On fully nonlinear PDEs derived from variational problems of \(L^{p}\) norms (Q2753562)

This paper considers the variational problem: NEWLINE\[NEWLINE \inf \biggl\{ \|D v \|^{p} - \int_{\Omega} f v dx \mid v \in W_{0}^{1,p} (\Omega) \biggr\}, NEWLINE\]NEWLINE where \(p > 1\), \(\Omega \subseteq \mathbb{R}^{n}\) is a bounded domain with a smooth boundary, \(Dv\) denotes the gradient of \(v\) and \(\|D v \|^{p}\) is a norm equivalent to the standard \(\|\cdot \|_{W^{1,p}(\Omega)}\) norm. Depending on the norm chosen one may associate with the minimizer of this problem an Euler-Lagrange equation. Denote this minimizer by \(\hat{u}^{p}\). NEWLINENEWLINENEWLINEThis paper investigates the conditions under which \(\hat{u}^{p}\) satisfies the associated Euler-Lagrange equation in the viscosity sense. When \(\min_{\overline{\Omega}} f(x) > 0\) the limiting function \(\hat{u}:= \lim_{p \to \infty} \hat{u}_{p}\) is then shown to be a viscosity solution of the corresponding limiting partial differential equation (PDE). This verification result immediately gives the existence of the viscosity solution of the limiting PDE. Problems arise when trying to prove the comparison principle for the limiting PDE due to discontinuities with respect to the \(Du\)-variables. The standard arguments for establishing the comparison principle do not apply so a local aconvexity or a local concavity assumption is invoked to overcome these difficulties. When \(\Omega\) is convex, sufficient conditions for power concavity of the solution are obtained.