Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

Rosa, Antonio; Kolasiński, Sławomir; Santilli, Mario

doi:10.1007/s00205-020-01562-y

Cited by 18 publications

(16 citation statements)

References 31 publications

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“…x, e n = a. This implies that, with the change of coordinate z = y−x, if y, e n < a, then z, e n < 0 and then we can write We refer the interested reader to [11,12,13] for the characterization of critical points of the isoperimetric porblem in other settings.…”

Section: T)jφ(y T)dxdymentioning

confidence: 99%

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A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Rosa¹,

Manna²

2021

CPAA

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We study a non local approximation of the Gaussian perimeter, proving the Gamma convergence to the local one. Surprisingly, in contrast with the local setting, the halfspace turns out to be a volume constrained stationary point if and only if the boundary hyperplane passes through the origin. In particular, this implies that Ehrhard symmetrization can in general increase the non local Gaussian perimeter taken into consideration.

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Section: T)jφ(y T)dxdymentioning

confidence: 99%

“…it has been proved in [1,Lemmata 7,11,12] that Γ n = ω n−1 . Hence we conclude the claimed inequality (2.7).…”

mentioning

confidence: 99%

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Rosa¹,

Manna²

2021

CPAA

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“…The general geometric approach used to prove Theorem 1.1 promises to have several other applications. For example it has been recently used to obtain the characterization of the critical points of the anisotropic isoperimetric functional in [5].…”

Section: Then S Can Be H M Almost Covered By a Countable Collection Of M Dimensional Submanifolds Of Classmentioning

confidence: 99%

“…and we say that x ∈ U (A) is a regular point of ξ A if and only if ξ A is approximately differentiable 5 at x with symmetric approximate differential and ap lim y→x ρ(A, y) ≥ ρ(A, x) > 1. The set of regular points of ξ A is denoted by R(A).…”

Section: Curvatures Of Arbitrary Closed Setsmentioning

confidence: 99%

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Second order rectifiability of varifolds of bounded mean curvature

Santilli

2021

Calc. Var.

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We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $$ {\mathscr {H}}^{m} $$ H m almost everywhere by a countable union of m dimensional submanifolds of class $$ {\mathcal {C}}^{2} $$ C 2 . The $$ {\mathcal {C}}^{2} $$ C 2 -regularity of the submanifolds is optimal.

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The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

De Philippis,

De Rosa

2023

Comm Pure Appl Math

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We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.

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Uniqueness of Critical Points of the Anisotropic Isoperimetric Problem for Finite Perimeter Sets

Cited by 18 publications

References 31 publications

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

A non local approximation of the Gaussian perimeter: Gamma convergence and Isoperimetric properties

Second order rectifiability of varifolds of bounded mean curvature

The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

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