Weighted Sobolev Spaces and Degenerate Elliptic Equations

We study a symmetric diffusion X on R d in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients a ω . The diffusion is formally associated with L ω u = ∇ · (a ω ∇u), and we make sense of it through Dirichlet forms theory. We prove for X a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.Nousétudions une diffusion symétrique X sur R d en forme de divergence dans un environnement aléatoire stationnaire et ergodique, dont les coefficients a ω sont mesurables et dégénérés. Cette diffusion qui est formellement engendrée par l'opérateur L ω u = ∇ · (a ω ∇u), peutêtre définieà l'aide de la théorie des formes de Dirichlet. Nous démontrons pour X un principe d'invariance presque sûr sous des conditions de moment de l'environnement; l'outil crucial est la sous-linéarité du correcteur obtenuà l'aide de l' itération introduite par J. Moser.

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“…can prove as in[3, Lemma 15] that lim ε→0 sup s∈[0,t]|ε 2 A ω s/ε 2 − sE µ [θ]| = 0, P ω x -a.s, a.a. x ∈ R d ,(5 5). …”

mentioning

confidence: 99%

Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium

Chiarini¹,

Deuschel²

2016

Ann. Inst. H. Poincaré Probab. Statist.

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“…Let us begin with the definition of the following weighted Sobolev space. Definition (see ). Let

k \geq 1 (k \in N)

and

1 \leq p < + \infty

.…”

Section: Introduction Preliminaries and Statement Of Resultsmentioning

confidence: 99%

The local superconvergence of the quadratic triangular element for the poisson problem in a polygonal domain

Cui

Zhu

2014

Numerical Methods Partial

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It is the first time for us to combine the local symmetric technique and the weak estimates to investigate the local superconvergence of the finite element method for the Poisson equation in a bounded domain with polygonal boundary where a uniform family of partitions is not required or the solution need not have high global smoothness. Combining a uniform family of triangulations in the interior of domain with a quasiuniform family of triangulations at the boundary of domain, we present a special family of triangulations. By the finite element theory of the derivative of the Green's function presented in this article, we combine the local symmetric technique and the weak estimates to obtain the local superconvergence of the derivative for the quadratic elements.

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“…In recent years, there are increasing interests to the models with degenerate variable. For equation (1) with degeneracy, the existence and uniqueness of solutions have been studied extensively, see for example, [3,6,15,16] for the elliptic case and [8,16,22] for the parabolic problem.…”

Section: Xin LI Chunyou Sun and Na Zhangmentioning

confidence: 99%

Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

Li¹,

Sun²,

Zhang³

2016

DCDS-A

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In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation ut − div(σ(x)∇u) + f (u) = g(x, t) defined on a bounded domain Ω ⊂ R N with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process U (t, τ) is continuous from L 2 (Ω) to D 1 0 (Ω, σ) w.r.t. initial data; And finally show that the known (L 2 (Ω), L 2 (Ω)) pullback D λ-attractor indeed can attract in D 1 0 (Ω, σ)-norm. Any differentiability on the forcing term is not required.

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Weighted Sobolev Spaces and Degenerate Elliptic Equations

Cited by 24 publications

References 18 publications

Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium

Invariance principle for symmetric diffusions in a degenerate and unbounded stationary and ergodic random medium

The local superconvergence of the quadratic triangular element for the poisson problem in a polygonal domain

Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

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