فضاء الطاقة لمؤثر هرميت في R^n وفضاءات سوبوليڤ موافقة by إبراهيم إبراهيم, طالب غريبة, آصف المحمد

“…We will see that   has similar properties as    for real numbers s > o, therefore we can construct new Hilbert spaces        which are the energy spaces of powers of . They are Sobolev spaces. We  can  also  generalize  those  spaces  to   for  . …”
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An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition by Zou Shuimu, Zhang Jun

“…First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. …”
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Contracting differential equations in weighted Banach spaces by Srinivasan, Anand, Slotine, Jean-Jacques

“…We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. …”
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Anticipating random periodic solutions I: SDEs with multiplicative linear noise by Feng, C, Wu, Y, Zhao, H

“…We then solve a localised forward–backward IHRIE in C9R,L^2loc (Ω)) using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener–Sobolev spaces in C([0,T], L^2 (Ω)) and Schauder's fixed point theorem. …”

Singularity formation, existence and regularity for degenerate Cahn-Hilliard equations by Pesce Reyes, CL

“…We complement this approach by adapting the known theory for degenerate parabolic partial differential equations and Sobolev spaces of radial functions to obtain the existence and regularity of a weak radially symmetric solution. …”

Boundary Regularity in Variational Problems by Kristensen, J, Mingione, G

“…Moreover, our approach allows for a treatment of systems and functionals with "rough" coefficients belonging to suitable Sobolev spaces of fractional order. © 2010 Springer-Verlag.…”

Stability for nonlinear diffusive PDEs by Alasio, L

“…The approach is typical of Analysis of PDEs, hence the concepts of weak solutions in Sobolev spaces, a priori estimates and well-posedness are crucial. …”

On a nonlocal boundary-value problem with constant coefficients for a multidimensional mixed type equation by Sirojiddin Z Dzhamalov

“…In this paper the unique solvability and smoothness of generalized solution of a nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind in Sobolev spaces $W_{2}^{l }(Q)$, ($2\le l $ is integer number), have been proved. …”
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On regular solutions to compressible radiation hydrodynamic equations with far field vacuum by Li Hao, Zhu Shengguo

“…When both shear and bulk viscosity coefficients depend on the mass density ρ\rho in a power law ρδ{\rho }^{\delta } (with 0<δ<10\lt \delta \lt 1), based on some elaborate analysis of this system’s intrinsic singular structures, we establish the local-in-time well-posedness of regular solution with arbitrarily large initial data and far field vacuum in some inhomogeneous Sobolev spaces by introducing some new variables and initial compatibility conditions. …”
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Local solutions for a hyperbolic equation by Aldo Trajano Lourêdo, Haroldo Rodrigues Clark, Manuel Milla Miranda, Marcondes Rodrigues Clark

“…As usual, restrictions on $\rho$ are considered in order to have the continuous embedding of Sobolev spaces.…”
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Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument by P.Ya. Pukach, S.M. Repetylo, M.M. Symotiuk, M.I. Vovk

“…These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. …”
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Homogenization and bounds for multiscale problems by Ang, Eng Hong

“…Next, we looked at Sobolev spaces which form the basis of weak solutions for the problem in its variational form. …”
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Generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds by Behrndt, T

“…By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities.…”

Radiation field for Einstein vacuum equations by Wang, Fang, Ph. D. Massachusetts Institute of Technology

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On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications by Chrusciel, P, Delay, E

“…Generalising an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted Sobolev spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^0$; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global Scri; we prove existence of initial data for many black holes which are exactly Kerr -- or exactly Schwarzschild -- both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localised, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{-m}$ terms, for any fixed $m$, and with multipole moments freely prescribable within certain ranges.…”

Stability of multidimensional thermoelastic contact discontinuities by Chen, G-QG, Secchi, P, Wang, T

“…We identify a stability condition on the piecewise constant background states and establish the linear stability of thermoelastic contact discontinuities in the sense that the variable coefficient linearized problem satisfies a priori tame estimates in the usual Sobolev spaces under small perturbations. Our tame estimates for the linearized problem do not break down when the strength of thermoelastic contact discontinuities tends to zero. …”

Parameter–Elliptic Fourier Multipliers Systems and Generation of Analytic and <i>C</i>^∞ Semigroups by Bienvenido Barraza Martínez, Jonathan González Ospino, Rogelio Grau Acuña, Jairo Hernández Monzón

“…As examples, we apply the theory to solve the heat equation, a linear thermoelastic plate equation, a structurally damped plate equation, and a generalized plate equation, all in the whole space, in the frame of Sobolev spaces.…”
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Asymptotic Behavior of Solutions to a Nonlinear Swelling Soil System with Time Delay and Variable Exponents by Mohammad M. Kafini, Mohammed M. Al-Gharabli, Adel M. Al-Mahdi

“…The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools for studying such problems. …”
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Complexity theory for spaces of integrable functions by Florian Steinberg

“…The family is extended to cover Sobolev spaces on the unit interval, where less basic operations like differentiation and some Sobolev embeddings are shown to be polynomial-time computable. …”
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A mixed-primal finite element approximation of a sedimentation-consolidation system by Alvarez, M, Gatica, G, Ruiz Baier, R

“…Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart-Thomas approximations of order k for the stress and piecewise continuous polynomials of order k + 1 for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. …”