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On the Isoperimetric Problem for the Laplacian with Robin and Wentzell Boundary Conditions

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On the Isoperimetric Problem for the Laplacian with Robin and Wentzell Boundary Conditions On the Isoperimetric Problem for the Laplacian with Robin and Wentzell Boundary Conditions James B. Kennedy A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics March 2010 Abstract We consider the problem of minimising the eigenvalues of the Laplacian with ∂u Robin boundary conditions ∂ν + αu = 0 and generalised Wentzell boundary condi- tions ∆u+β ∂u +γu = 0 with respect to the domain Ω RN on which the problem ∂ν ⊂ is defined. For the Robin problem, when α > 0 we extend the Faber-Krahn in- equality 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 do- mains of class C2. 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 Ω 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 α> 0. When α< 0, we prove that every eigenvalue behaves like α2 as α , provided only that Ω is bounded with C1 − → −∞ 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 β < 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 ∂Ω is smooth. We prove Faber-Krahn-type inequalities in the general case β,γ = 0, based on the Robin counterpart, and 6 for the “best” case β,γ > 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. ii Acknowledgements As is surely true of any such work, this thesis would not have been possible without the help and support of a large number of people. On the professional side, my warmest thanks go first and foremost to my supervisor, Dr. Daniel Daners, for all his support and guidance, for always taking the time to help me, even when busy, and even when the question was fatuous (an unfortunately frequent occurrence); for giving me a project that was both very appealing and within my abilities, but also giving me the freedom to explore new ideas and directions that seemed interesting. I would also like to thank my associate supervisor, Prof. E. N. Dancer, for also providing help and advice whenever it was needed, even though my project was perhaps somewhat outside his field of interest. I am indebted to Prof. Dr. Matthias Hieber of Darmstadt for suggesting the generalised Wentzell boundary conditions to me at a conference early in the course of my Ph.D., and to Prof. Dr. Wolfgang Arendt of Ulm for many helpful conversations, and in particular to whom the credit for Theorem 5.2.1 is due. My thanks also go to the referees for their thoughtful and thought-provoking comments on an earlier version of this thesis, as well as picking up a number of errors. As always, any errors that remain are entirely the fault of the author. Of course, this thesis would equally not have been possible without the support of a great number of family and friends; in particular I would like to thank my parents for their constant support, as well as all my colleagues and fellow post- graduate students at the School of Mathematics and Statistics for creating such a congenial environment. Finally, I gratefully wish to acknowledge the support of an Australian Postgraduate Award (APA) during my candidature, as well as an Australian Research Council (ARC) grant in my final year. iii Statement of originality This thesis contains no new material which has been accepted for the award of any other degree or diploma. All work in this thesis, except where duly acknowl- edged otherwise, is believed to be original. James Kennedy Sydney, 3 March, 2010 iv Contents Abstract ................................................... ................ ii Acknowledgements ................................................... .... iii Statement of originality .................................................. iv 1. Introduction ................................................... ....... 1 1.1. The isoperimetric problem . ............. 1 1.2. The Laplacian with Robin boundary conditions. ....... 7 1.3. Eigenvalue properties of the Robin Laplacian . ......... 12 2. An Inequality for the First Eigenvalue of the Robin Laplacian.. 20 2.1. The Faber-Krahn inequality........................... .............. 20 2.2. A functional HΩ of the level sets.................................... 22 2.3. An estimate of the first eigenvalue...................... ............. 28 2.4. Proof of the Faber-Krahn inequality.................... ............. 32 3. Variations and Applications ........................................ 39 3.1. The p-Laplacian with Robin boundary conditions. 40 3.2. An application to supersolutions . ............ 42 3.3. Cheeger’s inequality for the Robin Laplacian . ........ 44 3.4. Two open problems.................................. ............... 46 4. On the Higher Eigenvalues of the Robin Laplacian............... 50 4.1. The second eigenvalue............................... ................ 51 4.2. Proof of Theorem 4.1.1............................... ............... 55 4.3. Some remarks on the higher eigenvalues.................. ........... 62 4.4. The asymptotic behaviour as α .............................. 67 → −∞ 5. The Laplacian with Generalised Wentzell Boundary Conditions 77 5.1. Some background to the Wentzell problem ................ .......... 78 v Contents vi 5.2. Generation results when β > 0...................................... 81 5.3. The resolvent and spectrum when β < 0............................. 85 6. The Eigenvalues of the Wentzell Laplacian ........................ 96 6.1. General remarks.................................. .................. 96 6.2. The principal eigenvalues . ............... 99 6.3. The other eigenvalues............................... ................ 106 6.4. Variational and monotonicity properties . ............109 7. Inequalities for the Wentzell Laplacian ............................114 7.1. On the principal eigenvalues . ..............114 7.2. On the other eigenvalues............................. ............... 118 7.3. A variant of Cheeger’s inequality . ............ 121 Appendix A. Notation and Background Results.....................125 A1. General remarks on notation......................... ............... 125 A2. Classes of domains.................................. ................ 126 A3. Weak derivatives and Sobolev spaces.................... .............127 A4. Some properties of domains and Sobolev spaces. ..........130 A5. Operator theory and semigroups...................... ...............135 Appendix B. Background Results on the Functional HΩ ............138 Appendix C. On Harmonic Functions and Associated Operators ..143 C1. The normal derivative............................... ................143 C2. On the homogeneous Dirichlet problem.................... ..........144 C3. The Dirichlet-to-Neumann operator .................... ............. 151 References ................................................... ..............154 Chapter 1 Introduction No question is so difficult to answer as that to which the answer is obvious — George Bernard Shaw 1.1. The isoperimetric problem The application of isoperimetric inequalities to physical situations is, in a sense, a study of the obvious. That the shape of an object affects its physical properties is so trite as to be hardly worth saying. It is perhaps a more subtle observation that a physical optimum should be attained by an object satisfying some appropriate geometric optimum. The most basic isoperimetric result states that the ball has the least surface area of all objects of given volume. This result is even said to have been known in some form to the ancient Greeks. Physical intuition thus suggests that the object that minimises heat loss should be perfectly spherical, all other things being equal. Similarly, the fundamental frequency of a vibrating membrane should be lowest when the membrane is circular. This is the famous conjecture of Lord Rayleigh in the 19th Century [97]. But the word “conjecture” is telling. It is easy to make such a claim, and even to formulate it mathematically. However experience shows that such “obvious” conjectures are often extremely difficult to prove in all branches of mathematics — although possibly this is not quite what Shaw had in mind. The first full proofs of Rayleigh’s conjecture appeared almost 50 years after it was made, with the simultaneous but independent work of Faber [50] and Krahn [78] in the 1920s. Even then some residual questions remained about the validity of the method they used, and the issue was only completely resolved as late as 1951 with the seminal work of P´olya and Szeg¨o[96]. Such problems rank amongst the most interesting, and arguably most challenging, in mathematics, involving a fascinating interplay of analysis and geometry. 1 1.1. The isoperimetric problem 2 Mathematically, both Rayleigh’s conjecture
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