Morse theory and Hilbert’s 19th problem

Abstract

Let $\Omega \subset {\mathbb {R}}^n$ be a smooth $C^1$ compact domain, $\varphi : \Omega \rightarrow {\mathbb {R}}^N$ in $W^{1,k}(\Omega , {\mathbb {R}}^N)$ for all k. Furthermore let $F: \Omega \times {\mathbb {R}}^{nN} \rightarrow {\mathbb {R}}$, F(x, p),  be $C^0,$ differentiable with respect to p, and with $D_p F$ continuous on $\Omega \times {\mathbb {R}}^{nN}$ and strictly convex in p. Consider an $nN \times nN$ matrix $A^{ij}_{\alpha \beta } \in C^0(\Omega )$ satisfying

$$\begin{aligned} A^{i j}_{\alpha \beta }(x) \xi ^i_\alpha \xi ^j_{\beta } = A^{ji}_{\beta \alpha }(x) \xi ^i_\alpha \xi ^j_\beta \ge \lambda |\xi |^2,\quad \lambda >0 \end{aligned}$$

(0.1)

Suppose that

$$\begin{aligned} \lim _{|p| \rightarrow \infty } \tfrac{1}{|p|} \left( D_p F(x,p) - A(x) p \right) =0 \end{aligned}$$

(0.2)

uniformly in x. Consider the functional

$$\begin{aligned} J(u) := \int _\Omega F(x, D u(x)) \; dx \end{aligned}$$

(0.3)

for all u, ${ \left. u \phantom {\big |} \right| _{\partial \Omega } }= \varphi$, $u\in \varphi + W^{1,2}_0 (\Omega , {\mathbb {R}}^N)$. Then J has a unique minimum which is Hölder continuous up to the boundary for all Hölder exponents $\alpha$, $0<\alpha < 1$. We conclude with showing that our result is nearly optimal. Our approach is completely new, using for the first time, Morse theoretic ideas to prove, in one step, existence and regularity up to the boundary by minimizing the functional J within the Sobolev space $W^{1,k}$ for arbitrarily large k. Using the method of energy growth estimates, Mariano Giaquinta in his ETH lectures, showed the Hölder continuity of $W^{1,2}$ minimizers in the interior of $\Omega$ under the single condition $F(p)/|p|^2 \rightarrow 1$ (as $p\rightarrow \infty$) which, in our case, corresponds to ${A}^{ij}_{\alpha \beta } = \delta ^{ij}\delta _{\alpha \beta }$.