Sobolev Spaces Research Articles

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

Free scalar field theory on a Sobolev space over a bounded interval

This paper discusses several functional analytic issues relevant for field theories in the context of the Hamiltonian formulation for a free, massless, scalar field defined on a closed interval of the real line. The fields that we use belong to a Sobolev space with a scalar product. As we show this choice is useful because it leads to an explicit representation of the points in the fibers of the phase space (the cotangent bundle of the configuration space). The dynamical role of the boundary of the spatial manifold where the fields are defined is analyzed.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

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.

Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two Chi-squared statistic-based algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a dataset of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, analysis of variance for functional data, and geometric morphometrics. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Fractional double-phase nonlocal equation in Musielak-Orlicz Sobolev space

In this paper, we analyze the existence of solutions to a double-phase fractional equation of the Kirchhoff type in Musielak-Orlicz Sobolev space with variable exponents. Our approach is mainly based on the sub-supersolution method and the mountain pass theorem.

Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians

In this article, we examine variational inequalities of the form ⟨A(u),v−u⟩+⟨F(u),v−u⟩≥0,∀v∈Ku∈K,, where A is a generalized fractional Φ-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and F is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term F such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions.

Nonlinear Asymptotic Stability and Transition Threshold for 2D Taylor–Couette Flows in Sobolev Spaces

In this paper, we investigate the stability of the 2-dimensional (2D) Taylor–Couette (TC) flow for the incompressible Navier–Stokes equations. The explicit form of velocity for 2D TC flow is given by u=(Ar+Br)(-sinθ,cosθ)T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u=(Ar+\\frac{B}{r})(-\\sin \ heta , \\cos \ heta )^T$$\\end{document} with (r,θ)∈[1,R]×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(r, \ heta )\\in [1, R]\ imes \\mathbb {S}^1$$\\end{document} being an annulus and A, B being constants. Here, A, B encode the rotational effect and R is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor–Couette flow is asymptotically stable, even at high Reynolds number (Re∼ν-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Re\\sim \ u ^{-1}$$\\end{document}), with a sharp exponential decay rate of exp(-ν13|B|23R-2t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\ u ^{\\frac{1}{3}}|B|^{\\frac{2}{3}}R^{-2}t)$$\\end{document} as long as the initial perturbation is less than or equal to ν12|B|12R-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ^\\frac{1}{2} |B|^{\\frac{1}{2}}R^{-2}$$\\end{document} in Sobolev space. The powers of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} and B in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor–Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient B. The larger the |B|, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor–Couette flows.

Well-posedness theory for non-homogeneous incompressible fluids with odd viscosity

Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed “odd viscosity”, becomes non-dissipative. At the mathematical level, this fact translates into a loss of derivatives at the level of a priori estimates: while the odd viscosity term depends on derivatives of the velocity field, no parabolic smoothing effect can be expected.In the present paper, we establish a well-posedness theory in Sobolev spaces for a system of incompressible non-homogeneous fluids with odd viscosity. The crucial point of the analysis is the introduction of a set of good unknowns, which allow for the emerging of a hidden hyperbolic structure underlying the system of equations. It is exactly this hyperbolic structure which makes it possible to circumvent the derivative loss and propagate high enough Sobolev norms of the solution. The well-posedness result is local in time; two continuation criteria are also established.

Solving Abel integral equations by regularisation in Hilbert scales

Integral operators of Abel type of order a>0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a<1 to arbitrary a>0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori information associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori smoothness is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.

Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane 0\\\\ u\\left(0,\\mathbf{x}\\right) = u_{0}\\left(\\mathbf{x}\\right),\\mathbf{x\\in D},\\\\ u\\left\\vert _{x_{i} = 0}\\right. = h_{j}\\left(t,x_{j}\\right),\\ x_{j}>0,\\ j = 1,2,t>0, \\end{array} \\right. $?> {ut(t,x)−∇βu(t,x)=|u(t,x)|σu(t,x),x∈D,t>0u(0,x)=u0(x),x∈D,u|xi=0=hj(t,xj), xj>0, j=1,2,t>0, where D={x1>0,x2>0} , β∈(32,2), σ>0 and ∇β is a fractional Laplacian defined as ∇βu=∑j1Γ(2−β)∫0xjuyjyj(xj−yj)β−1dy. We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the L2− based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Existence and Uniqueness Solutions for Some Strongly Quasilinear Parabolic Problems in Anisotropic Sobolev Spaces

Volterra and Composition Inner Derivations on the Fock–Sobolev Spaces

On the Fock–Sobolev spaces, we study the range of Volterra inner derivations and composition inner derivations. The Volterra inner derivation ranges in the ideal of compact operators if and only if the induced function g is a linear polynomial. The composition inner derivation ranges in the ideal of compact operators if and only if the induced function φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is either identity or a contractive linear self-mapping of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document}. Moreover, we describe the compact intertwining relations for composition operators and Volterra operators between different Fock–Sobolev spaces. In this paper, our results are complement and in a sense extend some aspects of Calkin’s result (Ann Math 42:839–873, 1941) to the algebras of bounded linear operators on Fock–Sobolev spaces.

On consistency for wavelet estimator of regression function based on biased samples under extended negatively dependence

ABSTRACT For a nonparametric regression model with biased samples under extended negatively dependence, we concern with the estimation problem of the regression function over Sobolev space. On the basis of the wavelet kernel, we propose a wavelet estimator and investigate its consistency properties under mild conditions. For the proposed wavelet estimator, we establish the pth mean consistency based on moment inequality and get the complete consistency by virtue of exponential inequality and truncation technique. It should be pointed out that the complete consistency of wavelet estimator is a novel theoretical result in the nonparametric regression model with biased samples. In addition, the effectiveness of the proposed wavelet estimator is demonstrated on some numerical experiments.

Sobolev spaces on arbitrary domains and semigroups generated by the fractional Laplacian

We describe a procedure to introduce on an arbitrary domain of Rd Sobolev spaces of arbitrary order s∈R and with an intrinsic characterization via fractional powers of the Dirichlet Laplacian. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup generated by the fractional Dirichlet Laplacian.

Multiple Solutions for a Class of Biharmonic Nonlocal Elliptic Systems

We prove the existence of three distinct solutions for a biharmonic nonlocal elliptic system with singular terms under the Navier boundary conditions, by using variational methods and the theory of the variable exponent Sobolev space.

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

Continuity and regularity for local multi-fractional new maximal operators

In this paper, it is proved that the following form of local multilinear fractional new maximal operatorMφ,β,Ω(f→)(x)=sup0<r<dist(x,Ωc)⁡rβ(1+rn)γβnφ(|B(x,r)|)m|B(x,r)|m×∏i=1m∫B(x,r)|fi(y)|dy is bounded from W1,p1(Ω)×W1,p2(Ω)×⋯×W1,pm(Ω) into W1,q(Ω) by applying the pointwise gradient inequalities for Mφ,β,Ω, and it is also bounded on Sobolev space W1,q(Ω) as fi∈Lpi(Ω). Meanwhile, the pointwise gradient estimate and the Sobolev boundedness for the local multilinear new maximal operator Mφ,Ω are given. Moreover, the authors also investigate the continuity of Mφ,β,Ω and Mφ,Ω on Sobolev space W1,q(Ω). As applications, the bounds for Mφ,Ω and Mφ,β,Ω on Sobolev space with zero boundary values W01,q(Ω) will be showed.

Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s>12 , to those of BO in the deep-water limit (as δ→∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s>32 in the deep-water limit and s⩾2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂x(uk) for k⩾2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s⩾34 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.

On a class of obstacle problem via Young measure in generalized Sobolev space

On a class of obstacle problem via Young measure in generalized Sobolev space