2022/10/28 by Eric Bahuaud, Christine Guenther, Bahuaud, Eric +5 · 1 citation
Computer Science · Engineering · Mathematics · #35J05 #35K #35P #58J35 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2210.15886
openalex publication_date 2022/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present simple conditions which ensure that a strongly elliptic operator L generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that L is "sectorial", a condition that specifies the decay of the resolvent (λI - L)-1 as λ diverges from the Hölder spectrum of L. As one step, we prove existence of this resolvent if λ is sufficiently large, and on this general class of manifolds, use a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and e-tL we obtain can then be used to prove wellposedness of a wide class of nonlinear flows. We illustrate this by proving wellposedness on Hölder spaces of the flow associated to the ambient obstruction tensor on complete manifolds of bounded geometry.