James B. Kennedy

The Dirichlet-to-Neumann operator via hidden compactness

Abstract We show that to each symmetric elliptic operator of the form A = − ∑ ∂ k a k l ∂ l + c on a bounded Lipschitz domain Ω ⊂ R d one can associate a self-adjoint Dirichlet-to-Neumann operator on L 2 ( ∂ Ω ) , which may be multi-valued if 0 is in the Dirichlet spectrum of A . To overcome the lack of coerciveness in this case, we employ a new version of the Lax–Milgram lemma based on an indirect ellipticity property that we call hidden compactness. We then establish uniform resolvent convergence of a sequence of Dirichlet-to-Neumann operators whenever the underlying coefficients converge uniformly and the second-order limit operator in L 2 ( Ω ) has the unique continuation property. We also consider semigroup convergence.

Uniqueness in the Faber–Krahn Inequality for Robin Problems

We prove uniqueness in the Faber–Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that among all sufficiently smooth domains of fixed volume, the ball is the unique minimizer for the first eigenvalue. The method of proof, which avoids the use of any symmetrization, also works in the case of Dirichlet boundary conditions. We also give a characterization of all symmetric elliptic operators in divergence form whose first eigenvalue is minimal.

An isoperimetric inequality for the second eigenvalue of the Laplacian with Robin boundary conditions

We prove that the second eigenvalue of the Laplacian with Robin boundary conditions is minimized among all bounded Lipschitz domains of fixed volume by the domain consisting of the disjoint union of two balls of equal volume.

Eventually positive semigroups of linear operators

Abstract We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron–Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation. We also consider eventually positive semigroups on arbitrary Banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups.

On the spectral gap of a quantum graph

We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.

Eventually and asymptotically positive semigroups on Banach lattices

Abstract We develop a theory of eventually positive C 0 -semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on L p -spaces, the Dirichlet-to-Neumann operator on L 2 and the Laplacian with non-local boundary conditions on L 2 within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call “asymptotic positivity”, where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as t → ∞ . This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.

On the isoperimetric problem for the Laplacian with Robin and Wentzell boundary conditions

We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions ∂u ∂ν +αu = 0 and generalised Wentzell boundary conditions ∆u+β ∂u ∂ν +γu = 0 with respect to the domain Ω ⊂ R on which the problem is defined. For the Robin problem, when α > 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. 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 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.

The Cheeger constant of a quantum graph: The Cheeger constant of a quantum graph

We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.