Quadrature surfaces as free boundaries

Abstract

This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂Rⁿ be a bounded domain with a C² boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$

Applying simple techniques, we derive basic inequalities and show uniform boundedness for the set of solutions. Distance estimates as well as uniqueness results are obtained in special cases, e.g. we show that if ∂Ω and ∂D are two quadrature surfaces for a fixed measure μ and Ω is convex, then D⊂Ω. The main observation, however, is that if ∂Ω is a quadrature surface for μ≥0 and xε∂Ω, then the inward normal ray to ∂Ω at x intersects the convex hull of supp μ. We also study relations between quadrature surfaces and quadrature domains. D is said to be a quadrature domain with respect to a mesure μ if there is a solution to the following overdetermined Cauchy problem: $\Delta u = 1 - \mu in D, andu = |\nabla u| = 0 on \partial D.$

Finally, we apply our results to a problem of electrochemical machining.