Generalized Sobolev Spaces Research Articles

Variable exponent Sobolev spaces associated with Jacobi expansions

In this paper we define variable exponent Sobolev spaces associated with Jacobi expansions. We prove that our generalized Sobolev spaces can be characterized as variable exponent potential spaces and as variable exponent Triebel-Lizorkin type spaces.

A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

Weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation with nonstandard growth

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in [45] as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

Existence and multiplicity of solutions for nonlocal Neumann problem with non-standard growth

In this paper, we are concerned with questions of existence of solution for a nonlocal and non-homogeneous Neumann boundary value problems involving the $p(x)$-Laplacian in which the non-linear terms have critical growth. The main tools we will use are the generalized Sobolev spaces and the Mountain Pass Theorem.

Fractional Continuous Wavelet Transform on Some Function Spaces

Boundedness results for the fractional continuous wavelet transform on some test function spaces are obtained. The fractional continuous wavelet transform is also studied on some generalized Sobolev spaces.

Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces

We address the global regularity of solutions to the Boussinesq equations with zero diffusivity in two spatial dimensions. Previously, the persistence in the space $H^{1+s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ for all $s\ge 0$ has been obtained. In this paper, we address the persistence in general Sobolev spaces, establishing it on a time interval which is almost independent of the size of the initial data. Namely, we prove that if $(u_0,\rho_0)\in W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $s\in(0,1)$ and $q\in[2,\infty)$, then the solution $(u(t),\rho(t))$ of the Boussinesq system stays in $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $t\in[0,T^*)$, where $T^*$ depends logarithmically on the size of initial data. If we furthermore assume that $sq>2$, then we get the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$. Moreover, we prove the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for the initial data with compact support, as well us for data in $W^{1+s,q}(\mathbb{T}^2)\times W^{s,q}(\mathbb{T}^2)$, without any restriction on $s\in(0,1)$ and $q\in[2,\infty)$.

Sobolev spaces associated with a Dunkl type differential-difference operator on the real line and applications

In this paper we study the Sobolev spaces associated with a Dunkl type differential-difference operator on the real line. As applications we study the theory of reproducing kernels to the Tikhonov regularization, on the generalized Sobolev spaces. We give some properties including some estimates for the solution of the generalized Schrodinger equation.

The heat equation associated with Laguerre operator

In this paper we study a generalized Sobolev spaces Hs of exponential type associated with Laguerer operator based on the space of test functions and employ its properties further to estimate the heat equation's solution in the setting of the Laguerre hypergroup .

On a necessary condition in the calculus of variations in Sobolev spaces with variable exponent

We prove an approximation theorem in generalized Sobolev spaces with variable exponent $W^{1,p(\cdot )}(\varOmega )$ and we give an application of this approximation result to a necessary condition in the calculus of variations.

Differentiability of the Energy Functional of a Class of Quasi-Linear Elliptic

To study the differentiability of a class of quasi-linear elliptic equations energy functional by using the embedding theorems and some other properties of generalized Sobolev spaces. According to the variation principle the energy functional of equations is expressed in some appropriate generalized Sobolev spaces, the various parts of differentiability of the energy functional of equations are discussed under certain conditions. Using Lebesgue dominated convergence theorem and embedding theorem, the energy functional of equations is proved. Finally, the overall differentiability is proved. The conclusions lay the foundation for the next step to prove the existence of the critical point of the energy functional, that is, the existence of solutions of the equations.

A compact embedding theorem for generalized Sobolev spaces

We give an elementary proof of a compact embedding theorem in abstract Sobolev spaces. The result is first presented in a general context and later specialized to the case of degenerate Sobolev spaces defined with respect to nonnegative quadratic forms on R n . Although our primary interest concerns degenerate quadratic forms, our result also applies to nondegener- ate cases, and we consider several such applications, including the classical Rellich-Kondrachov compact embedding theorem and results for the class of s-John domains in R n , the latter for weights equal to powers of the distance to the boundary. We also derive a compactness result for Lebesgue spaces on quasimetric spaces unrelated to R n and possibly without any notion of gradient. The Rellich-Kondrachov theorem is frequently used to study the existence of solutions to elliptic equations, a famous example being subcritical and critical Yamabe equations, resulting in the solution of Yamabe's problem; see (Y), (T), (A), (S). Further applications lie in proving the existence of weak solutions to Dirichlet problems for elliptic equations with rough boundary data and coefficients; see (GT). In a sequel to this paper, we will apply our compact embedding results to study the existence of solutions for some classes of degenerate equations.

On Existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index

We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress is concentration dependent. Namely, we consider a coupled system of the generalized Navier–Stokes equations and convection–diffusion equation with non-linear diffusivity. We prove the existence of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized Sobolev spaces with variable exponent. Such a framework is involved since the function spaces, where we look for the weak solution, are “dependent” of the solution itself, and thus, we a priori do not know them. This leads us to the principal a priori assumptions on the model parameters that ensure the Hölder continuity of the variable exponent. We present here a constructive proof based on the Galerkin method that allows us to obtain the result for very general class of models.

Optimal Regularity Properties of the Generalized Sobolev Spaces

We prove optimal embeddings of the generalized Sobolev spaces <svg style="vertical-align:-0.1638pt;width:34.75px;" id="M1" height="16.9625" version="1.1" viewBox="0 0 34.75 16.9625" width="34.75" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path id="x1D44A" d="M1004 650l-6 -29q-54 -6 -71 -19.5t-51 -74.5l-271 -539h-33l-98 506h-3l-258 -506h-30l-78 532q-10 67 -21.5 80t-66.5 21l6 29h241l-8 -29l-26 -5q-34 -6 -41 -16t-3 -47l59 -425h4l251 510h31l102 -510h2q150 299 198 423q14 40 8.5 48.5t-47.5 16.5l-28 5l7 29h231z&#xA;" /></g> <g transform="matrix(.012,-0,0,-.012,17.338,8.537)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g> <g transform="matrix(.017,-0,0,-.017,23.938,16.7)"><path id="x1D438" d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5&#xA;l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" /></g> </svg>, where <svg style="vertical-align:-0.0pt;width:10.8625px;" id="M2" height="11.175" version="1.1" viewBox="0 0 10.8625 11.175" width="10.8625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D438"/></g> </svg> is a rearrangement invariant function space, into the generalized H&#xf6;lder-Zygmund space <svg style="vertical-align:-0.20474pt;width:28.575001px;" id="M3" height="12.0625" version="1.1" viewBox="0 0 28.575001 12.0625" width="28.575001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x1D49E" d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5&#xA;q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57&#xA;q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z" /></g><g transform="matrix(.017,-0,0,-.017,13.611,11.738)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g> </svg> generated by a function space <svg style="vertical-align:-0.0pt;width:15.0375px;" id="M4" height="11.175" version="1.1" viewBox="0 0 15.0375 11.175" width="15.0375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D43B"/></g> </svg>.

Algebra properties for Sobolev spaces — applications to semilinear PDEs on manifolds

In this work, we aim to prove algebra properties for generalized Sobolev spaces W s,p ∩ L ∞ on a Riemannian manifold (or more general homogeneous type space as graphs), where W s,p is of Bessel-type W s,p := (1+L)−s/m (L p ) with an operator L generating a heat semigroup satisfying off-diagonal decays. We do not require any assumption on the gradient of the semigroup. Instead, we propose two different approaches (one by paraproducts associated to the heat semigroup and another one using functionals). We also study the action of nonlinearities on these spaces and give applications to semi-linear PDEs. These results are new on Riemannian manifolds (with a non-bounded geometry) and even in euclidean space for Sobolev spaces associated to second order uniformly elliptic operators in divergence form.

Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.

L&lt;SUP align=right&gt;p&lt;/SUP&gt; estimates of solutions to some non-linear wave equations in one space dimension

Decay estimates like a negative power of t are studied in some general Sobolev spaces for a solution of the one-dimensional semi-linear wave equation with power-like damping term and Dirichlet homogeneous boundary conditions. Upper estimates of this kind are valid for smooth initial data, and we establish also weaker estimates from below on the time-derivative which differ from the above upper estimates by a 3/2 factor with respect to the power of t.

Nonlinear and multiplicative inequalities in Sobolev spaces associated with Lie algebras

This paper is devoted to multiplicative inequalities in some generalized Sobolev spaces associated with Lie algebras. These Lie algebras are generated by the differential operator of variable coefficients or by pseudo-differential operators having non-regular symbols. Under geometrical assumptions we show that the norms of two suitable classes of generalized Sobolev spaces are equivalent. This leads to the proof that the composition operator u → | u | p acts on such spaces.

On a class of almost hypoelliptic operators in generalized Sobolev spaces

The paper considers differential operators in the generalized Sobolev spaces. Differring necessary and sufficient conditions on the polyhedron ℜ are found, under which the function h ℜ s satisfies the Beurling condition for s great enough. They coincide (become necessary and sufficient) for polyhedrons in ℝ + 2 . The regularity of a class of differential equations is investigated.

Existence Results for Strongly Nonlinear Elliptic Equations of Infinite Order

In this work, generalized Sobolev spaces and Sobolev spaces of infinite order are considered. Existence of solutions for strongly nonlinear equation of infinite order of the form Au+g(x,u)=f is established. Here A is an elliptic operator from a functional space of Sobolev type to its dual and g(x,s) is a lower order term satisfying a sign condition on s.

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo---Nirenberg's type in generalized Sobolev spaces, we show that u belongs to $$L^2(-a, 0, H^2(B(\sigma), {\mathbb{R}}^N)) \cap H^1(-a, 0, L^2(B(\sigma), {\mathbb{R}}^N))$$ (see Theorem 3).