Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

The L p Minkowski problem for q-capacity

In the present paper, we first introduce the concepts of the L p q-capacity measure and L p mixed q-capacity and then prove some geometric properties of L p q-capacity measure and a L p Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the L p Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and L p Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the L p Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on L p Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of L p Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.