## Dr. Karsten Bohlen

### Researcher

Universität Regensburg

E-mail: karsten.bohlen(at)mathematik.uni-regensburg.de

Telephone: +49 941 943-2766

## Project

**3**Geometric operators on a class of manifolds with bounded geometry

## Publications within SPP2026

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains. Our class of domains contains the class of domains with isolated oscillating conical singularities. Our results thus generalize the classical results of Kondratiev on domains with conical singularities. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier results on manifolds with boundary and bounded geometry.

Journal | Comptes Rendus Mathématique Sér. I 357 487-493 (2019) |

Link to preprint version | |

Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

Let M be a smooth manifold with (smooth) boundary ∂M and bounded geometry and ∂<sub>D</sub>M ⊂ ∂M be an open and closed subset. We prove the well-posedness of the mixed Robin boundary value problem Pu = f in M, u = 0 on ∂<sub>D</sub>M, ∂<sup>P</sup><sub>ν</sub> u + bu = 0 on ∂M \ ∂<sub>D</sub>M under the following assumptions. First, we assume that P satisfies the strong Legendre condition (which reduces to the uniformly strong ellipticity condition in the scalar case) and that it has totally bounded coefficients (that is, that the coefficients of P and all their derivatives are bounded). Let ∂<sub>R</sub>M ⊂ ∂M \ ∂<sub>D</sub>M be the set where b≠ 0.

Journal | Rev. Roumaine Math. Pures Appl. 64 85-111 (2019) |

Link to preprint version | |

Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry

Let M be a manifold with boundary and bounded geometry. We assume that M has "finite width", that is, that the distance from any point to the boundary is bounded uniformly. Under this assumption, we prove that the Poincaré inequality for vector valued functions holds on M. We also prove a general regularity result for uniformly strongly elliptic equations and systems on general manifolds with boundary and bounded geometry. By combining the Poincaré inequality with the regularity result, we obtain-as in the classical case-that uniformly strongly elliptic equations and systems are well-posed on M in Hadamard's sense between the usual Sobolev spaces associated to the metric. We also provide variants of these results that apply to suitable mixed Dirichlet-Neumann boundary conditions. We also indicate applications to boundary value problems on singular domains.

Journal | Mathematische Nachrichten |

Publisher | Wiley |

Link to preprint version | |

Link to published version |

**Related project(s):****3**Geometric operators on a class of manifolds with bounded geometry