Let Ω ⊂⊂ Cn be a bounded pseudoconvex domain with Lipschitz boundary. Diederich and Fornaess have shown that when the boundary of Ω is C2, there exists a constant 0 < η < 1 and a defining function ρ for Ω such that −(−ρ)η is a plurisubharmonic function on Ω. In this paper, we show that the result of Diederich and Fornaess still holds when the boundary is only Lipschitz.

We denote by (P ) the problem given by the equation (1), the boundary condition (2) and the initial condition (3). We assume that the function c satisfies the hypothesis (Hc) c is a continuous nondecreasing function such that c ′ ∈ Lloc(R); and that the initial condition u0 and the boundary data u D satisfy the hypotheses (H0) u0 ∈ L∞(Ω) and we define U0 := ‖u0‖L∞(Ω); (HD) u D is Lipschitz cont...

The rendering of an implicit surface requires many function evaluations of the underlying implicit function. The time efficiency of the rendering is dominated by the number of function evaluations times the efficiency of these evaluations. If the implicit surface is created by means of a CSG model the function evaluations can be made more efficient if the underlying function is a Lipschitz func...

The Jacobi polynomials induce a translation operator on function spaces on the interval [−1, 1]. For any homogeneous Banach space B w.r.t. this translation, we can study the according little and big Lipschitz spaces, lipB(λ) and LipB(λ), respectively. The big Lipschitz spaces are not homogeneous themselves. Therefore we introduce semihomogeneous Banach spaces w.r.t. Jacobi translation, of which...

We show differentiability of a class of Geroch’s volume functions on globally hyperbolic manifolds. Furthermore, we prove that every volume function satisfies a local anti-Lipschitz condition over causal curves, and that locally Lipschitz time functions which are locally antiLipschitz can be uniformly approximated by smooth time functions with timelike gradient. Finally, we prove that in stably...

Many interesting real functions on Euclidean space are differentiable almost everywhere. All Lipschitz functions have this property, but so, for example, does the spectral abscissa of a matrix (a non-Lipschitz function). In practice, the gradient is often easy to compute. We investigate to what extent we can approximate the Clarke subdifferential of such a function at some point by calculating ...

Explicit sharp estimates for the Green function of the Laplacian in C domains were completed in 1986 by Zhao [42]. Sharp estimates of the Green function of Lipschitz domains were given in 2000 by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heat kernel in C domains were established in 2002 by Zhang [41]. Qualitatively sharp heat kernel estimates in Lipschitz domains were...

(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...