Morse theory and Hilbert’s 19th problem - Tomi, Friedrich; Tromba, Anthony J. (Calculus of variations and partial differential equations, 2019)
Let Ω⊂RnΩ⊂RnΩ \subset {\mathbb {R}}^n be a smooth C1C1C^1 compact domain, φ:Ω→RNφ:Ω→RNφ : Ω → {\mathbb {R}}^N in W1,k(Ω,RN)W1,k(Ω,RN)W^{1,k}(Ω , {\mathbb {R}}^N) for all k. Furthermore let F:Ω×RnN→RF:Ω×RnN→RF: Ω × {\mathbb {R}}^{nN} → {\mathbb {R}}, F(x, p), be C0,C0,C^0, differentiable with respect to p, and with DpFDpFD_p F continuous on Ω×RnNΩ×RnNΩ × {\mathbb {R}}^{nN} and strictly convex in p. Consider an nN×nNnN×nNnN × nN matrix Aijαβ∈C0(Ω)Aαβij∈C0(Ω)A^{ij}_{α β } \in C^0(Ω ) satisfying Aijαβ(x)ξiαξjβ=Ajiβα(x)ξiαξjβ≥λ|ξ|2,λ>0Aαβij(x)ξαiξβj=Aβαji(x)ξαiξβj≥λ|ξ|2,λ>0\begin{aligned} A^{i j}_{α β }(x) ξ ^i_α ξ ^j_{β } = A^{ji}_{β α }(x) ξ ^i_α ξ ^j_β \ge λ |ξ |^2,\quad λ >0 \end{aligned} (0.1)Suppose that lim|p|→∞1|p|(DpF(x,p)−A(x)p)=0lim|p|→∞1|p|(DpF(x,p)−A(x)p)=0\begin{aligned} \lim _{|p| → ∞ } \tfrac{1}{|p|} \left( D_p F(x,p) - A(x) p \right) =0 \end{aligned} (0.2)uniformly in x. Consider the functional J(u):=∫ΩF(x,Du(x))dxJ(u):=∫ΩF(x,Du(x))dx\begin{aligned} J(u) := \int _Ω F(x, D u(x)) \; dx \end{aligned} (0.3)for all u, u∣∣∣∣∂Ω=φu||∂Ω=φ{ \left. u \phantom {\big |} \right| _{\partial Ω } }= φ , u∈φ+W1,20(Ω,RN)u∈φ+W01,2(Ω,RN)u\in φ + W^{1,2}_0 (Ω , {\mathbb {R}}^N). Then J has a unique minimum which is Hölder continuous up to the boundary for all Hölder exponents ααα , 0<α<10<α<10<α < 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 W1,kW1,kW^{1,k} for arbitrarily large k. Using the method of energy growth estimates, Mariano Giaquinta in his ETH lectures, showed the Hölder continuity of W1,2W1,2W^{1,2} minimizers in the interior of ΩΩΩ under the single condition F(p)/|p|2→1F(p)/|p|2→1F(p)/|p|^2 → 1 (as p→∞p→∞p→ ∞ ) which, in our case, corresponds to Aijαβ=δijδαβAαβij=δijδαβA^{ij}_{α β } = δ ^{ij}δ _{α β }.