Measure estimates of nodal sets of bi-harmonic functions

In this paper, we define a frequency for bi-harmonic functions. By using this frequency, we give the measure estimates of nodal sets of bi-harmonic functions. We also show that the frequency has some interesting properties similar to the frequency of harmonic functions.     

Characterization of octonionic analytic functions

It is shown that there is only one possible way to define the O-analytic functions. A simple way to construct the O-analytic functions is also given.     

Real analytic generalized functions

Real analytic generalized functions are introduced and investigated. The analytic singular support and analytic wave front of a generalized function in are introduced and described. Authors’ addresses: S. Pilipović, Department of Mathematics and Informatics, University of Novi Sad, Trg D. Obradovića 4, 21000 Novi Sad, Serbia; D. Scarpalezos, U.F.R. de Mathématiques, Université Paris 7, 2 Place Jussieu, Paris 5^e, 75005, France; V. Valmorin, Université des Antilles et de la Guyane, Département Math-Info, Campus de Fouillole, 97159 Pointe á Pitre Cedex, France     

The Third Problem for the Laplace Equation with a Boundary Condition from L p

The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in L_p and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C¹ boundary the explicit solution of the problem is calculated.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000     

The Oblique Derivative Problem for the Laplace Equation in a Plain Domain

The oblique derivative problem for the Laplace equation is studied in a planar multiply connected domain. The boundary condition has a form where is the unit normal vector, is the unit tangential vector and is a fixed real number. If is a Hölderian function and the corresponding domain has Ljapunov boundary then the classical problem is studied. If on the boundary and the domain has a locally Lipschitz boundary then a solution, which fulfils the boundary condition in the sense of a nontangential limit, is studied. If is a real measure on the boundary and the domain has bounded cyclic variation then a solution in a sense of distributions is studied. The solution is looked for in a form of a linear combination of a single layer potential and an angular potential.     

Integral means and Dirichlet integral for analytic functions

For normalized analytic functions f in the unit disk, the estimate of the integral means is important in certain problems in fluid dynamics, especially when the functions are non‐vanishing in the punctured unit disk . We consider the problem of finding the extremal function f which maximizes the integral means for f belong to certain classes of analytic functions related to sufficient conditions of univalence. In addition, for certain subclasses of the class of normalized univalent and analytic functions, we solve the extremal problem for the Yamashita functional where denotes the area of the image of under . The first problem was originally discussed by Gromova and Vasil'ev in 2002 while the second by Yamashita in 1990.     

A symmetry property for polyharmonic functions vanishing on equidistant hyperplanes

Let be a polyharmonic function of order N defined on the strip satisfying the growth condition (0.1) for and any compact subinterval K of , and suppose that vanishes on equidistant hyperplanes of the form for and Then it is shown that is odd at t ₀, i.e. that for . The second main result states that u is identically zero provided that u satisfies 0.1 and vanishes on 2N equidistant hyperplanes with distance c .     

Gleason's problem for harmonic Bergman and Bloch functions on half-spaces

On the setting of the half-spaceR ^n–1×R ₊, we investigate Gleason's problem for harmonic Bergman and Bloch functions. We prove that Gleason's problem for the harmonicL ^p -Bergman space is solvable if and only ifp>n. We also prove that Gleason's problem for the harmonic (little) Bloch space is solvable.     

Weighted Sobolev L2 estimates for a class of Fourier integral operators

In this paper we develop elements of the global calculus of Fourier integral operators in ${{\mathbb R}^n}$ under minimal decay assumptions on phases and amplitudes. We also establish global weighted Sobolev L² estimates for a class of Fourier integral operators that appears in the analysis of global smoothing problems for dispersive partial differential equations. As an application, we exhibit a new type of weighted estimates for hyperbolic equations, where the decay of data in space is quantitatively translated into the time decay of solutions.     

Numerical evaluation of analytic functions by Cauchy's theorem

The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.     

On the convergence of the Galerkin method for nonsmooth solutions of integral equations

Summary It is shown that the stability region of the Galerkin method includes solutions not lying in the conventional energy space. Optimal order error estimates for these nonsmooth solutions are derived. The new result is compared with the classical statement by means of the basic potential problem.     

Liouville-type theorems for real sub-Laplacians

An inequality generalizing the classical Liouville and Harnack Theorems for real sub-Laplacians ℒ is proved. A representation formula for functions $u$ for which ℒu is a polynomial is also showed. As a consequence, some conditions are given ensuring that u is a polynomial whenever ℒu is a polynomial. Finally, an application of this last result is given: if ψ is a C ² map commuting with ℒ, then any of its component is a polynomial function. Received: 3 November 2000     

High-frequency asymptotics of waves trapped by underwater ridges and submerged cylinders

As is well-known, underwater ridges and submerged horizontal cylinders can serve as waveguides for surface water waves. For large values of the wavenumber k   in the direction the ridge, there is only one trapped wave (this was proved in Bonnet-Ben Dhia and Joly [Mathematical analysis of guided water waves, SIAM J. Appl. Math. 53 (1993) 1507–1550]. We construct asymptotics of these trapped waves and their frequencies as k→∞$k\to \infty$ by means of reducing the initial problem to a pair of boundary integral equations and then by applying the method of Zhevandrov and Merzon [Asymptotics of eigenfunctions in shallow potential wells and related problems, Amer. Math. Soc. Trans. 208 (2) (2003) 235–284], in order to solve them.     

Heat kernel estimates for jump processes of mixed types on metric measure spaces

In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity where ν is a probability measure on , c(α, x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c ₀(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between γ₁ and γ₂, where either γ₂ ≥ γ₁ > 0 or γ₁ = γ₂ = 0. This example contains mixed symmetric stable processes on as well as mixed relativistic symmetric stable processes on . We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes. Dedicated to Professor Masatoshi Fukushima on the occasion of his 70th birthday. The research of Zhen-Qing Chen is supported in part by NSF Grants DMS-0303310 and DMS-06000206. The research of Takashi Kumagai is supported in part by the Grant-in-Aid for Scientific Research (B) 18340027.     

Square function estimates on layer potentials for higher‐order elliptic equations

In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m , with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.     

Envelopes,widths, and landau problems for analytic functions

Fix a positive integerr and letA denote those analytic functions on the open unit disk whoserth derivative is bounded by 1 and which are real-valued on the interval (–1, 1). For a positive integern letx ₀,,x _n+r be distinct points in (–1, 1) and set

Wavelet moment method for the Cauchy problem for the Helmholtz equation

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method.