Daniel Daners

Heat Kernel Estimates for Operators with Boundary Conditions

We prove Gaussian upper bounds for kernels associated with non– symmetric, non – autonomous second order parabolic operators of divergence form subject to various boundary conditions. The growth of the kernel in time is determined by the boundary conditions and the geometric properties of the domain. The theory gives a unified treatment for Dirichlet, Neumann and Robin boundary conditions, and the existence of a Gaussian type bound is essentially reduced to verifying some properties of the Hilbert space in the weak formulation of the problem.

Dirichlet problems on varying domains

Abstract The aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with Dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain. Necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators. Examples are given to illustrate that most results are optimal.

A PRIORI ESTIMATES FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS

In this paper we prove a priori estimates for a class of quasi-linear elliptic equations. To make the proofs clear and transparent we concentrate on the p-Laplacian. We focus on L p -estimates for weak solutions of the problem with all standard boundary conditions on non-smooth domains. As an application we prove existence, continuity and compactness of the resolvent operator. We finally prove estimates for solutions to equations with non-linear source and show that, under suitable growth conditions, all solutions are globally bounded.

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.

Domain Perturbation for Linear and Semi-Linear Boundary Value Problems

Abstract This is a survey on elliptic boundary value problems on varying domains and tools needed for that. Such problems arise in numerical analysis, in shape optimisation problems and in the investigation of the solution structure of nonlinear elliptic equations. The methods are also useful to obtain certain results for equations on non-smooth domains by approximation by smooth domains. Domain independent estimates and smoothing properties are an essential tool to deal with domain perturbation problems, especially for non-linear equations. Hence we discuss such estimates extensively, together with some abstract results on linear operators. A second major part deals with specific domain perturbation results for linear equations with various boundary conditions. We completely characterise convergence for Dirichlet boundary conditions and also give simple sufficient conditions. We then prove boundary homogenisation results for Robin boundary conditions on domains with fast oscillating boundaries, where the boundary condition changes in the limit. We finally mention some simple results on problems with Neumann boundary conditions. The final part is concerned about non-linear problems, using the Leray-Schauder degree to prove the existence of solutions on slightly perturbed domains. We also demonstrate how to use the approximation results to get solutions to nonlinear equations on unbounded domains.

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.

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.

The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions

In this paper we investigate the singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions subject to Dirichlet boundary conditions at the boundary. We show that the positive periodic solution of the diffusion model tends to the periodic solution of the purely kinetic model as the diffusion coefficient goes to zero, uniformly in time on compact subsets of the domain.

CHANGE OF STABILITY FOR SCHRODINGER SEMIGROUPS

when the parameter X varies in the positive real axis. Here, m is some bounded and continuous weight function being positive somewhere and u0 an initial condition. The stability is understood as stability in the || • || -norm. We point out that by duality and interpolation we also get the stability in Lp(R ), 1 g p < oo. The question of stability of the zero solution is closely related to the existence of a principal eigenvalue for the elliptic eigenvalue problem

A Liouville theorem for p-harmonic functions on exterior domains

We prove Liouville type theorems for