Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems

Amann, Herbert

doi:10.1007/978-3-0348-0939-9_4

Cited by 22 publications

(31 citation statements)

References 28 publications

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“…See Section 3 for the precise definition. The L p theory of uniformly strongly ρ-elliptic operators has been established by H. Amann in [4].…”

Section: Introductionmentioning

confidence: 99%

“…This is, in fact, one of the most challenging tasks in the form operator approach. One of the most important features of this article is that with the assistance of the theory for function spaces and differential operators on singular manifolds established in [2,3,4], we can find a precise characterization for the domains of the second order (ρ, λ)-singular elliptic operators.…”

Section: Introductionmentioning

confidence: 99%

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Singular parabolic equations of second order on manifolds with singularities

Shao

2016

Journal of Differential Equations

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Abstract. The main aim of this article is to establish an Lp-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the singular ends of the manifolds. Such a theory is of importance for the study of elliptic and parabolic equations on non-compact, or even incomplete manifolds, with or without boundary.

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“…See Section 3 for the precise definition. The L p theory of uniformly strongly ρ-elliptic operators has been established by H. Amann in [4].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Singular parabolic equations of second order on manifolds with singularities

Shao

2016

Journal of Differential Equations

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“…A 0 := A | M0 is uniformly strongly ρ-elliptic in the sense of [4, formula (5.1)]. It is proved in [4,Theorem 5.2] that…”

Section: By the Previous Discussion And Proposition 33 We Infer Thatmentioning

confidence: 99%

The Yamabe flow on incomplete manifolds

Shao

2018

J. Evol. Equ.

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“…Remark 6.14. As is well-known, coercivity estimates lead to solutions of evolution equations [6,7,46,59]. Let H 1 0 (M ; E) ⊂ V ⊂ H 1 (M ; E) be the space defining our variational boundary value problem, see 4.1, Equation (12), and Remark 4.13.…”

Section: A Uniform Agmon Conditionmentioning

confidence: 98%

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

Große

Nistor

2019

Potential Anal

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We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P, C) is equivalent to the uniform regularity of the natural family (Px, Cx) of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincaré inequality of (Ammann-Große-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, wellknown theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition. 12 4. Variational boundary conditions and regularity 13 4.

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Parabolic Equations on Uniformly Regular Riemannian Manifolds and Degenerate Initial Boundary Value Problems

Cited by 22 publications

References 28 publications

Singular parabolic equations of second order on manifolds with singularities

Singular parabolic equations of second order on manifolds with singularities

The Yamabe flow on incomplete manifolds

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

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