Field Value Language dc.contributor.author Kennedy, James Bernard dc.date.accessioned 2010-03-17 dc.date.available 2010-03-17 dc.date.issued 2010-03-17 dc.identifier.uri http://hdl.handle.net/2123/5972 dc.description Doctor of Philosophy en_AU dc.description.abstract We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu} + \alpha u = 0$ and generalised Wentzell boundary conditions $\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$ with respect to the domain $\Omega \subset \mathbb R^N$ on which the problem is defined. For the Robin problem, when $\alpha > 0$ we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class $C^2$. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain $\Omega$ and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin $p$-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of $\alpha > 0$. When $\alpha < 0$, we prove that every eigenvalue behaves like $-\alpha^2$ as $\alpha \to -\infty$, provided only that $\Omega$ is bounded with $C^1$ boundary. This generalises a result of Lou and Zhu [Pacific J. Math. 214 (2004), 323--334] for the first eigenvalue. For the Wentzell problem, we (re-)prove general operator properties, including for the less-studied case $\beta < 0$, where the problem is ill-posed in some sense. In particular, we give a new proof of the compactness of the resolvent and the structure of the spectrum, at least if $\partial \Omega$ is smooth. We prove Faber-Krahn-type inequalities in the general case $\beta, \gamma \neq 0$, based on the Robin counterpart, and for the best'' case $\beta, \gamma > 0$ establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case. en_AU dc.rights The author retains copyright of this thesis. dc.rights.uri http://www.library.usyd.edu.au/copyright.html dc.subject Elliptic partial differential equations en_AU dc.subject Laplacian en_AU dc.subject Robin boundary conditions en_AU dc.subject Wentzell boundary conditions en_AU dc.subject Isoperimetric inequality en_AU dc.subject Shape optimisation en_AU dc.title On the isoperimetric problem for the Laplacian with Robin and Wentzell boundary conditions en_AU dc.type Thesis en_AU dc.date.valid 2010-01-01 en_AU dc.type.thesis Doctor of Philosophy en_AU usyd.faculty Faculty of Science, School of Mathematics and Statistics en_AU usyd.degree Doctor of Philosophy Ph.D. en_AU usyd.awardinginst The University of Sydney en_AU
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