Application of De Giorgi's Theorem To Hilbert's Problem
Hilbert's problem asks whether the minimizers w of an energy functional such as
are analytic. Here w is a function on some compact set U of Rn, Dw is the vector of its first derivatives, and L is the Lagrangian, a function of the derivatives of w that satisfies certain growth, smoothness, and convexity conditions. The smoothness of w can be shown using De Giorgi's theorem as follows. The Euler-Lagrange equation for this variational problem is the non-linear equation
and differentiating this with respect to xk gives
This means that u=wxk satisfies the linear equation
with
so by De Giorgi's result the solution w has Hölder continuous first derivatives.
Once w is known to have Hölder continuous (n+1)st derivatives for some n ≥ 0, then the coefficients aij have Hölder continuous nth derivatives, so a theorem of Schauder implies that the (n+2)nd derivatives are also Hölder continuous, so repeating this infinitely often shows that the solution w is smooth.
Read more about this topic: Hilbert's Nineteenth Problem
Famous quotes containing the words application of, application, theorem and/or problem:
“Courage is resistance to fear, mastery of fear—not absence of fear. Except a creature be part coward it is not a compliment to say it is brave; it is merely a loose application of the word. Consider the flea!—incomparably the bravest of all the creatures of God, if ignorance of fear were courage.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“It is part of the educator’s responsibility to see equally to two things: First, that the problem grows out of the conditions of the experience being had in the present, and that it is within the range of the capacity of students; and, secondly, that it is such that it arouses in the learner an active quest for information and for production of new ideas. The new facts and new ideas thus obtained become the ground for further experiences in which new problems are presented.”
—John Dewey (1859–1952)