Brief communications

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

A. V. Pokrovskii

Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine

Abstract: It is shown that, for any compact set $K\subset\mathbb{R}^n$ ($n\ge 2$) of positive Lebesgue measure and any bounded domain $G\supset K$, there exists a function in the Hölder class $C^{1, 1}(G)$ that is a solution of the minimal surface equation in $G\setminus K$ and cannot be extended from $G\setminus K$ to $G$ as a solution of this equation.

Keywords: minimal surface equation, Hölder class, removable set, nonlinear mapping, Schauder theorem, fixed point.

UDC: 517.956

Received: 16.05.2016

DOI: 10.4213/faa3446

 English version: Functional Analysis and Its Applications, 2018, 52:1, 62–65

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