Hilbert’s twenty-third problem (Math 540)

Hilbert’s twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. In contrast with Hilbert’s other 22 problems, his 23rd is not so much a specific “problem” as an encouragement towards further development of the calculus of variations. His statement of the problem is a summary of the state-of-the-art (in 1900) of the theory of calculus of variations, with some introductory comments decrying the lack of work that had been done of the theory in the mid to late 19th century.

The first part to solving this problem, is correctly stating it.

So far, I have generally mentioned problems as definite and special as possible…. Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture-which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, it is due—I mean the calculus of variations.

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.[a] Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points.[1] Weierstrass’s demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were denounced by contemporaries: Henri Poincaré famously described them as “monsters” and called Weierstrass’ work “an outrage against common sense”, while Charles Hermite wrote that they were a “lamentable scourge”. The functions were impossible to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of Brownian motion necessitated infinitely jagged functions (nowadays known as fractal curves).[2]

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat’s principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet’s principlePlateau’s problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

Real analysis, as we now understand it, acquired a different emphasis from classical analysis in the second half of the nineteenth century. In his famous remark to Thomas Stieltjes, Charles Hermite wrote: “I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives”. Nowadays, non-regular functions and their generalizations are of fundamental importance in mathematics and physics. The study of non-regular functions requires a better theory of integration; historically, that was the main motivation for the development of measure theory.

The course will cover the prescribed syllabus:

Abstract measure theory, outer measure, Lebesgue measure on the real line, measurable functions, Egorov and Lusin theorems, the Lebesgue integral, differentiation, L-spaces, elementary Hilbert space theory and trigonometric series. Other topics will be included as time permits.

Texts (required): H.L. Royden and Patrick Fitzpatrick, Real Analysis (4th edition), Pearson, 2010.
G. B. Folland, Real Analysis, John Wiley & Sons.W.
Rudin, Real and Complex Analysis, McGraw-Hill.
Princeton Lecture Series in Analysis

HOMEWORK: https://math.berkeley.edu/~brent/files/104_weierstrass.pdf