A Relaxation of a Min-Cut Problem in an Anisotropic Continuous Network

Strang [18] introduced optimization problems on a Euclidean domain which are closely related with problems in mechanics and noted that the problems are regarded as continuous versions of famous max-flow and min-cut problems. In [15] we generalized the problems and called the generalized problems max-flow and min-cut problems of Strang's type. In this paper we formulate a relaxed version of the min-cut problem of Strang's type and prove the existence of optimal solutions under some suitable conditions. The conditions are essential. In fact, there is an example of the relaxed version which has no optimal solutions if the conditions are not fulfilled. We give such an example in the final section. Accepted 8 October 1998