Recently, by Z. Shen, resolvent estimates for the Stokes operator were established in L^p(Ω) when Ω is a Lipschitz domain in R^d, with d≥3 and |1/p-1/2|<1/(2d)+ε. This result implies that the Stokes operator generates a bounded analytic semigroup in L^p(Ω) in the case that Ω is a three-dimensional Lipschitz domain and 3/2-ε<p<3+ε. To fully understand the work of Z. Shen, a lot of background information is needed. In this thesis the resolvent estimates are studied in detail in the case d=3. In the end the results of Shen are extended to resolvent estimates in L^p(w,Ω), where Ω is a three-dimensional Lipschitz domain, |1/p-1/2|<1/6, and w∈A_2p/3∩RH_3/(3-p) is a weight function that belongs to an intersection of a Muckenhoupt weight class and satisfies a reverse Hölder inequality. ... Read more

The NV-center in diamond is a promising system for quantum information processing. The typical qubits are the electron spin in the NV-center and the surrounding 13C nuclear spins. Fast and precise control of many 13C spins is desirable but challenging due to crosstalk. Crosstalk is the unwanted effect of a control pulse on the other qubits. The coupled NV-13C system is explored, and numerical techniques to optimize the control of multiple and single qubits are used. In this report the effect of the Bloch-Siegert shift in the weak driving regime is investigated first. This is done by studying the effect of a π-pulse when two carbon qubits are driven at the same time on resonance. It turns out that in this weak driving regime the effect of the off resonance is very small, and changes the fidelity not more than 0.3%. Furthermore the effect of a square πpulse on 4 nearby qubits is studied. If the pulse is not optimized (thus square), the pulse cannot be done in less than 100µs while retaining at least 99% fidelity. By optimizing the envelope of the driving field using constrained optimization the pulse can be done in 22µs while retaining at least 99% fidelity. ... Read more