The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

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Rigorous probabilistic analysis of equilibrium crystal shapes

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \\Bbb R^3.

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Proof of the Double Bubble Conjecture

In this article, we present a diffusion-generated approach for evolving volume-preserving motion by mean curvature. Our algorithm alternately diffuses and sharpens characteristic functions to produce a normal velocity which equals the mean curvature minus the average mean curvature. This simple algorithm naturally treats topological mergings and breakings and can be made very fast even when the volume constraint is enforced to double precision (or more). Two dimensional numerical studies are provided to demonstrate the convergence of the method for smooth and nonsmooth problems.

A Simple Scheme for Volume-Preserving Motion by Mean Curvature

We study the equilibrium energies of two-dimensional (2D) noncoarsening fluid foams, which consist of bubbles with fixed areas. The equilibrium states correspond to local minima of the total perimeter. We present a theoretical derivation of energy minima; experiments with ferrofluid foams, which can be either highly distorted, locally relaxed, or globally annealed; and Monte Carlo simulations using the extended large-Q Potts model. For a dry foam with small size variance we develop physical insight and an electrostatic analogy, which enables us to (i) find an approximate value of the global minimum perimeter, accounting for (small) area disorder, the topological distribution, and physical boundary conditions; (ii) conjecture the corresponding pattern and topology: small bubbles sort inward and large bubbles sort outward, topological charges of the same signs "repel" while charges of the opposite signs "attract;" (iii) define local and global markers to determine directly from an image how far a foam is from its ground state; (iv) conjecture that, in a local perimeter minimum at prescribed topology, the pressure distribution and thus the edge curvature are unique. Some results also apply to 3D foams.

Equilibrium states and ground state of two-dimensional fluid foams.

Of course the circle is the least-perimeter way to enclose a region of prescribed area in the plane. This paper proves that a certain standard «double bubble» is the least-perimeter way to enclose and separate two regions of prescribed areas. The solution for three regions remains conjectural

https://msp.org/pjm/1993/159-1/pjm-v159-n1-p03-p.pdf

Nickelous Hodges

Papers

The standard double soap bubble in R2 uniquely minimizes perimeter