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The Robin Mean Value Equation II: Asymptotic Hölder Regularity

Potential Analysis (2022)Cite this article

Abstract

We show that solutions to the Robin mean value equations (RMV), introduced in Lewicka and Peres (2022), converge uniformly in the limit of the vanishing radius of averaging, to the unique solution of the Robin-Laplace boundary value problem (RL), posed on any \(\mathcal {C}^{1,1}\)-regular domain and with any bounded Borel right hand side. When compared with the case of continuous right hand side, analyzed in Lewicka and Peres (2022), the present more general setting presents significant technical challenges. Along the way, we prove the asymptotic Hölder equicontinuity of solutions to (RMV): Lipschitz in the interior and \(\mathcal {C}^{0,\alpha }\) up to the boundary, for any α ∈ (0,1). Our proofs employ martingale techniques, where (RMV) is interpreted as the dynamic programming principle for a discrete stochastic process, interpolating between the reflecting and the stopped-at-exit Brownian walks.

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Acknowledgments

The authors are grateful to Dorin Bucur for bringing to their attention questions related to the Robin boundary condition. M.L. acknowledges partial support from the NSF grant DMS-1613153 and support through visits to Microsoft Research in Redmond.

Funding

M.L. acknowledges partial support from the NSF grant DMS-1613153 and support through visits to MSR (Microsoft Research) in Redmond.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, PA, 15260, USA

    Marta Lewicka

  2. Kent, OH, USA

    Yuval Peres

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  1. Marta Lewicka
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  2. Yuval Peres
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Correspondence to Marta Lewicka.

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Lewicka, M., Peres, Y. The Robin Mean Value Equation II: Asymptotic Hölder Regularity. Potential Anal (2022). https://doi.org/10.1007/s11118-022-10014-z

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  • DOI: https://doi.org/10.1007/s11118-022-10014-z

Keywords

  • Robin problem
  • Third boundary value problem
  • Oblique boundary conditions
  • Dynamic programming principle
  • Random walks
  • Finite difference approximations
  • Viscosity solutions

Mathematics Subject Classification (2010)

  • 35J05
  • 35J25
  • 60G50