Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

Eldredge, Nathaniel; Saloff‐Coste, Laurent

doi:10.1007/s10959-013-0484-1

Cited by 10 publications

(9 citation statements)

References 29 publications

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“…In particular, we explain at the end of this section that under a very natural assumption on existence of cut-off functions, and when we consider the right-hand side f to be locally in L 2 (I → F ), our choice of definition of local weak solutions agrees with the definition used in other papers. This is proved by adapting the proof of Lemma 1 in [13].…”

Section: Function Spaces Associated With (E F)mentioning

confidence: 99%

Time regularity for local weak solutions of the heat equation on local Dirichlet spaces

Hou¹,

Saloff‐Coste²

2019

Preprint

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We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak L 2 Gaussian type upper-bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to the local weak solutions of a uniformly elliptic symmetric divergence form second order operator with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of [8] and [31].

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Section: Function Spaces Associated With (E F)mentioning

confidence: 99%

Time regularity for local weak solutions of the heat equation on local Dirichlet spaces

Hou¹,

Saloff‐Coste²

2019

Preprint

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“…To save space, we refer the reader to [3,12,19,20,21] for the definition of local weak solutions of the heat equation in an open cylindrical domain (a, b)×U ⊂ R×Ω, in the context of the strictly local regular Dirichlet space (W 1 0 (Ω), Ω g(∇f, ∇f )dµ). Because such weak solutions are automatically smooth in time, one can be a bit cavalier with the details of such definitions.…”

Section: Local and Global Solutions Of The Heat Equationmentioning

confidence: 99%

Heat kernel estimates on manifolds with ends with mixed boundary condition

Dautenhahn¹,

Saloff‐Coste²

2021

Preprint

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We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and Saloff-Coste by allowing for Dirichlet boundary condition. The proof requires the construction of a global harmonic function which is then used in the h-transform technique. introductionIn [6], Alexander Grigor'yan and the second author initiated the study of twosided heat kernel estimates on weighted complete Riemannian manifolds with finitely many nice ends,The components M i of this connected sum are, themselves, assumed to be weighted complete Riemannian manifolds. The main assumption is that, on each M i , the heat kernel p Mi (t, x, y), is well understood in the sense that it satisfies a classical-looking two-sided Gaussian estimate, uniformly at all times and locations. Equivalently ([4, 17, 18]), the volume functions of these manifolds, M i , 1 ≤ i ≤ k, are uniformly doubling at all scales and locations AND their geodesic balls satisfy a Neumann-type Poincaré inequality, uniformly at all all scales and locations. These are very strong hypotheses, and, in certain cases, additional more technical hypotheses are needed. The results of [6] are sharp twosided estimates on the heat kernel of M. The most basic case illustrating these results is when M i = R N for some N, and, more generally, M i = R ni × S N −ni , for some N and n i , 1 ≤ n i ≤ N . These basic cases were new and already plenty challenging at the time [6] was published. They are richer than they appear if one takes into consideration the variation afforded by the weight functions. In addition, the results hold without change when the term "complete Riemannian manifold" is interpreted in the context of manifolds with boundary. Complete, then, means metrically complete, and the heat equations and heat kernels on M and on the M i , 1 ≤ i ≤ k, are all taken with Neumann boundary condition. So, for instance, the results of [6, 10] cover the solid three-dimensional body in Figure 1. (This figure created by A. Grigor'yan appears in [10].)The aim of the present work is to initiate the study of the case when the heat equation on the complete manifold M (with boundary) above is taken with mixed boundary condition: Neumann on some part of the boundary and Dirichlet on the rest of the boundary.

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“…There are different ways of defining local weak solutions and these are not exactly equivalent. See, e.g., [39,13,3,9].…”

Section: 31mentioning

confidence: 99%

“…In the last equality, we applied the chain rule for L * with Φ(x) = x p for x ≥ 0 and Φ(x) = 0 for x < 0 (for 0 = p < 2), and Φ(x) = x p (for p ≥ 2). Combining (8) and (9), and applying the chain rule and the Leibniz rule for L * , we obtain…”

Section: This Shows Thatmentioning

confidence: 99%