The infinity-potential in the square

Abstract A representation formula for the solution of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">∞</m:mi> </m:math> {\infty} -Laplace equation is constructed in a punctured square, the prescribed boundary values being <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {u=0} on the sides and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {u=1} at the centre. This so-called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">∞</m:mi> </m:math> {\infty} -potential is obtained with a hodograph method. The heat equation is used and one of Jacobi’s Theta functions appears. The formula disproves a conjecture.