A uniqueness proof for the Wulff Theorem

SynopsisThe Wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of (small) elastic crystals. It consists in minimising the (generally anisotropic) surface energy among sets of given volume. A solution of this problem is given by a geometric construction due to Wulff. In the class of sets of finite perimeter this was first shown by J. E. Taylor who, using methods of geometric measure theory, also proved uniqueness. Here a more analytic uniqueness proof is presented. The main ingredient is a sharpened version of the Brunn–Minkowski inequality.

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ON THE MEASURE-THEORETIC FOUNDATIONS OF THE SECOND LAW OF THERMODYNAMICS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202502001866 ◽

2002 ◽

Vol 12 (05) ◽

pp. 721-736 ◽

Cited By ~ 3

Author(s):

ALFREDO MARZOCCHI ◽

ALESSANDRO MUSESTI

Keyword(s):

Measure Theory ◽

Geometric Measure Theory ◽

Second Law Of Thermodynamics ◽

Second Law ◽

Weak Version ◽

Balance Laws ◽

Geometric Measure ◽

Sets Of Finite Perimeter ◽

Finite Perimeter ◽

Theoretic Foundations

Balance laws of the type of entropy are treated in the framework of geometric measure theory, and a weak version, although conceptually simple, of the Second Law of Thermodynamics is introduced, allowing extensions to measure-valued entropy productions and to sets of finite perimeter as subbodies.

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What is a reduced boundary in general relativity?

International Journal of Modern Physics D ◽

10.1142/s0218271821500504 ◽

2021 ◽

pp. 2150050

Author(s):

Emmanuele Battista ◽

Giampiero Esposito

Keyword(s):

General Relativity ◽

Integral Formula ◽

Measure Theory ◽

Apparent Horizon ◽

Geometric Measure Theory ◽

Einstein Equations ◽

Theoretic Approach ◽

Geometric Measure ◽

Finite Perimeter ◽

Riemannian Geometries

The concept of boundary plays an important role in several branches of general relativity, e.g. the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e. the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in–out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.

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