Variational problems in involving semilinear second order differential operators p, li { white-space: pre-wrap; }

Abstract

For an elliptic, semilinear differential operator of the form S ( u ) = A : D 2 u + b ( x , u , Du ), consider the functional E ∞ ( u ) = ess sup Ω , | 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.