Bounded imaginary powers of cone differential operators on higher order Mellin–Sobolev spaces and applications to the Cahn–Hilliard equation

Abstract

Extending earlier results on the existence of bounded imaginary powers for cone differential operators on weighted L p -spaces H 0,γ p (B) over a manifold with conical singularities, we show how the same assumptions also yield the existence of bounded imaginary powers on higher order Mellin-Sobolev spaces H s,γ p (B), s ≥ 0. As an application we consider the Cahn-Hilliard equation on a manifold with (possibly warped) conical singularities. Relying on our work for the case of straight cones, we first establish R-sectoriality (and thus maximal regularity) for the linearized equation and then deduce the existence of a short time solution with the help of a theorem by Clément and Li. We also obtain the short time asymptotics of the solution near the conical point.

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

“…Lemma 7.2 implies thatB −1 ∈ L(H 2,γ p (B), D(∆ 2 )) ֒→ L(H 2,γ p (B), H 4,γ p (B)). Hence, we have that B −1 ∈ L(H s,γ p (B)) and the result follows by the result for the case of s = 2 and by[37, Theorem 4.1]. Iteration then shows the assertion.We are now in a position to prove the main result of this section.Proof of Theorem 1.2.…”

“…We regard ∆ as a cone differential operator or a Fuchs type operator and recall some basic facts and results from the related underlined pseudodifferential theory, which is called cone calculus, towards the direction of the study of nonlinear partial differential equations. For more details we refer to [6], [13], [14], [25], [28], [36], [37], [38], [39], [40], [41], [42], [43], [44] and [45].…”

“…Therefore, by [43, Theorem 4.1] for each p ∈ (1, ∞) there exists some r 0 > 0 such that R p (λ) exists for λ ∈ S θ , |λ| ≥ r 0 , and is equal to R 2 (λ), in the sense that R p (λ) is the restriction of R 2 (λ) and vice versa. Furthermore, by [37,Theorem 4.2] we know that R p (λ) exists for all λ ∈ S θ \{0} and that for each fixed δ 0 > 0, R p (λ) L(H 0,γ p (B)) is uniformly bounded by K 0 when λ ∈ S θ , |λ| ≥ δ 0 , for certain K 0 > 0. Thus, (5.43) and by Neumann series we get that…”

“…It is well known that when ∆ is considered as an unbounded operator in H s,γ p (B), it admits several closed extensions; each of these extensions corresponds to a subspace of a finite dimensional space E ∆,γ that is determined explicitly by the metric h, see Section 3 for details. Moreover, if we denote by C ω the space of smooth functions on B that are locally constant close to the singularities, see Definition 3.2, it is known that under appropriate choice of the weight γ in terms of the dimension and the local geometry, the map ∆ : H s+2,γ+2 p (B) ⊕ C ω → H s,γ p (B) defines a closed extension ∆ s of ∆ in H s,γ p (B) such that c 0 − ∆ s ∈ R(θ) for each c 0 > 0 and θ ∈ [0, π), see [37,Theorem 4.2] or [41,Theorem 6.7]. By studying the nature of the pole zero of the resolvent of the above realization we show the following.…”

Smoothness and long time existence for solutions of the porous medium equation on manifolds with conical singularities

“…. A standard estimate for cone Sobolev spaces, see [19,Corollary 2.9], then implies that, for a suitable constant c depending only on B and p,…”