JournalsrmiVol. 34, No. 2pp. 767–810

The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition

  • Martin Dindoš

    Edinburgh University, UK
  • Sukjung Hwang

    Yonsei University, Seoul, Republic of Korea
The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition cover

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Abstract

We establish LpL^p, 2p2\le p\le\infty, solvability of the Dirichlet~boundary value problem for a parabolic equation utdiv(Au)Bu=0u_t- \mathrm {div}(A\nabla u) - \boldsymbol{B}\cdot\nabla u =0 on time-varying domains with coefficient matrices A=[aij]A=[a_{ij}] and B=[bi]\boldsymbol{B} = [b_{i}] that satisfy a small Carleson condition. The results are sharp in the following sense. For a given value of 1<p<1 < p < \infty there exists operators that satisfy Carleson condition but fail to have LpL^p solvability of the Dirichlet problem. Thus the assumption of smallness is sharp. Our results complements results of Hofmann, Lewis and Rivera-Noriega, where solvability of parabolic LpL^p (for some large pp) Dirichlet boundary value problem for coefficients that satisfy large Carleson condition was established. We also give a new (substantially shorter) proof of these results.

Cite this article

Martin Dindoš, Sukjung Hwang, The Dirichlet boundary problem for second order parabolic operators satisfying a Carleson condition. Rev. Mat. Iberoam. 34 (2018), no. 2, pp. 767–810

DOI 10.4171/RMI/1003