The Yamabe flow on incomplete manifolds

Abstract

This article is concerned with developing an analytic theory for second order nonlinear parabolic equations on singular manifolds. Existence and uniqueness of solutions in an Lp-framework is established by maximal regularity tools. These techniques are applied to the Yamabe flow. It is proven that the Yamabe flow admits a unique local solution within a class of incomplete initial metrics.

Wellposedness of a nonlocal nonlinear diffusion equation of image processing

“…Proof. This theorem is a consequence of the work in [21,23]. We would like to refer the reader to these two papers for more details, and thus only necessary explanations will be pointed out here.…”

“…Here for F ∈ {BC, W p },F s,ϑ (Ū δ k \ Γ) is defined as the closure of D(U δ \ Γ) in F s,ϑ π (Q \ Γ). One can show that the semigroup {e −tA (0) } t≥0 is positive by means of the same argument as in step (iii) of the proof for [23,Theorem 4.8].…”

“…[9, Proposition 4.2], we infer that A (w) ∈ RS(X 0 (δ)) with φ R A (w) < π/2. The last step is to use a standard decomposition and gluing procedure as in step (v)-(vii) of the proof for [23,Theorem 4.8], and we can prove that for some ω ≥ 0 ω + A (w) ∈ RS(L 2ε+ϑ p,π (Q \ Γ)) with φ R A (w) < π/2. Then the assertion follows from [9,Theorem 4.4].…”

“…The precise degeneration type, however, depends on the exact properties of the chosen non-smooth initial datum. The construction of a unique solution proposed here is therefore based on the use of function spaces defined around a single singular function (in order to fix the degenration type) and of recently developed results for parabolic equations on singular Riemannian manifolds which provide a tool for analyzing degenerate parabolic equations with fixed degeneration; see [2,4,21,23]. While the results of this paper do not resolve the general uniqueness/nonuniqueness question, they appear to be the first delivering non-trivial existence results of solution to non-smooth initial data and uniqueness in a restricted class of functions which share a common singularity.…”

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…In [38] it was shown existence, uniqueness and maximal L q -regularity for the short time solutions, where in [39] this result was improved to long time existence and smoothness. Moreover, concerning the case of singular manifolds in the sense of H. Amann [1], in [46] it was shown existence, uniqueness and maximal continuous regularity for the short time solutions and in [47] global existence of L 1 -mild solutions; see also [48] and [49] for similar problems on such spaces. For the case of the hyperbolic space, or more generally for Riemannian manifolds with nonpositive sectional curvature, we refer to [15], [16], [19] and [53].…”