## Abstract

We study the zero exterior problem for the elliptic equation ∆^{α/}^{2}u − λu = f, x ∈ D ; u|_{D}c = 0 as well as for the parabolic equation ut = ∆^{α/}^{2}u + f, t > 0, x ∈ D ; u(0, ·)|_{D} = u0, u|_{[0},T]×_{D}c = 0. Here, α ∈ (0, 2), λ ≥ 0 and D is a C^{1,1} open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.

Original language | English |
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Pages (from-to) | 3338-3377 |

Number of pages | 40 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 43 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2023 Sept |

### Bibliographical note

Publisher Copyright:© 2023 American Institute of Mathematical Sciences. All rights reserved.

## Keywords

- Dirichlet problem
- Hölder estimates
- Non-local elliptic and parabolic equations
- Sobolev regularity theory
- fractional Laplacian

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

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