Lower semicontinuity of integrals of the calculus of variations in Cheeger–Sobolev spaces

Hafsa, Omar Anza; Mandallena, Jean Philippe

doi:10.1007/s00526-020-1702-1

Cited by 3 publications

(1 citation statement)

References 28 publications

(24 reference statements)

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“…In general, lower semicontinuity of integral functionals in metric measure spaces is not yet completely understood; the most classical example of a lower semicontinuous functional is the Cheeger energy [4]. Some positive results for more general functionals can be found in [24,25] and [26].…”

Section: We Rewrite Them Respectively Asmentioning

confidence: 99%

Weak solutions to gradient flows in metric measure spaces

Górny

Mazón

2023

Proc Appl Math and Mech

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Due to the fact that in a metric space there is (in general) no notion of directional derivatives, the definition of solutions to gradient flows in metric measure spaces necessarily avoid their direct use. The most classical example is the heat flow, or more generally the p‐Laplacian evolution equation, which has been studied as the gradient flow in L2 of the p‐Cheeger energy. Typically, solutions are defined using the semigroup theory through a subdifferential of the energy. Other popular approaches include variational solutions and the weighted energy‐dissipation formalism. In this paper, using the first‐order differential structure on a metric measure space introduced by Gigli, we characterize the subdifferential in L2 of the p‐Cheeger energy. This gives rise to a new notion of solutions to the p‐Laplacian evolution equation in metric measure spaces. In the case p = 1, we introduce a metric analogue of the Anzellotti pairing and obtain a Green‐Gauss formula, which is then used in place of Gigli's structure to characterise the 1‐Laplacian operator and study the total variation flow.

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Section: We Rewrite Them Respectively Asmentioning

confidence: 99%

Weak solutions to gradient flows in metric measure spaces

Górny

Mazón

2023

Proc Appl Math and Mech

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Integral representation and relaxation of local functionals on Cheeger–Sobolev spaces

Hafsa

Mandallena

2022

Nonlinear Analysis

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The ε-ε property and the boundedness of isoperimetric sets with different monomial weights

Abreu¹,

Fernandes²,

Ramirez³

2023

ESAIM: COCV

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We consider a class of monomial weights $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$ in $\mathbb{R}^{N}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the $\varepsilon-\varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for $N=2$, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.

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Lower semicontinuity of integrals of the calculus of variations in Cheeger–Sobolev spaces

Cited by 3 publications

References 28 publications

Weak solutions to gradient flows in metric measure spaces

Weak solutions to gradient flows in metric measure spaces

Integral representation and relaxation of local functionals on Cheeger–Sobolev spaces

The ε-ε property and the boundedness of isoperimetric sets with different monomial weights

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