## Illinois Journal of Mathematics

### The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data

Minh Hoang Nguyen

#### Abstract

In this paper, we study the Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$ with possible infinite boundary data, where $\mathrm{Sol}_{3}$ is the non-Abelian solvable $3$-dimensional Lie group equipped with its usual left-invariant metric that makes it into a model space for one of the eight Thurston geometries. Our main result is a Jenkins–Serrin type theorem which establishes necessary and sufficient conditions for the existence and uniqueness of certain minimal Killing graphs with a non-unitary Killing vector field in $\mathrm{Sol}_{3}$.

#### Article information

Source
Illinois J. Math., Volume 58, Number 4 (2014), 891-937.

Dates
Received: 28 January 2014
Revised: 9 June 2015
First available in Project Euclid: 6 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1446819293

Digital Object Identifier
doi:10.1215/ijm/1446819293

Mathematical Reviews number (MathSciNet)
MR3421591

Zentralblatt MATH identifier
1328.53077

#### Citation

Nguyen, Minh Hoang. The Dirichlet problem for the minimal surface equation in $\mathrm{Sol}_{3}$, with possible infinite boundary data. Illinois J. Math. 58 (2014), no. 4, 891--937. doi:10.1215/ijm/1446819293. https://projecteuclid.org/euclid.ijm/1446819293

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