Properties of some balayage operators, with applications to quadrature domains and moving boundary problems

IN THIS PAPER we define and derive some properties of a fairly wide class of balayage operators. We also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems (e.g. for Hele-Shaw flows). Actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool. Our balayage operators are denoted F = Fp = Fp,R and they depend on an open set R c I?" and a measure p in R; actually we always assume that p has a density (also denoted p) in L”(R). What F does is replace a given measure p by the nearest one (in the energy norm) v = F(p) which satisfies v 5 p in R. This presupposes that fi has finite energy. If this is not the case, there is another definition, which is the one we will actually use: among all distributions u in IRN satisfying u I 17’ (the Newtonian potential of p) in IRN and -Au I p in R there is a largest one, u = VP. Then F(p) = -AI/‘” (by definition). The operator of interest in the above-mentioned applications is F with R = IRN, p = 1. The definition of F(y) when ,U has finite energy can be formulated as an elliptic variational inequality (or linear complementarity problem), and this is one of the standard tools when dealing with free boundary problems of the above kind (see [l-4]). However, we think that the balayage point of view is quite natural in our context. The reason for considering the balayage operators for more general R and p than R = RN, p = 1 is, firstly, that we need them in the proof of one of our theorems (theorem 4.1). Secondly, and perhaps more important, is that we think that the more general operators have quite a lot of intrinsic interest, in particular as they turn out to contain “classical” balayage (of positive measures) as a special case (namely with R bounded and p = 0). Recently, we also learnt that discrete (or numerical) versions of these operators have been developed and applied in geophysics during the last three decades by Dimiter Zidarov, who calls the balayage process “(partial) gravi-equivalent mass scattering”. See [5], and also, for example, [6]. The paper is organized as follows. In Section 2 we define and establish the basic properties of our operators. At the end we also explain their relation to quadrature domains and certain moving boundary problems. Section 3 is devoted to one single theorem (theorem 3.1) and its corollaries. These results concern the F with R = lRN, p = 1 and give rather good geometric