The hyperplane is the only stable, smooth solution to the Isoperimetric Problem in Gaussian space

This chapter deals with the gluing of solutions and the relevant theorem (Theorem 12.1), which states the condition for a Hölder continuous solution to exist. By taking a Galilean transformation if necessary, the solution can be assumed to have zero total momentum. The cut off velocity and pressure form a smooth solution to the Euler-Reynolds equations with compact support when coupled to a smooth stress tensor. The proof of Theorem (12.1) proceeds by iterating Lemma (10.1) just as in the proof of Theorem (10.1). Applying another Galilean transformation to return to the original frame of reference, the theorem is obtained.

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Boundary conditions in topological AdS4/CFT3

Journal of High Energy Physics ◽

10.1007/jhep02(2021)156 ◽

2021 ◽

Vol 2021 (2) ◽

Author(s):

Pietro Benetti Genolini ◽

Matan Grinberg ◽

Paul Richmond

Keyword(s):

Field Theory ◽

Free Energy ◽

Boundary Conditions ◽

Smooth Solution ◽

Three Dimensional ◽

Field Theories ◽

Legendre Transformation ◽

Generating Functional ◽

Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

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Pansu–Wulff shapes in ℍ1

Advances in Calculus of Variations ◽

10.1515/acv-2020-0093 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Julián Pozuelo ◽

Manuel Ritoré

Keyword(s):

Mean Curvature ◽

Isoperimetric Problem ◽

Variational Formula ◽

Geodesic Curvature ◽

Vertical Axis ◽

Singular Part ◽

Tangent Plane ◽

Curvature Function ◽

Invariant Norm ◽

The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

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Periodic Problems with the Scalar p-Laplacian Resonant at any Eigenvalue via Critical Point Methods

Advanced Nonlinear Studies ◽

10.1515/ans-2013-0309 ◽

2013 ◽

Vol 13 (3) ◽

Author(s):

Sophia Th. Kyritsi ◽

Donal O’ Regan ◽

Nikolaos S. Papageorgiou

Keyword(s):

Critical Point ◽

Morse Theory ◽

Critical Point Theory ◽

Smooth Solution ◽

Point Theory ◽

Reaction Term ◽

Variational Techniques ◽

Periodic Problems

AbstractWe consider nonlinear periodic problems driven by the scalar p-Laplacian with a Carathéodory reaction term. Under conditions which permit resonance at infinity with respect to any eigenvalue, we show that the problem has a nontrivial smooth solution. Our approach combines variational techniques based on critical point theory with Morse theory.

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AN ISOPERIMETRIC RESULT FOR THE GAUSSIAN MEASURE AND UNCONDITIONAL SETS

Bulletin of the London Mathematical Society ◽

10.1017/s0024609301008141 ◽

2001 ◽

Vol 33 (4) ◽

pp. 408-416 ◽

Cited By ~ 6

Author(s):

F. BARTHE

Keyword(s):

Gaussian Measure ◽

Isoperimetric Problem ◽

Boundary Measure

The paper studies an isoperimetric problem for the Gaussian measure and coordinatewise symmetric sets. The notion of boundary measure corresponding to the uniform enlargement is considered, and it is proved that symmetric strips or their complements have minimal boundary measure.

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