A Hadamard variational formula for p-capacity of convex bodies in R^n is established when 1 < p < n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge–Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1 < p < n and existence and regularity when 1 < p < 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).
THE HADAMARD VARIATIONAL FORMULA AND THE MINKOWSKI PROBLEM FOR p-CAPACITY / Andrea Colesanti; Kaj Nystrom; Paolo Salani; Jie Xiao; Deane Yang; Gaoyong Zhang. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - STAMPA. - 285:(2015), pp. 1511-1588. [10.1016/j.aim.2015.06.022]
THE HADAMARD VARIATIONAL FORMULA AND THE MINKOWSKI PROBLEM FOR p-CAPACITY
COLESANTI, ANDREA;SALANI, PAOLO;
2015
Abstract
A Hadamard variational formula for p-capacity of convex bodies in R^n is established when 1 < p < n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge–Ampère type equation. Uniqueness for the Minkowski problem for p-capacity is established when 1 < p < n and existence and regularity when 1 < p < 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).
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