On the approximation of solutions of one singular differential equation on the axis by A.S. Kassym, L.K. Kussainova

“…The weighted Sobolev space corresponding to the posed problem is determined. …”
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Extension of the Hodge theorem to certain non-compact manifolds by Shapiro, Yakov (Yakov Mikhaylovich)

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Global flows with invariant measures for the inviscid modified SQG equations by Nahmod, Andrea R, Pavlović, Nataša, Totz, Nathan, Staffilani, Gigliola

“…For this family of equations we construct an invariant measure on a rough L[superscript 2]-based Sobolev space and establish the existence of solutions of arbitrarily large lifespan for initial data in a set of full measure in the rough Sobolev space.…”
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Expected integration approximation under general equal measure partition by Xiaoda Xu, Dianqi Han, Zongyou Li, Xiangqin Lin, Zhidong Qi, Lai Zhang

“…In this paper, we first use an L2−discrepancy bound to give the expected uniform integration approximation for functions in the Sobolev space H1(K) equipped with a reproducing kernel. …”
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Global stability solution of the 2D incompressible anisotropic magneto-micropolar fluid equations by Ru Bai, Tiantian Chen, Sen Liu

“…Our results reveal that any perturbations near a background magnetic field is globally stable in Sobolev space $ H^2({\mathbb{R}}^2) $.…”
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A Note on the Relaxation-Time Limit of the Isothermal Euler Equations by Fang Daoyuan, Xu Jiang

“…We show that Coulombel-Goudon's results (2007) can hold in the <it>weaker</it> and <it>more general</it> Sobolev space of fractional order. The method of proof used is the Littlewood-Paley decomposition.…”
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The Boundary Value Problem of the Equations with Nonnegative Characteristic Form by Limei Li, Tian Ma

“…Then by using the inverse H&#246;lder inequalities, we obtain the regularity of the weak solution in the anisotropic Sobolev space.…”
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Wavelets for nonuniform non-stationary MRA on $H^s(\mathbb{K})$ by Ashish Pathak, Guru P. Singh

“… In this paper, we are defined the nonuniform non-stationary multiresolution analysis (NUNSMRA) on Sobolev space over local fields ($H^s(\mathbb{K})$) and with help of NUNSMRA orthonormal wavelets are constructed. …”
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The Informational Complexity of Learning from Examples by Niyogi, Partha

“…The tasks chosen vary from learning functions in a Sobolev space using radial basis function networks to learning grammars in the principles and parameters framework of modern linguistic theory. …”
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Functions of Bounded kth p-Variation and Continuity Modulus by Odalis Mejía, Pilar Silvestre

“…A scale of spaces exists connecting the class of functions of bounded kth p-variation in the sense of Riesz-Merentes with the Sobolev space of functions with p-integrable kth derivative. …”
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A randomized trapezoidal quadrature by Wu, Y

“…Indeed, we consider functions in some fractional Sobolev space. Various error bounds for the randomized rule are established while an error bound for classical trapezoidal quadrature is obtained for comparison. …”

On a New Parabolic Sobolev Embedding Map by El Aidi Mohammed

“…The purpose of the present article is to provide a new parabolic Sobolev embedding map between a parabolic weighted Sobolev space and the space of square-integrable functions on a cylinder. …”
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The local strong solution and wave breaking feature to a Camassa–Holm type equation by Jin Hong, Shaoyong Lai

“…Certain estimates in Sobolev space, which are derived from the equation itself, are obtained. …”
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Bounded variation and the strength of Helly's selection theorem by Alexander P. Kreuzer

“…For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. …”
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Resolvent estimates for elliptic systems in function spaces of higher regularity by Robert Denk, Michael Dreher

“…We consider parameter-elliptic boundary value problems and uniform a priori estimates in $L^p$-Sobolev spaces of Bessel potential and Besov type. The problems considered are systems of uniform order and mixed-order systems (Douglis-Nirenberg systems). …”
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On coercive variational integrals by Kristensen, J, Chen, C

“…<p>It is well-known that sequential weak lower semicontinuity of a variational integral</p> <p>𝕱(u, Ω) = ∫Ω F (∇u(x)) dx</p> <p>on the Sobolev space W^1,p (Ω, ℝ^N) under a p-growth condition on the integrand F is equivalent to quasiconvexity in the sense of Morrey. …”

Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions by Yeguo Sun

“…Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. …”
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On nonlinear evolution variational inequalities involving variable exponent by Mingqi Xiang

“…In this paper, we discuss a class of quasilinear evolution variational inequalities with variable exponent growth conditions in a generalized Sobolev space. We obtain the existence of weak solutions by means of penalty method. …”
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Existence of weak solutions for a nonuniformly elliptic nonlinear system in R^N by Nguyen Thanh Chung

“…We study the nonuniformly elliptic, nonlinear system $$displaylines{ - hbox{div}(h_1(x) abla u)+ a(x)u = f(x,u,v) quad ext{in } mathbb{R}^N,cr - hbox{div}(h_2(x) abla v)+ b(x)v = g(x,u,v) quad ext{in } mathbb{R}^N. }$$ Under growth and regularity conditions on the nonlinearities $f$ and $g$, we obtain weak solutions in a subspace of the Sobolev space $H^1(mathbb{R}^N, mathbb{R}^2)$ by applying a variant of the Mountain Pass Theorem.…”
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On the Morse–Sard property and level sets of Wn,1 Sobolev functions on ℝn by Bourgain, J, Korobkov, M, Kristensen, J

“…We establish Luzin N and Morse–Sard properties for functions from the Sobolev space Wn,1(Rn). Using these results we prove that almost all level sets are finite disjoint unions of C1-smooth compact manifolds of dimension n - 1. …”