FR
F.B. Roodenburg
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In this thesis we consider the incompressible and stationary Stokes problem with Navier-slip boundary conditions on an infinite two-dimensional wedge with opening angle θ. As is common for differential equations on domains with corners, the problem is decomposed into a singular expansion near the corner (polynomial problem) and a regular remainder (smooth problem). We prove existence and uniqueness of solutions to the smooth problem related to the Stokes equation which is given by -PΔu = f, where P is the Helmholtz projection. By means of the Lax-Milgram theorem it is found that this problem has a unique strong solution in a certain class of weighted Sobolev spaces if the opening angle θ is small enough. Direct application of the Lax-Milgram theorem would normally only yield a weak solution. However, by introducing additional bilinear forms we gain control on all second order derivatives and therewith obtain a strong solution. Finally, we touch upon the time-dependent Stokes problem and the polynomial problem.
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In this thesis we consider the incompressible and stationary Stokes problem with Navier-slip boundary conditions on an infinite two-dimensional wedge with opening angle θ. As is common for differential equations on domains with corners, the problem is decomposed into a singular expansion near the corner (polynomial problem) and a regular remainder (smooth problem). We prove existence and uniqueness of solutions to the smooth problem related to the Stokes equation which is given by -PΔu = f, where P is the Helmholtz projection. By means of the Lax-Milgram theorem it is found that this problem has a unique strong solution in a certain class of weighted Sobolev spaces if the opening angle θ is small enough. Direct application of the Lax-Milgram theorem would normally only yield a weak solution. However, by introducing additional bilinear forms we gain control on all second order derivatives and therewith obtain a strong solution. Finally, we touch upon the time-dependent Stokes problem and the polynomial problem.
This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued analytically to the whole complex plane except at s = 1. Two proofs of this continuation were given by Bernhard Riemann in his famous article ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' from 1859. Those proofs are studied in detail in this thesis after introducing all the required foreknowledge on the gamma function. The prime counting function π(x) counts the number of primes less than or equal to x. An explicit formula for π(x) in terms of the nontrivial zeros of the zeta function will be constructed in a similar way as Riemann did in his article. Finally, the prime number theorem will be proved. This theorem describes the asymptotic distribution of the primes among the natural numbers. Using the analytic continuation of the zeta function and a Tauberian theorem, the prime number theorem can be proved quite easily with only basic theory from complex analysis.
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This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued analytically to the whole complex plane except at s = 1. Two proofs of this continuation were given by Bernhard Riemann in his famous article ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' from 1859. Those proofs are studied in detail in this thesis after introducing all the required foreknowledge on the gamma function. The prime counting function π(x) counts the number of primes less than or equal to x. An explicit formula for π(x) in terms of the nontrivial zeros of the zeta function will be constructed in a similar way as Riemann did in his article. Finally, the prime number theorem will be proved. This theorem describes the asymptotic distribution of the primes among the natural numbers. Using the analytic continuation of the zeta function and a Tauberian theorem, the prime number theorem can be proved quite easily with only basic theory from complex analysis.