Function spaces on singular manifolds

Abstract

A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weighted function spaces. Besides of the derivation of Sobolev-type embedding results, sharp trace theorems, point-wise multiplier properties, and interpolation characterizations particular emphasize is put on spaces distinguished by boundary conditions. This work is the fundament for the analysis of time-dependent partial differential equations on singular manifolds. *

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

“…As unit normal vector field for we choose the outer unit normal field. Similar definitions were considered by . We will show in Section 2.3 that our definition coincides with [, Definition 2.2].…”

“…Recall the notation of Equation ( We have then the following proposition that is a consequence of Remark 3.5 and Theorem 3.9 in [33]. See also [3,4,7,10,11,29,45,65,69] for related results, in particular, for the use of partitions of unity.…”

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

“…We consider the L p -realization A B and remark that it is well known that A B is sectorial with angle larger than (ii) We consider (A B ) s/m and the scale generated by A B . By real interpolation, we have for k ∈ N 0 and θ ∈ (0, 1) the identities We remark that the exceptional cases s = km + m j + 1 p arise due to the real interpolation results, see the discussion in Amann [6], Amann [7].…”

Eikonal equations in metric spaces

“…There are several works on elliptic and parabolic equations in a singular manifold. The reader is referred to [2] and references theirin. The theory for a gradient flow in a general complete metric space is discussed in [4].…”