Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

By Gerd Grubb. 2014

111 citations recorded by the source corpus.

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown recently by the author, the condition assures solvability of Dirichlet-type boundary problems for elliptic $P$ in full scales of Sobolev spaces with a singularity $d^{\mu -k}$, $d(x)=\operatorname{dist}(x,\partial\Omega)$. Examples include fractional Laplacians $(-\Delta)^a$ and complex powers of strongly elliptic PDE. We now introduce new boundary conditions, of Neumann type or more general nonlocal. It is also shown how problems with data on $R^n\setminus \Omega $ reduce to problems supported on $\bar\Omega$, and how the so-called "large" solutions arise. Moreover, the results are extended to general function spaces $F^s_{p,q}$ and $B^s_{p,q}$, including H\"older-Zygmund spaces $B^s_{\infty ,\infty}$. This leads to optimal H\"older estimates, e.g. for Dirichlet solutions of $(-\Delta)^au=f\in L_\infty (\Omega)$, $u\in d^aC^a(\bar\Omega)$ when $0

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/6abc6fb5f91c71f13f6af6d31f80021b1b1ca3a5

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In Applied Mathematics

  1. Xavier Ros-Oton
  2. Nicola Abatangelo
  3. J. Serra
  4. Gerd Grubb
  5. Enrico Valdinoci
  6. Nicola Garofalo
  7. Van Hoang Nguyen
  8. Mahamadi Warma

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