12.2 Geometric measure theory and the calculus of variations

Geometric measure theory bridges the gap between geometry and analysis, providing tools to study complex shapes and surfaces. It extends classical calculus of variations to handle irregular objects, enabling us to tackle tricky problems in minimal surfaces and shape optimization.

This powerful framework lets us analyze everything from soap films to black hole horizons. By introducing concepts like currents and varifolds, we can work with weird, non-smooth surfaces and still get meaningful results about their properties and behavior.

Geometric Measure Theory and Variational Problems

Relationship between Geometric Measure Theory and Variational Problems

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• Geometric measure theory provides a rigorous framework for studying variational problems in a geometric setting
  • Allows for the analysis of objects with irregular or singular behavior (fractals, non-smooth surfaces)
• Variational problems involve finding extrema (minima or maxima) of functionals
  • Functionals are often defined on spaces of functions or more general geometric objects (surfaces, curves)
• The calculus of variations deals with finding extrema of functionals
  • Geometric measure theory extends these ideas to geometric objects
• Geometric measure theory introduces concepts to describe and analyze geometric objects
  • Hausdorff measures quantify the size of sets in a way that is compatible with their dimension
  • Rectifiable sets are sets that can be approximated by smooth objects (Lipschitz functions)
  • Currents are generalized oriented surfaces with integer multiplicities
• Tools of geometric measure theory allow for the computation of geometric quantities associated with variational problems
  • Area and coarea formulas compute volumes and areas of sets and their projections
  • Federer-Fleming compactness theorem ensures the existence of minimizers for certain variational problems

Applications of Geometric Measure Theory to Variational Problems

• Geometric measure theory provides a framework for studying the regularity of minimizers in variational problems
  • Regularity refers to the smoothness properties of minimizers (continuity, differentiability)
• Geometric measure theory allows for the analysis of minimal surfaces
  • Minimal surfaces are surfaces that locally minimize area among all surfaces with the same boundary (soap films)
• Currents and varifolds are generalized notions of surfaces used in geometric measure theory
  • Enable the study of variational problems with singularities or non-orientable behavior
• Geometric measure theory has applications in various fields
  • Minimal surface theory studies surfaces that minimize area subject to boundary constraints
  • Harmonic maps are maps between Riemannian manifolds that minimize energy functionals
  • Phase transitions in materials science involve the study of interfaces between different phases of matter

Regularity of Minimizers in Variational Problems

Regularity Theory in Geometric Measure Theory

• Regularity theory aims to understand the smoothness properties of minimizers or critical points of variational problems
• Minimizers are often expected to exhibit some degree of regularity
  • Continuity: minimizers are continuous functions
  • Differentiability: minimizers have well-defined derivatives up to a certain order
  • Higher-order smoothness: minimizers have continuous derivatives of higher order (C^k regularity)
• Geometric measure theory provides tools to study regularity in a general setting
  • Allows for the presence of singularities or low regularity (corners, edges)
• Rectifiability plays a crucial role in regularity theory
  • Rectifiable sets can be approximated by smooth objects (Lipschitz graphs)
  • Often appear as minimizers of variational problems (minimal surfaces, soap films)

Techniques for Studying Regularity

• Blow-up analysis studies the local behavior of minimizers by zooming in on a point
  • Rescales the minimizer and takes a limit to obtain a tangent object (tangent plane, tangent cone)
  • Provides information about the local structure of the minimizer
• Monotonicity formulas relate geometric quantities at different scales
  • Example: monotonicity formula for minimal surfaces relates the area of a minimal surface to the area of its projection onto a plane
  • Gives insight into the growth behavior of the minimizer
• Excess decay estimates quantify the deviation of a minimizer from a simpler object (plane, cone)
  • Measure how quickly the minimizer approaches the simpler object as the scale decreases
  • Used to derive regularity results by iteratively improving the approximation

Applications of Regularity Theory

• Minimal surface theory: regularity of minimal surfaces
  • Plateau problem: finding a surface of minimal area spanning a given boundary curve
  • Bernstein problem: classifying entire minimal graphs in Euclidean space
• Harmonic maps between Riemannian manifolds
  • Regularity of energy-minimizing maps
  • Eells-Sampson theorem on the existence of harmonic maps in certain homotopy classes
• Phase transitions in materials science
  • Regularity of interfaces between different phases of matter
  • Allen-Cahn equation describing the motion of phase boundaries

Minimal Surfaces and Geometric Measure Theory

Existence and Properties of Minimal Surfaces

• Minimal surfaces are surfaces that locally minimize area among all surfaces with the same boundary
  • Examples: soap films, catenoids, helicoids
• Existence of minimal surfaces can be established using variational methods
  • Direct method of the calculus of variations: minimizes the area functional over a suitable class of surfaces
  • Compactness results from geometric measure theory ensure the existence of minimizers (Federer-Fleming compactness theorem)
• Properties of minimal surfaces can be studied using geometric measure theory
  • Regularity theory investigates the smoothness of minimal surfaces
  • Monotonicity formula relates the area of a minimal surface to the area of its projection onto a plane
  • Second variation formula and Jacobi fields analyze the stability and instability of minimal surfaces

Generalized Minimal Surfaces

• Geometric measure theory allows for the study of more general classes of minimal surfaces
• Varifolds are generalized unoriented surfaces with real multiplicities
  • Can represent non-smooth or singular surfaces (soap films with singularities)
  • Varifold formulation of the Plateau problem: finding a varifold of minimal mass spanning a given boundary
• Integral currents are generalized oriented surfaces with integer multiplicities
  • Suitable for studying variational problems with geometric constraints (area, volume)
  • Federer-Fleming compactness theorem ensures the existence of minimizing integral currents
• Regularity theory for varifolds and integral currents
  • Allard regularity theorem: regularity of varifolds with bounded first variation
  • Almgren-Federer regularity theorem: regularity of minimizing integral currents

Applications of Minimal Surfaces

• Architecture and design: minimal surfaces as optimal structures
  • Soap films and bubbles as inspiration for lightweight and efficient designs
  • Munich Olympic Stadium roof: based on minimal surface principles
• Materials science: minimal surfaces in self-assembly and phase separation
  • Block copolymers: self-assemble into structures resembling minimal surfaces
  • Gyroid and Schwarz P surfaces: appear in nanoscale structures
• General relativity: minimal surfaces as models for black hole horizons
  • Event horizon of a black hole: minimal surface in spacetime
  • Penrose inequality relates the area of the event horizon to the mass of the black hole

Currents and Varifolds in the Calculus of Variations

Currents as Generalized Surfaces

• Currents are linear functionals on the space of differential forms
  • Generalize the notion of oriented surfaces with integer multiplicities
  • Can represent non-smooth or singular surfaces (fractals, soap films with singularities)
• Space of currents is equipped with the flat norm topology
  • Allows for the study of convergence and compactness properties of sequences of currents
  • Flat norm measures the cancellation of mass between positive and negative parts of a current
• Integral currents are a special class of currents
  • Have finite mass and are rectifiable (can be approximated by Lipschitz graphs)
  • Suitable for studying variational problems with geometric constraints (area, volume)
• Boundary operator on currents generalizes the boundary of a surface
  • Stokes' theorem relates the boundary of a current to the exterior derivative of a differential form
  • Allows for the formulation of variational problems with boundary conditions

Varifolds as Generalized Measures

• Varifolds are measures on the product space of a manifold and its Grassmannian bundle
  • Represent generalized unoriented surfaces with real multiplicities
  • Can model surfaces with singularities or non-orientable behavior (Möbius strip)
• Space of varifolds is equipped with the weak-* topology
  • Provides a framework for studying convergence and compactness of sequences of varifolds
  • Convergence of varifolds corresponds to the convergence of their generalized area measures
• First variation of a varifold generalizes the notion of mean curvature
  • Allard regularity theorem: varifolds with bounded first variation are regular (smooth) almost everywhere
• Varifold formulation of variational problems
  • Plateau problem: finding a varifold of minimal mass spanning a given boundary
  • Isoperimetric problem: finding a varifold enclosing a given volume with minimal surface area

Applications of Currents and Varifolds

• Plateau problem: finding a surface of minimal area spanning a given boundary curve
  • Can be formulated and solved using the theory of currents or varifolds
  • Existence and regularity of solutions studied using geometric measure theory
• Isoperimetric problem: finding a region of given volume with minimal surface area
  • Formulated using currents or varifolds to allow for non-smooth or singular solutions
  • Regularity and stability of isoperimetric regions investigated using geometric measure theory
• Shape optimization: finding optimal shapes under geometric constraints
  • Currents and varifolds used to represent and manipulate shapes in a flexible way
  • Applications in engineering, design, and computer graphics (3D printing, computer-aided design)
• Image segmentation and analysis: extracting meaningful structures from images
  • Currents and varifolds used to represent and match shapes in images
  • Applications in medical imaging, computer vision, and pattern recognition