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A New H2 Regularity Condition of the Solution to Dirichlet Problem of the Poisson Equation and Its Applications

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Abstract

We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain. A new sufficient condition, uniformly positive reach is introduced. Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach, the H2 regularity of the solution of the Poisson equation is established. In particular, this includes all star-shaped domains whose closures are of positive reach, regardless if they are Lipschitz domains or non-Lipschitz domains. Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.

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Authors and Affiliations

  1. Department of Mathematics, University of Idaho, Moscow, ID, 83844, USA

    Fu Chang Gao

  2. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Ming Jun Lai

Authors
  1. Fu Chang Gao
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  2. Ming Jun Lai
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Correspondence to Fu Chang Gao or Ming Jun Lai.

Additional information

The first author is partially supported by Simons collaboration (Grant No. 246211) and the National Institutes of Health (Grant No. P20GM104420), the second author is partially supported by Simons collaboration (Grant No. 280646) and the National Science Foundation under the (Grant No. DMS 1521537)

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Gao, F.C., Lai, M.J. A New H2 Regularity Condition of the Solution to Dirichlet Problem of the Poisson Equation and Its Applications. Acta. Math. Sin.-English Ser. 36, 21–39 (2020). https://doi.org/10.1007/s10114-019-8015-3

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  • DOI: https://doi.org/10.1007/s10114-019-8015-3

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