Abstract
We prove a local Hölder estimate for any exponent \(0<\delta <\frac {1}{2}\) for solutions of the dynamic programming principle
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
where λ1(D2u) ≤⋯ ≤ λn(D2u) are the eigenvalues of the Hessian.
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Acknowledgements
The authors would like to thank Julio D. Rossi for useful discussions.
Funding
Open Access funding provided by University of Jyväskyä (JYU). J. H. was supported by NRF-2021R1A6A3A14045195. P. B. and M. P. were partly supported by the Academy of Finland project 298641.
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Blanc, P., Han, J., Parviainen, M. et al. Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian. Potential Anal 59, 1995–2015 (2023). https://doi.org/10.1007/s11118-022-10037-6
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DOI: https://doi.org/10.1007/s11118-022-10037-6
Keywords
- Dynamic programming principle
- Hölder estimate
- Viscosity solution
- Eigenvalue of the Hessian
- Fully nonlinear PDEs