Quasilinear hyperbolic operators with Log-Lipschitz coefficients

We know that the Cauchy problem for a linear strictly hyperbolic operator with Log-Lipschitz in time coefficients is well posed in C^∞. Here we show that the same result is valid also in the case of a quasilinear operator, but only locally in time.     

Quasilinear Burgers-Hopf equation and Stasheff polytopes

The generating function for the numbers of faces of the nth Stasheff polytope is shown to satisfy the quasilinear Burgers-Hopf equation. Applications of this result are given.     

Quasilinear equation in moving domain

In this article we are concerned with the existence and uniqueness of global weak solutions of a mixed problem associated with one-dimensional damped elastic stretched string equation when the supports of the ends have small displacements. In addition, we show that the energy decays exponentially. In previous investigations about string equation in moving domain, local or global solutions for increasing domain with the growth of the time has been shown. Here, thanks to the internal strong damping we eliminate this hypothesis.     

The wave equation for Dunkl operators

Let k = (k_α)_αε, be a positive-real valued multiplicity function related to a root system , and Δ_k be the Dunkl-Laplacian operator. For (x, t) ε ^N, × , denote by u_k(x, t) the solution to the deformed wave equation Δ_ku_k,(x, t) = δ_ttu_k(x, t), where the initial data belong to the Schwartz space on ^N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σ_αε+k_α ε . Here ⁺ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of u_k(x, t) is contained in the conical shell {(x, t), ε ^N × | |t| − R x |t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of u_k is, for large |t|, partitioned into equal potential and kinetic parts.     

An algebraic equation for linear operators

Let T be a linear operator on a vector space V, possibly of infinite dimension, over a general field K. We solve the functional equation p(T) = F where p ∈ K[x] and F, an algebraic operator on V, are given. For nilpotent F we give an explicit linear system which determines the solutions by their similarity classes. The method is based on a canonical decomposition theorem.     

Quasilinear iterative scheme for a fourth-order differential equation with retardation and anticipation

In this paper, the estimations of error between the approximate solution and the solution for a fourth-order differential equation with retardation and anticipation are given by employing the quasilinear iterative scheme and the approximate quasilinear iterative scheme.     

Boundary integral operators for the heat equation

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.Parts of this work were done while the author had visiting positions at the Carnegie Mellon University, Pittsburgh, USA, and at the Université de Nantes, France, or was supported by the DFG-Forschergruppe KO 634/32-1.     

A functional equation related to symmetry of operators

In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\). We require no additional assumptions on the function \(f\). Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein–Uhlenbeck process.     

Wave operators for pairs of spaces and the Klein-Gordon equation

We give a new criterion for the existence of wave operators for pairs of self adjoint operators acting in different Hilbert spaces. We apply this method to obtain results for the Klein-Gordon equation.     

一类混合单调算子方程的求解及应用

利用锥理论和半序方法证明了Banach空间中一类非线性二元算子方程的解的存在唯一性定理,并给出迭代序列收敛于解的误差估计,作为应用,讨论了不具有单调性的算子方程的可解性,改进并推广了已有的一些结果.     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.