Duality of boundary value problems for minimal and maximal surfaces

In 1966, Jenkins and Serrin gave existence and uniqueness results for infinite boundary value problems of minimal surfaces in the Euclidean space, and after that such solutions have been studied by using the univalent harmonic mapping theory. In this paper, we show that there exists a one-to-one correspondence between solutions of infinite boundary value problems for minimal surfaces and those of lightlike line boundary problems for maximal surfaces in the Lorentz-Minkowski spacetime. We also investigate some symmetry relations associated with the above correspondence together with their conjugations, and observe function theoretical aspects of the geometry of these surfaces. Finally, a reflection property along lightlike line segments on boundaries of maximal surfaces is discussed.