Differential Tests for Plurisubharmonic Functions and Koch Curves

Abstract

We study minimum sets of singular plurisubharmonic functions and their relation to upper contact sets. In particular we develop an algorithm checking when a naturally parametrized curve is such a minimum set. The case of Koch curves is studied in detail. We also study the size of the set of upper non-contact points. We show that this set is always of Lebesgue measure zero thus answering an open problem in the viscosity approach to the complex Monge-Ampère equation. Finally, we prove that similarly to the case of convex functions, strictly plurisubharmonic lower tests yield existence of upper tests with a control on the opening.

Plurisubharmonic envelopes and supersolutions

“…We refer the reader to [EGZ11, W12, Ze13] for basic properties of viscosity sub/super-solutions to degenerate complex Monge-Ampère equations. We stress that viscosity subsolutions are θ-psh [EGZ11] and admit upper tests at almost every point by [DD16]. The analogous properties for viscosity supersolutions are far less obvious.…”

Fractal Analysis of the Fracture Evolution of Freeze-Thaw Damage to Asphalt Concrete

“…In order to simulate the fractal characteristics of fracture structures, the Koch curve is used. The Koch curve is a kind of typical fractal curve, which was first proposed by Koch, H.von in 1904 [36] and is now widely used to analyze specific and complex engineering problems. The construction process of the Koch fractal curve is shown in Figure 10.…”

Alexandrov-Bakelman-Pucci type estimate for plurisubharmonic functions

“…As one application of this generalization we mention that the following theorem was proved in [DD19] (Theorem 31) with the extra assumption, that u is continuous.…”

“…The only place where we needed the continuity was Lemma 29 in [DD19], which now can be substituted by Corollary 18 below, so the continuity assumption can be dropped.…”