The Bj\"orling problem and its solution is a well known result for minimal surfaces in Euclidean three-space. The minimal surface equation is similar to the Born-Infeld equation, which is naturally studied in physics. In this semi-expository article, we ask the question of the Bj\"orling problem for Born-Infeld solitons. This begins with the case of locally Born-Infeld soliton surfaces and later moves on to graph-like surfaces. We also present some results about their representation formulae.
The Dirichlet problem for the minimal surface equation in $\rm Sol_3$, with possible infinite boundary data
The Dirichlet problem for the minimal surface equation -with possible infinite boundary data- over domains in a Riemannian surface
The Dirichlet problem for the minimal surface equation on unbounded helicoidal domains of $\mathbb{R}^{m}$
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