the fundamental solution of a linear partial differential equation with a given functional. The functional performs as a distribution in the convolution. Convolution exists if the linear functional is finite. Using this fact, we prove that the representation of a finite linear functional via the boundary element is identical to the representation via the extreme function.

Contraction analysis of Volterra integral equations via realization theory and frequency-domain methods

&lt;p style='text-indent:20px;'&gt;Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the contraction analysis of a class of nonlinear Volterra integral equations in some Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in some weighted Sobolev space which has the same stability properties as the Volterra equation.&lt;/p&gt;

The a posteriori error estimate in fractional differential equations using generalized Jacobi functions

&lt;abstract&gt;&lt;p&gt;In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators.&lt;/p&gt;&lt;/abstract&gt;

Strong and weak associativity of weighted Sobolev spaces of the first order

В работе дан краткий обзор недавних результатов по проблеме характеризации ассоциированных и дважды ассоциированных пространств к функциональным классам, включающим как идеальные, так и неидеальные структуры. К числу последних относятся двухвесовые пространства Соболева первого порядка на положительной полуоси. Показано, что, в отличие от понятия двойственности, ассоциированность может быть "сильной" и "слабой". При этом дважды ассоциированные пространства делятся еще на три типа. В этом контексте установлено, что пространство Соболева функций с компактным носителем обладает слабой ассоциированной рефлексивностью, а сильно ассоциированное к слабому ассоциированному пространству состоит только из нуля. Аналогичными свойствами обладают весовые классы типа Чезаро и Копсона, для которых проблема изучена полностью и установлена их связь с пространствами Соболева со степенными весами. В качестве приложения рассмотрена задача об ограниченности преобразования Гильберта из весового пространства Соболева в весовое пространство Лебега. Библиография: 49 названий.