Concentration of Eigenfunctions Near a Concave Boundary

This article concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L ^p norms of the restrictions of eigenfunctions to broken geodesics.     

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.     

Randwertaufgaben in der verallgemeinerten Potentialtheorie

The Neumann and Dirichlet boundary value problem of generalized potential theory is considered. Based on a compact imbedding result, existence and uniqueness theorems are obtained for Riemannian manifolds with compact boundary (‘interior and exterior domains’).     

Nodal volumes for eigenfunctions of analytic regular elliptic problems

We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions of a regular elliptic problem on a bounded domain Ω ⊆ ℝ ⁿ with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary.     

Dirichlet to Neumann operator on differential forms

We define the Dirichlet to Neumann operator on exterior differential forms for a compact Riemannian manifold with boundary and prove that the real additive cohomology structure of the manifold is determined by the DN operator. In particular, an explicit formula is obtained which expresses Betti numbers of the manifold through the DN operator. We express also the Hilbert transform through the DN map. The Hilbert transform connects boundary traces of conjugate co-closed forms.     

A posteriors error bounds for inhomogeneous linear dirichlet and neumann problems

Second-order inhomogeneous linear Dirichlet and Neumann problems in divergent form on a simply-connected to-dimensional domain with Lipschitz-continious boundary of finite length are considered. Conjugate problems, that is, a pair of one Dirichlet and one Neumann problem the minima of energies of which add to a known constant, are introduced. From the conceppt of conjugate problems, two-sided bounds for the energy of the exact solution of any given Dirichlet or Neumann problem are constructed. These two-sided bounds for the energy at the exact solution are in turn used to obtain easily calculable a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the given problem     

A synthetic characterization of the hemisphere

We show that round hemispheres are the only compact two-dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp isoperimetric inequality for surfaces with boundary such that every pair of geodesics has at most one interior intersection point.

     

A posteriori error bounds for quasi-linear Dirichlet and Neumann problems in full divergent form

Second-order quasi-linear Dirichlet and Neumann problems in four-term divergent form on a simply connected domain with a Lipschitz-continuous boundary of finite length are considered. Derivatives and primitives of distributions on the boundary are defined in such a way that for sufficiently smooth boundary distributions, these derivatives and primitives coincide with derivatives and primitives with respect to arc length on the boundary. Using these concepts, conjugate problems, that is, a pair of one Dirichlet and one Neumann problem, the minima of the energies of which add to zero, are introduced. From the concept of conjugate problems, two-sided bounds for the energy of the exact solution of any given Dirichlet or Neumann problem are constructed. These two-sided bounds for the energy at the exact solution are in turn used to obtain a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the problem. These a posteriori bounds consist of a constant times the sum of the energies of the approximate solutions of the conjugate Dirichlet and Neumann problems and are easily constructed numerically.     

Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s=0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523-540], to a noncompact case.     

Quantum Ergodic Restriction Theorems. I: Interior Hypersurfaces in Domains with Ergodic Billiards

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions \({\phi_j |_H}\) to H of Δ-eigenfunctions of Riemannian manifolds (M, g) with ergodic geodesic flow are quantum ergodic on H. We prove two kinds of results: First (i) for any smooth hypersurface H in a piecewise-analytic Euclidean domain, the Cauchy data \({(\phi_j|H,\partial_{\nu}^H \phi_j|H)}\) is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly, (ii) we give conditions on H so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincaré maps for H. The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an ‘almost-orthogonality’ result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow.     

Numerical solutions of the Dirichlet problem via a density theorem

Under mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for L^p boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.     

Dirichlet and neumann spectrum

It is well known that the critical points of the Rayleigh quotient for the Dirichlet form relative to the square integral norm are eigenfunctions of the Laplace operator with Neumann boundary conditions and eigenvalue equal to the critical value. Clearly then, if we fix a critical point and consider the corresponding value and restrict the Rayleigh quotient to the space of eigenfunctions of the Laplace operator with that given eigenvalue, then the critical point remains a critical point of the restriction. In this paper we answer the converse question. We show that if a given number occurs as a critical value on the subspace of eigenfunctions with that same number as eigenvalue, then either it is a critical value for the quotient on the entire space or it is a critical value on the subspace of all functions satisfying Dirichlet boundary conditions. Moreover, a precise relation holds between the corresponding critical points. A similar relation holds for any elliptic operator in divergence form. The relation stems from a lattice identity induced by a certain indefinite quadratic form, involving certain subspaces. This theory has been successfully applied to calculating the cuspidal spectrum in a certain range for the Laplace-Beltrami operator for certain finite volume hyperbolic surfaces, addressing a conjecture of Roelcke and Selberg, and recent conjectures of Sarnak; see [1]. The numerical method used can be applied quite generally, typically to equations related to underlying symmetries where decompositions of the group give rise to lots of formal solutions.     

Full asymptotics and Laurent series of layer potentials for Laplace's equation on the half‐space

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Ein Hodge-Satz für Mannigfaltigkeiten mit nicht-glattem Rand

In this paper the question of determining the dimension of the space of harmonic Dirichlet and Neumann differential forms on a Riemannian manifold with non-smooth boundary is answered for a wide class of boundaries. The admissible boundaries can be characterized using a generalized “global segment property”. The well-known relation between the Betti numbers and the dimension of these spaces is established in this more general case, too. Bounded and non-bounded manifolds are treated (“exterior and interior domains”).     

Maximum principle for hypersurfaces

We give an intrinsic proof and a generalization of the interior and boundary maximum principle for hypersurfaces in Riemannian and Lorentzian manifolds. Moreover, we show some new applications to manifolds with lower Ricci curvature bounds. E.g. we prove a local and a Lorentzian version of Cheng's maximal diameter theorem and a non-existence result for closed minimal hypersurfaces.     

Analyticity of solutions for semilinear elliptic systems of second order

This paper deals with systems , , where the right hand side is a -valued, real analytic function. We prove that a solution of such a system can be continued across a straight line segment , if one prescribe certain nonlinear, mixed boundary conditions on , which are assumed to be real analytic too. This continuation will be constructed by solving certain hyperbolic initial boundary value problems, generalizing an idea of H. Lewy. We apply this result to surfaces of prescribed mean curvature and to minimal surfaces in Riemannian manifolds spanned into a regular Jordan curve : Supposing analyticity of all data, we show that both types of surfaces can be continued across . Received: 29 December 2000 / Accepted: 11 July 2001 / Published online: 29 April 2002     

Complex Monge-Ampère equations and totally real submanifolds

We study the Dirichlet problem for complex Monge-Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kähler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampère (HCMA) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kähler metrics.     

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L^p-decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.     

Lower bounds for the blow-up time in a non-local reaction–diffusion problem

For a non-local reaction–diffusion problem with either homogeneous Dirichlet or homogeneous Neumann boundary conditions, the questions of blow-up are investigated. Specifically, if the solutions blow up, lower bounds for the time of blow-up are derived.