The purpose of this paper is to calculate the first variation of capacity and of the lowest eigenvalue for the Dirichlet problem in convex domains in R N . These formulas are well known in the smooth case and are due to Poincare and Hadamard, respectively. The point is to prove them in sufficient generality to make it possible to apply the direct method of the calculus of variations to variational problems involving these functionals. This project was proposed in [CJL] where uniqueness (up to translation) for the variational problem associated to capacity was proved.
The analogous first variation formula for volume was proved by Alexandrov [S, Lemma 6.5.3]. It yields a direct variational proof of the classical theorem of Minkowski. The more typical approach to existence in Minkowski-type problems is to prove it for a dense family of measures and to pass to the limit. In the case of capacity, such an existence proof was carried out in [J].
We begin the paper with an outline of the direct variational approach to the classical Minkowski problem and the statement of the corresponding problem and results for capacity in dimensions N 3. The third section presents the main technical lemma, an elementary lemma for perturbations of support functions. We then deduce the generalized variational formula for capacity and existence in the Minkowski-type variational problem. In the sixth section we present the two-dimensional case, in which the quantity replacing the area is the logarithmic capacity or the transfinite diameter. In the final section we discuss the eigenvalue problem.
It should be mentioned that the proof of the generalized variational formula given here depends on the variational formula for capacity for convex combinations of convex sets proved in [J]. (See Lemma 4.3 below.) It would be nice to have a more direct argument. The argument in [J] leading to this preliminary variational formula is special in several ways. It depends indirectly on Brunn Minkowski-type inequalities of [B] and on article no.