Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators

Abstract

Some linear integro-differential operators have old and classical representations as the Dirichlet-to-Neumann operators for linear elliptic equations, such as the 1/2-Laplacian or the generator of the boundary process of a reflected diffusion. In this work, we make some extensions of this theory to the case of a nonlinear Dirichlet-to-Neumann mapping that is constructed using a solution to a fully nonlinear elliptic equation in a given domain, mapping Dirichlet data to its normal derivative of the resulting solution. Here we begin the process of giving detailed information about the Lévy measures that will result from the integro-differential representation of the Dirichlet-to-Neumann mapping. We provide new results about both linear and nonlinear Dirichlet-to-Neumann mappings. Information about the Lévy measures is important if one hopes to use recent advancements of the integro-differential theory to study problems involving Dirichlet-to-Neumann mappings.

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Acknowledgments

The authors all acknowledge partial support from the NSF leading to the completion of this work: N. Guillen DMS-1201413 and DMS-1700307; J. Kitagawa DMS-1700094; R. Schwab DMS-1665285. They would like to thank Rodrigo Bañuelos and Renming Song for helpful information on background results appearing in Section ??. They would also like to thank the anonymous referee for suggesting some helpful changes to the presentation of the results.

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Correspondence to Nestor Guillen.

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Guillen, N., Kitagawa, J. & Schwab, R.W. Estimates for Dirichlet-to-Neumann Maps as Integro-differential Operators. Potential Anal 53, 483–521 (2020). https://doi.org/10.1007/s11118-019-09776-w

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Keywords

  • Dirichlet-to-Neumann
  • Integro-differential
  • Nonlocal
  • Elliptic equation
  • Boundary process
  • Fully nonlinear
  • Levy measures
  • Boundary operators