6533b874fe1ef96bd12d6393

RESEARCH PRODUCT

Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

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AbstractWe prove a local Hölder estimate for any exponent $0<\delta <\frac {1}{2}$ 0 < δ < 1 2 for solutions of the dynamic programming principle $$ \begin{array}{@{}rcl@{}} u^{\varepsilon} (x) = \sum\limits_{j=1}^{n} \alpha_{j} \underset{\dim(S)=j}{\inf} \underset{|v|=1}{\underset{v\in S}{\sup}} \frac{u^{\varepsilon} (x + \varepsilon v) + u^{\varepsilon} (x - \varepsilon v)}{2} \end{array} $$ u ε ( x ) = ∑ j = 1 n α j inf dim ( S ) = j sup v ∈ S | v | = 1 u ε ( x + ε v ) + u ε ( x − ε v ) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $$ \sum\limits_{i=1}^{n} \alpha_{i} \lambda_{i}(D^{2}u)=0, $$ ∑ i = 1 n α i λ i ( D 2 u ) = 0 , where λ1(D2u) ≤⋯ ≤ λn(D2u) are the eigenvalues of the Hessian.