Variational problems in $L^{\infty }$ involving semilinear second order differential operators

Abstract

For an elliptic, semilinear differential operator of the form $S(u)=A:D^{2}u+b(x,u,Du)$, consider the functional $E_{\infty }(u)={\mathrm{esssup}}_{\Omega }|S(u)|$. We study minimisers of $E_{\infty }$ 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.

21 pages, journal: ESAIM-COCV

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