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Register a new userEvolution PDEs and augmented eigenfunctions. Half-line
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The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the {em unified transform} introduced by Fokas in the 90s. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is {em no} suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of {em augmented eigenfunctions}, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.
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On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation of the ground state with respect to the Robin parameter is not a concave function. We conclude from this that the Robin ground stat is not log-concave (and indeed even has some superlevel sets which are non-convex) for small Robin parameter on polyhedral convex domains outside a special class, and hence also on arbitrary convex domains which approximate these in Hausdorff distance.
We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Omega subset mathbb{R}^N$. By means of topological arguments, we show how symmetries of $Omega$ help to construct subsets of $W_0^{1,p}(Omega)$ with suitably high Krasnoselskiu{i} genus. In particular, if $Omega$ is a ball $B subset mathbb{R}^N$, we obtain the following chain of inequalities: $$ lambda_2(p;B) leq dots leq lambda_{N+1}(p;B) leq lambda_ominus(p;B). $$ Here $lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $lambda_ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $lambda_2(p;B)=lambda_ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=infty$ are also considered.
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $phi (Omega)$ parametrized by Lipschitz homeomorphisms $phi $ defined on a fixed reference domain $Omega$. Given two open sets $phi (Omega)$, $tilde phi (Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $|tilde phi -phi |_{W^{1,p}(Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.
We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension $n$. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile-adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.
The linearization of the classical Boussinesq system is solved explicitly in the case of nonzero boundary conditions on the half-line. The analysis relies on the unified transform method of Fokas and is performed in two different frameworks: (i) by exploiting the recently introduced extension of Fokass method to systems of equations; (ii) by expressing the linearized classical Boussinesq system as a single, higher-order equation which is then solved via the usual version of the unified transform. The resulting formula provides a novel representation for the solution of the linearized classical Boussinesq system on the half-line. Moreover, thanks to the uniform convergence at the boundary, the novel formula is shown to satisfy the linearized classical Boussinesq system as well as the prescribed initial and boundary data via a direct calculation.
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