TY - JOUR

T1 - Bounded H∞ -calculus for a degenerate elliptic boundary value problem

AU - Krietenstein, Thorben

AU - Schrohe, Elmar

N1 - Funding Information: The results in this article are based on the second author’s thesis, see []. His work was partly supported by Deutsche Forschungsgemeinschaft through the research training group GRK 1463. The authors thank K. Taira and Ch. Walker for helpful discussions and the referees for their suggestions that led to an improvement of the results.

PY - 2022/8

Y1 - 2022/8

N2 - On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form T= φγ+ φ 1γ 1. Here γ and γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that φ, φ 1≥ 0 , and φ+ φ 1≥ c, for some c> 0 , where either φ0,φ1∈Cb∞(∂X) or φ= 1 and φ 1= φ 2 for some φ∈ C 2 + τ(∂X) , τ> 0. We also assume that the highest order coefficients of A belong to C τ(X) and the lower order coefficients are in L ∞(X). We show that the L p(X) -realization of A with respect to the boundary operator T has a bounded H ∞-calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

AB - On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form T= φγ+ φ 1γ 1. Here γ and γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that φ, φ 1≥ 0 , and φ+ φ 1≥ c, for some c> 0 , where either φ0,φ1∈Cb∞(∂X) or φ= 1 and φ 1= φ 2 for some φ∈ C 2 + τ(∂X) , τ> 0. We also assume that the highest order coefficients of A belong to C τ(X) and the lower order coefficients are in L ∞(X). We show that the L p(X) -realization of A with respect to the boundary operator T has a bounded H ∞-calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

KW - math.AP

KW - math.FA

KW - 47A60, 58J32, 35S05

UR - http://www.scopus.com/inward/record.url?scp=85112765444&partnerID=8YFLogxK

U2 - 10.1007/s00208-021-02251-1

DO - 10.1007/s00208-021-02251-1

M3 - Article

VL - 383

SP - 1597

EP - 1646

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 3-4

ER -