Mathematics 675-2 Modern Problems in Calculus of Variations M-675-2. Modern Problems in Calculus of Variations

### Instructor Andrej Cherkaev

Office: JWB 225
Telephone: 581-6822
E-mail: cherk@math.utah.edu

Summary

David Hilbert

## Course description

The course introduces classical methods of Calculus of Variations, Legendre transform, conservation laws and symmetries. The attention is paid to variational problems with unstable (highly oscillatory) solutions, especially in multidimensional problems. These problems arrive in large number of applications, including structural optimization, phase transitions, composites, inverse problems, etc., where an optimal layout are characterized by short scale inhomogenuities: patterns of unknown shapes. We discuss methods of effective description of such solutions. They are called relaxation methods and are based on theory of quasiconvexity.

### Topics to be covered:

• Basic techniques of Calculus of Variations. Euler equations, Legendre and Weierstrass tests, direct methods.
• Noether theory: symmetries and invariants.
• Duality and Legendre transform. Two sides bounds.
• Traditional and novel variational principles.
• Ill-posed variational problems for multiple integrals:
• Quasiconvexity of Lagrangian and existence of a solution; relaxation of ill-posed problems:
• Translation method (sufficient conditions),
• Weierstrass conditions (necessary conditions),
• minimizing sequences.
Clear and elegant methods of modern Calculus of Variations allow to solve large number of problems in Science and Engineering. Originated by Bernoulli, Newton, Euler, and systematically developed beginning from XVIII century, these days Calculus of Variations attracts attention of mathematicians and provides new tools to find the best possible solutions, and to understand the essence of optimality.

#### Text:

Instructors notes (will be distributed).

1. Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover, 1974.
2. Charles Fox. An Introduction to the calculus of variations. Dover, 1987.
3. Young.
4. Dacorogna

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Introduction

The main question of the classical theory of Calculus of Variation is: What curve or surface is in a sense `the best one'? For example,

• What curve with the fixed length covers maximal area? (of course, the circle!)
• What curve corresponds to the fastest glide along it? (the brachistochrone problem)
• What surface with a fixed boundary has the smallest area? (the minimal surface problem)
• What variable thickness of a plate maximizes its stiffness? (an optimal design problem).

In these problems, the extremal property is attributed to an entire curve (function). A group of methods aimed to find `optimal' functions is called Calculus of Variations.

From ancient times, geometers noticed extremal properties of symmetric figures and bodies. The circle has maximal area among all figures with fixed perimeter; the right triangular and the square have maximal area among all triangles and quadrangles with fixed perimeter, respectively, etc. However, regular proofs of these extremal properties usually are not easy. Even more difficult task was to develop a regular theory that was able to search for optimal curves and surfaces. This theory of extremal problem has been actively developed for the last three centuries.

Calculus of Variations has been originated by Bernoulli, Newton, Euler; systematically developed beginning from XVIII century; it still attracts attention of mathematicians and it helps scientists and engineers to find the best possible solutions. The study of Calculus of Variations was fruitful for mathematics: it leaded to development of Analysis, Harmonic analysis, Operator theory and distributions, and other important branches of math.

Modern branches of Calculus of Variations include Control theory, Optimal design theory or Structural Optimization, Differential games theory, Operation research, Programming, etc. The goal of the theory is to analyse extremal trajectories. In the last decades, interest to these approaches grows thanks to advances in numerical methods: it becomes possible to solve equations for extremals and to use them in many scientific and engineering applications.

Extremal problems are attractive due to human's natural desire to find perfect solutions, they also root in natural laws of physics. The last ones can often be formulated as extremal problems: the true trajectory delivers minimum of a functional called `the energy' among all admissible trajectories. In mechanics, a real motion minimizes the mechanical energy; in thermodynamics, the dissipative processes maximizes the dissipation rate, etc. This phenomenon looks mysterious, and it has caused a lot of philosophical speculations.

The course covers classical techniques of Calculus of Variations, discusses natural variational principles in classical and continuum mechanics, and introduces modern applications.

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Contents of lectures
1. Introduction. Examples ( Linear algebra. Dido problem. Minimal surface. Bureaucratic activity. )
2. Symmetrization and the Isoperimetric problem. Convexity.
3. Variations. Euler Equation. First integrals
4. Problems of geometric optics. Ferma's principle. Generalization of Euler equation (several dependent variables, various boundary conditions, free boundary points).

5. Restricted extremum ( Isoperimetric problem. Convexity and min-max theorem. Algebraic or differential restrictions. Self-adjoint operators. Null-Lagrangian.)
6. Multidimensional integrals. ( Green's theorem. The Bolza problem. Euler equations, boundary conditions. Variable domain. )
7. Multidimensional integrals. ( Differential constraints, duality. Null-Lagrangians.

8. Second Variation. Weierstrass test

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Homework
1. Show by symmetrization that the square has the minimal perimeter among all quadrangles.
2. Can the symmetrization be applied for similar problem for pentagons?
3. Solve the problems of brachistochrone, of minimal surface of the surface of revolution.
4. Find a brachistochrone between a point and a curve g(x, y)=0

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