Abstract. We develop a very general operator-valued functional calculus for operators with an H ∞ −calculus. We then apply this to the joint functional calculus of two commuting sectorial operators when one has an H ∞ calculus. Using this we prove theorem of Dore-Venni type on sums of commuting sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting. 1.

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

this paper maximal L p -regularity is shown in case the semigroup e generated by \GammaA admits heat kernel estimates. The proof relies on Calderon-Zygmund theory, Interpolation and on the theorem of Benedek, Calder'on and Panzone. In Pruss [18] this approach is generalized to fractional evolution equations of parabolic type, using Poisson estimates for the kernel of the resolvent ( + A) of A rather than heat kernel estimates

Abstract. We give a negative solution to the problem of the L p-maximal regularity on various classes of Banach spaces including L q-spaces with 1 < q ̸ = 2 < +∞.

Abstract. A recent result of Kalton and Weis is extended to the case of noncommuting operators, employing the commutator condition of Labbas and Terreni, or of Da Prato and Grisvard. Under appropriate assumptions it is shown that the sum of two non-commuting operators admits an H ∞-calculus. The main results are then applied to a parabolic problem on a wedge domain. 1.

Dedicated to the memory of Pascal Laubin Abstract. We study an elliptic differential operator A on a manifold with conic points. Assuming A to be defined on the smooth functions supported away from the singularities, we first address the question of possible closed extensions of A to Lp Sobolev spaces and then explain how additional ellipticity conditions ensure maximal regularity for the operator A. Investigating the Lipschitz continuity of the maps f(u) = |u | α, α ≥ 1, and f(u) = u α, α ∈ N, and using a result of Clément and Li, we finally show unique solvability of a quasilinear equation of the form (∂t − a(u)∆)u = f(u) in suitable spaces. Contents

Abstract. We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B, 1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A z, z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Contents

Abstract. We derive conditions that ensure the existence of a bounded H∞-calculus in weighted Lp-Sobolev spaces for closed extensions A T of a differential operator A on a conic manifold with boundary, subject to differential boundary conditions T. In general, these conditions ask for a particular pseudodifferential structure of the resolvent (λ − AT) −1 in a sector Λ ⊂ C. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem ( A). T Examples concern the Dirichlet and Neumann Laplacians. Contents

Abstract. We show that if L is a second-order uniformly elliptic operator in divergence form on R d, then C1(1 + |α|) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + |α|) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ C ∞ (R d), aij = aji for 1 ≤ i, j ≤ d and that κI ≤ (aij) ≤ τI for some positive constants κ and τ. We define a positive self-adjoint operator L on L 2 (R d) by the formula (1) L = − ∑ ∂iaij∂j. We refer readers to [8] for the precise definition and basic properties of L. In particular, L

Abstract. The two-phase free boundary value problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. We extract the boundary symbol which is crucial for the dynamics of the free boundary and present an analysis of this symbol. Of particular interest are its singularities and zeros which lead to refined mapping properties of the corresponding operator.