It is proved that the Stokes operator on a bounded domain, an exterior domain, or a perturbed half-space admits a bounded H ∞ -calculus on L q ( ) if q ∈ (1, ∞).
The notion of capacity of a subspace which was introduced in [16] is used to prove new estimates on the shift of the eigenvalues which arises if the form domain of a self-adjoint and semibounded operator is restricted to a smaller subspace. The upper bound on the shift of the spectral bound given in [16] is improved and another lower bound is proved which leads to a generalization of Thirring's inequality if the underlying Hilbert space is an L 2 -space. Moreover we prove a similar capacitary upper bound for the second eigenvalue. The results are applied to elliptic constant coefficient differential operators of arbitrary order. Finally it is given a capacitary characterization for the shift of the spectral bound being positive which works for operators with spectral bound of arbitrary type.
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The significance of the bottom eigenvalue in mathematical physicsIn this section we explain the significance of the bottom eigenvalue of a self-adjoint operator in various areas of mathematical physics, so this will be a motivation for investigating and estimating the bottom eigenvalue. The main role is played by the Laplacian because this is one of the most important self-adjoint operators in mathematical physics as it plays a fundamental role in quantum mechanics, theory of heat, theory of vibrations and other areas.___ Qn the other hand, higher-order differential operators are important as well.For instance the dynamics of the clamped plate is described by the bi-potential equationwhere 6 2 is the biharmonic operator, subject to Dirichlet boundary conditions.Another example of a relevant higher-order differential operator is the analysis of the vertices of incompressible fluids which also leads to the biharmonic equation. Most of the known estimates for the bottom eigenvalue only apply to secondorder differential operators and some of themcan only handle the Laplacian. This is due to the fact that in the case of secondorder differential operators there is an interplay between analysis and stochastics via the theory of Dirichlet forms. One can prove that for certain second-order differential operators, see Appendix C.l for a precise statement, there is a stochastic process associated to the operator. It turns out that the existence of such a process is closely related to positivity preserving properties of the semigroup (and the resolvent) and to the maximum principle, [BH86,FOT94,MR92,DC]. This allows us to treat analytic problems, e.g., estimating eigenvalues, with stochastic methods or by using the powerful tools of potential analysis and vice versa.It is known that in contrast to the second-order case there is no such interplay for higher-order differential operators. Therefore the proofs of the eigenvalue estimates which rely on stochastic processes or the maximum principle typically do not carryover to the higher-order case.
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