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
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