Weighted Sobolev spaces and regularity for polyhedral domains

We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ). We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that. Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

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“…Then it is proved in [3] that T : H r → H r−m is bounded, for any T of order m. In [1] it is proved using partitions of unity that T : …”

Section: Identifies With the Domain Of P With The Graph Topology And Hmentioning

confidence: 99%

Pseudodifferential operators on manifolds with a Lie structure at infinity

2007

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“…Recall that we defined the set S = {(1, 0), (1, l), [0, y], 0 ≤ y ≤ l} in Section 2. It has been shown in [2,3,14] that the trace or restriction of u ∈ K m a on the boundary follows…”

Section: The Well-posedness and Regularity Of The Solutionmentioning

confidence: 99%

“…Therefore, u has no singularity on Ω\P , and no further study is needed in general on this region. In fact, Lemma 2.11 is particular for the analysis of the solution near x = 0, while one can refer to [3,14,31] for the solution around the vertices (1, 0), (1, l).…”

Section: This Relation Also Holds Formentioning

confidence: 99%

A-priori analysis and the finite element method for a class of degenerate elliptic equations

2008

Math. Comp.

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Abstract. Consider the degenerate elliptic operatorWe prove well-posedness and regularity results for the degenerate elliptic equation L δ u = f in Ω, u| ∂Ω = 0 using weighted Sobolev spaces K m a . In particular, by a proper choice of the parameters in the weighted Sobolev spaces K m a , we establish the existence and uniqueness of the solution. In addition, we show that there is no loss of K m a -regularity for the solution of the equation. We then provide an explicit construction of a sequence of finite dimensional subspaces V n for the finite element method, such that the optimal convergence rate is attained for the finite element solution u n ∈ V n , i.e., ||uwith C independent of f and n.

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“…□ Next, by changing problem (2) - (4) into the Dirichlet problem for second order elliptic depending on time parameter, we can apply the results for this problem in polyhedral domains (cf. [4,5]) and our previous ones to deal with the regularity with respect to both of time and spatial variables of the solution. …”

Section: Then (13) Impliesmentioning

confidence: 99%

“…Next, by this method we obtain the regularity in time of the solution. Finally, we apply the results for elliptic boundary value problems in polyhedral domains given in [4,5] and former our results to deal with the global regularity of the solution.…”

Section: Introductionmentioning

confidence: 99%