On the Lawson-Osserman conjecture

Abstract

Abstract We prove that if $u : B_{1} \subset \mathbb{R}^{2} \rightarrow \mathbb{R}^{n}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> is a Lipschitz critical point of the area functional with respect to outer variations, then $u$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> </mml:math> is smooth. This solves a conjecture of Lawson and Osserman from 1977 in the planar case.