Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL ^p (m), p1 andm not necessarily -finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL ^q (m), qp. It turns out that, in the limitq,A satisfies the positive maximum principle. If the test functionsC _c D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Lévy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL ^p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL ² (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL ² (m) with explicitly given Beurling-Deny formula.     

Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function

In this work a class of nondissipative curves in Hilbert spaces whose correlation functions have a limit ast± is presented. These curves correspond to a class of nondissipative basic operators that are a coupling of a dissipative operator and an antidissipative one. The wave operators and the scattering operator for the couple (A ^*, A) ( ) are obtained. The present work is a continuation and a generalization of the investigations of K.Kirchev and V.Zolotarev [1, 2, 3] on the model representations of curves in Hilbert spaces where the respective semigroup generator is a dissipative operator. This article includes four parts. A new form of the triangular model of M.S. Livic ([4, 5]) for the considered operators is introduced in the first part by the help of a suitable representation of the selfadjoint operatorL. This allows us to describe the studied class of nondissipative curves. The second part studies some results concerning the application of the analogue for multiplicative integrals of the well-known Privalov's theorem ([6]) about the limit values in the scalar case. This analogue is a reconstruction of measure by limit values in Stieltjes-Perron's style and it is obtained by L.A. Sakhnovich ([7]). Another problem, considered in the second part is the analogue inC ^m of the classical gamma-function and several properties for further consideration. In the third part the asymptotics of the studied curves corresponding to the nondissipative operators-couplings of a dissipative and an antidissipative operator with absolutely continuous real spectra and the limits of their correlation functions are obtained In the fourth part a scattering theory of a couple (A ^*, A) with a nondissipative operatorA from is constructed as in the selfadjoint case ([8, 9, 10]) and in the dissipative case ([7]). These results show an interesting new effect: the studied nondissipative case is near to the dissipative one.Partially supported by Grant MM-810/98 of MESC     

A Note on π-Engel Conditions

In this paper we show that the weakly -Engel conditions are closely related to the existance of normal -complements; while the -Engel conditions are closely related to the -nilpotent groups.AMS Subject Classification (2000): 20D20     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

Lifting chains of prime ideals to paravaluation rings

Résumé  SoitR ⊂T une extension des anneaux commutatifs et soit {P _α :α ∈I} une cha?ne croissante des idéaux premiers deR (I étant un ensemble totalement ordonné, peut-être infini). Alors il existe un anneau de paravaluationV deT et une cha?ne {Q _α} des idéaux premiers deV de sorte queR ⊂V etQ _α ∩R =P _α pour toutα ∈I. Tout d’abord, on établit le cas spécial dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas général découle comme conséquence. Dans le cas général, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.      

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Exact Internal Controllability of Maxwell's Equations

In this paper we obtain two exact internal controllability results of Maxwell's equations in a general region by using multiplier techniques. The first one is exact controllability in a short time, in which we obtain the ``optimal" (observability) estimates when the location and the shape of the controller is fixed. What happens if we allow the controller to change? Under some conditions, we show that by doing that the system can be exactly controllable within any given time duration, which is our second exact controllability result. Accepted 30 September 1998     

The essential norm of Hankel operator on the Bergman spaces of strongly pseudoconvex domains

Let D be a bounded strongly pseudoconvex domain with smooth boundary in Cⁿ and let fL²(D). For the Hankel operator H_f on the Bergman space A²(D), it is shown that the essential norm of H_f in L²(D) is comparable to the distance norm from H_f to compact Hankel operators. The result extends the previous corresponding version in the disc proved by Lin and Rochberg in Integ.Equat.Oper.Theory 361–372,17 (1993).     

A trace formula for Weyl transforms with radial symbols

A trace formula for Weyl transforms onL ² () with radial symbols is given.This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.