Integral distance on a Lipschitz Riemannian manifold

“…The next two propositions show that the upper semicontinuity property of the length functional L ϕ plays a role in this issue. These results are essentially known [13][14][15]; they have been restated here for the reader's convenience. Proof.…”

Section: Lemma 32mentioning

confidence: 99%

Monge solutions for discontinuous Hamiltonians

Briani

Davini

2005

ESAIM: COCV

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“…This means that X * has a manifold structure with locally Lipschitz transition maps, and that it is equipped with a Riemannian metric g so that in coordinate charts, g and g −1 are in L ∞ loc . In addition, d X is compatible with the metric d X * on X * coming from g [8], in the sense that d X and d X * coincide on some neighborhood of the diagonal in X * × X * . In particular, if F is a function with compact support in a coordinate neighborhood of X * , and F is Lipschitz in terms of the coordinates, then F is a Lipschitz function on X.…”

Section: Differential Form Laplacian On An Alexandrov Spacementioning

confidence: 99%

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

Lott

2018

Math. Ann.

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We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.

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“…We recall here the definition of intrinsic distance introduced by De Cecco and Palmieri and its main properties (for details, see [24], [25], [26], [27]). …”

Section: Intrinsic Distancesmentioning

confidence: 99%

Homogenization of two-phase metrics and applications

Davini

Ponsiglione

2007

J Anal Math

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We consider two-phase metrics of the form phi( x, xi) := alpha chi B-alpha(x) vertical bar xi vertical bar + beta chi B-alpha(x) |xi|, where alpha, beta are fixed positive constants and B-alpha, B-beta are disjoint Borel sets whose union is R-N, and prove that they are dense in the class of symmetric Finsler metrics phi satisfying alpha vertical bar xi vertical bar <= phi(x,xi) <= beta vertical bar xi vertical bar on R-N x R-N. Then we study the closure Cl(M-theta(alpha, beta)) of the class M-theta(alpha, beta) of two-phase periodic metrics with prescribed volume fraction theta of the phase alpha. We give upper and lower bounds for the class Cl(M-theta(alpha, beta)) and localize the problem, generalizing the bounds to the non-periodic setting. Finally, we apply our results to study the closure, in terms of Gamma-convergence, of two- phase gradient- constraints in composites of the type f (x, del u) <= C(x), with C(x) is an element of {alpha, beta} for almost every x

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Integral distance on a Lipschitz Riemannian manifold

Cited by 32 publications

References 1 publication

Monge solutions for discontinuous Hamiltonians

Monge solutions for discontinuous Hamiltonians

Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

Homogenization of two-phase metrics and applications

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