A minimum principle for superharmonic functions subject to interface conditions

Abstract

Let D be a bounded domain in R 2 with smooth boundary. Let B 1, …, B m be non-intersecting smooth Jordan curves contained in D, and let D′ denote the complement of ∪ i − 1 m B i respect to D. Suppose that u ϵ C 2(D′) ∩ C( D ̄ ) and Δu ⩽ 0 in D′ (where Δ is the Laplacian), while across each “interface” B i , i = 1,…, m, there is “continuity of flux” (as suggested by the theory of heat conduction). It is proved here that the presence of the interfaces does not alter the conclusions of the classical minimum principle (for Δu ⩽ 0 in D). The result is extended in several regards. Also it is applied to an elliptic free boundary problem and to the proof of uniqueness for steady-state heat conduction in a composite medium. Finally this minimum principle (which assumes “continuity of flux”) is compared with one due to Collatz and Werner which employs an alternative interface condition.