Dirichlet Operators Research Articles

Delocalization of quasimodes on the disk

This note deals with semiclassical measures associated with (sufficiently accurate) quasimodes (uh) for the Laplace–Dirichlet operator on the disk. In this time-independent set-up, we simplify the statements of [3] and their proofs. We describe the restriction of semiclassical measures to every invariant torus in terms of two-microlocal measures. As corollaries, we show regularity and delocalization properties for limit measures of |uh|2dx: these are absolutely continuous in the interior of the disk and charge every open set intersecting the boundary.

Nonlinear parabolic SPDEs involving Dirichlet operators

We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet forms. In proofs we combine the methods of backward doubly stochastic differential equations with those of probabilistic potential theory and Dirichlet forms.

Dirichlet operators: A priori estimates and uniqueness problems II

We study some basic properties of the Dirichlet operator in weighted Lp spaces. We work under rather general assumptions on weights that destroy minimal local Sobolev regularity of the form domain.

Elliptic differential operators on Lipschitz domains and abstract boundary value problems

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi-boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non-smooth domains. A key feature is the extension of the boundary maps by continuity to the duals of certain range spaces, which directly leads to a description of all self-adjoint extensions of the underlying symmetric operator with the help of abstract boundary values. In the second part of the paper a complete description is obtained of all self-adjoint realizations of the Laplacian on bounded Lipschitz domains, as well as Kreĭn type resolvent formulas and a spectral characterization in terms of energy dependent Dirichlet-to-Neumann maps. These results can be viewed as the natural generalization of recent results by Gesztesy and Mitrea for quasi-convex domains. In this connection we also characterize the maximal range spaces of the Dirichlet and Neumann trace operators on a bounded Lipschitz domain in terms of the Dirichlet-to-Neumann map. The general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains, and particular attention is paid to the second order case which is illustrated with various examples.

Spectral properties of the Dirichlet operator $\sum _{i=1}^{d}(-\partial _i^2)^{s}$∑i=1d(−∂i2)s on domains in d-dimensional Euclidean space

In this article, we investigate the distribution of eigenvalues of the Dirichlet pseudo-differential operator \documentclass[12pt]{minimal}\begin{document}$\sum _{i=1}^{d}(-\partial _i^2)^{s}, \, s\in (0,1]$\end{document}∑i=1d(−∂i2)s,s∈(0,1] on an open and bounded subdomain \documentclass[12pt]{minimal}\begin{document}$\Omega \subset \mathbb {R}^d$\end{document}Ω⊂Rd and predict bounds on the sum of the first N eigenvalues, the counting function, the Riesz means, and the trace of the heat kernel. Moreover, utilizing the connection of coherent states to the semi-classical approach of quantum mechanics, we determine the sum for moments of eigenvalues of the associated Schrödinger operator.

Pointwise estimates for ground states of some classes of positivity preserving operators

We establish pointwise estimates for the ground states of some classes of positivity preserving operators. The considered operators are negatively perturbed (by measures) strongly local Dirichlet operators. These estimates will be written in terms of the Green's kernel of the considered operators, whose existence will be proved. In many circumstances our estimates are even sharp so that they recover known results about the subject. The results will deserve to obtain large time heat kernel estimates for the related operators.

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $\Omega\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(\Omega)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partial\Omega)$.

Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

We restrict a quantum particle under a Coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three-dimensional tubes along the x axis and diameter ɛ, and study the confining limit ɛ → 0. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.

An Inverse Problem of Calderón Type with Partial Data

A generalized variant of the Calderón problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension n ≥ 2. The following two results are shown: (i) The selfadjoint Dirichlet operator associated with an elliptic differential expression on a bounded Lipschitz domain is determined uniquely up to unitary equivalence by the knowledge of the Dirichlet-to-Neumann map on an open subset of the boundary, and (ii) the Dirichlet operator can be reconstructed from the residuals of the Dirichlet-to-Neumann map on this subset.

Tunneling for a Class of Difference Operators

We analyze a general class of difference operators \({H_\varepsilon = T_\varepsilon + V_\varepsilon}\) on \({\ell^2((\varepsilon \mathbb {Z})^d)}\) where \({V_\varepsilon}\) is a multi-well potential and \({\varepsilon}\) is a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we shall treat the eigenvalue problem for \({H_\varepsilon}\) as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix, similar to the analysis for the Schrödinger operator [see Helffer and Sjöstrand in Commun Partial Differ Equ 9:337–408, 1984], and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.

Strong uniqueness for both Dirichlet operators and stochastic dynamics to Gibbs measures on a path space with exponential interactions

We prove L p -uniqueness of Dirichlet operators for Gibbs measures on the path space C ( R , R d ) associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors. We also give an SPDE characterization of the corresponding dynamics. In particular, for the first time, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous. As examples, to which our results apply, we mention the stochastic quantization of P ( ϕ ) 1 -, exp ( ϕ ) 1 -, and trigonometric quantum fields in infinite volume. In particular, our results imply essential self-adjointness of the generator of the stochastic dynamics for these models. Finally, as an application of the strong uniqueness result for the SPDE, we prove some functional inequalities for diffusion semigroups generated by the above Dirichlet operators.

Minimization of the Ground State for Two Phase Conductors in Low Contrast Regime

In this article we consider the problem of the optimal distribution of two conducting materials with given volume inside a fixed domain, in order to minimize the first eigenvalue (the ground state) of a Dirichlet operator. It is known, when the domain is a ball, that the solution is radial, and it was conjectured that the optimal distribution of the materials consists of putting the material with the highest conductivity in a ball around the center. We show that this conjecture is not true in general. For this, we consider the particular case where the two conductivities are close to each other (low contrast regime) and we perform an asymptotic expansion with respect to the difference of conductivities. We find that the optimal solution is the union of a ball and an outer ring when the amount of the material with the higher density is large enough.

Dirichlet operators with convex potentials on the half-line

We consider the Dirichlet operator H t =−d 2/dx 2+q(x) on L 2([t,∞)), where q is a convex potential with q(x)→∞ as x→∞. We show that the eigenvalue gap Γ(t) of H t is monotone increasing as t increases from −∞ to ∞. We also show that Γ(t) is strictly increasing if q is not linear at infinity. An asymptotic estimate of Γ(t) for quadratic potentials is obtained.

Hilbert, Dirichlet and Fejér families of operators arising from C 0-groups, cosine functions and holomorphic semigroups

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejer operators (defined by convolution with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different approaches of other authors whose starting points were usually C 0-groups or cosine functions. We present relations with holomorphic semigroups. We characterize the geometric property of UMD spaces in terms of the Dirichlet and Fejer operators. To end the paper, we give examples to illustrate our results.

Sloshing of a layered fluid with a free surface as a Hamiltonian system

The aim of this paper is the investigation of a layered sloshing fluid system using both a new Hamiltonian mathematical model and new laboratory experiments. The mathematical model is defined for a cylindrical tank with an arbitrary shape and subjected to an arbitrary rigid motion. The model consists of a pure evolution system of partial differential first order equations in the canonical four unknowns: water elevation at the upper free surface and at the separation surface and the gap in momentum potential density computed at each fluid surface. The system of equations is obtained by avoiding the construction of the Hamiltonian and its variational derivatives. An important advantage of this formulation, with respect to the Lagrangian formulation, is that the nonevolution constraint, which imposes for each fluid the same velocity component along the normal direction of the separation surface, is fulfilled by the model itself. The model implementation needs to define the so-called Neumann–Dirichlet operators, which are computed by an efficient algorithm, for any instantaneous configuration of the two fluid domains. A numerical integration of the model is performed by a suitable Galerkin projection of the evolution equations. New laboratory experiments, simulating the sloshing of a layered fluid system, inside a tank with a squared cross section, were performed. The experiments, with a two-dimensional sloshing, were carried out by varying the forcing frequency, the oscillation amplitude and the ratio of the two fluids’ depths. In some experiments, a traveling wave, with a shape similar to a moving hydraulic jump, was observed at the separation surface. Measurements of the space-time evolution of both the free and the separation surfaces were performed and compared with the model’s predictions. A good agreement between the model predictions and laboratory measurements is found, even for strong nonlinear cases such as the traveling wave.

Shape derivative for a two-phase eigenvalue problem and optimal configurations in a ball

In this article we deal with the problem of distributing two conducting materials in a given domain, with their proportions being xed, so as to minimize the rst eigenvalue of a Dirichlet operator. When the design region is a ball, it is known that there is an optimal distribution of materials which does not involve the mixing of the materials. However, the optimal conguration even in this simple case is not known. As a step in the resolution of this problem, in this paper, we develop the shape derivative analysis for this two-phase eigenvalue problem in a general domain. We also obtain a formula for the shape derivative in the form of a boundary integral and obtain a simple expression for it in the case of a ball. We then present some numerical calculations to support our conjecture that the optimal distribution in a ball should consist in putting the material with higher conductivity in a concentric ball at the centre.

ON THE STRONG UNIQUENESS OF SOME FINITE DIMENSIONAL DIRICHLET OPERATORS

We prove essential self-adjointness of a class of Dirichlet operators in ℝnusing the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.

Reflection positivity and monotonicity

We prove general reflection positivity results on Riemannian manifolds for both scalar fields and Dirac fields, and comment on applications to quantum field theory. As another application, we prove the inequality CD⩽CN between Dirichlet and Neumann covariance operators on a manifold with a reflection.

$L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators

$L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators

Sharp estimates for large coupling convergence with applications to Dirichlet operators

Let H be a nonnegative selfadjoint operator, E the closed quadratic form associated with H, and P a nonnegative quadratic form such that E + P is closed and D ( P ) ⊃ D ( H ) . For every β > 0 let H β be the selfadjoint operator associated with E + β P . The pairs ( H , P ) satisfying L ( H , P ) : = lim inf β → ∞ β ‖ ( H β + 1 ) −1 − lim β ′ → ∞ ( H β ′ + 1 ) −1 ‖ < ∞ are characterized. A sufficient condition for convergence of the operators ( H β + 1 ) −1 within a Schatten–von Neumann class of finite order is derived. It is shown that L ( H , P ) = 1 , if E is a regular conservative Dirichlet form with the strong local property and P the killing form corresponding to the equilibrium measure of a closed set with finite capacity and nonempty interior. An example is given where L ( H , P ) is finite, H is a regular Dirichlet operator and P the killing form corresponding to a measure which has infinite mass and a support with infinite capacity.