Variational Problems in L∞ Involving Semilinear Second Order Differential Operators

Abstract

For an elliptic, semilinear differential operator of the form S(u) = A : D2u + b(x, u, Du), consider the functional E∞(u) = ess supΩ, |S(u)|. We study minimisers of E∞ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem. The theory of partial differential equations therefore becomes available for the study of a large class of variational problems in L∞ for the first time.