An Application of the Theory of Viscosity Solutions to Higher Order Differential Equations

Abstract

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,\alpha}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian equation on a ball $B_R\subset \mathbb{R}^n$: $$\Delta_\infty^2 u:=(\Delta u)^3 |D(\Delta u)|^2 =f(x)$$ with Navier Boundary conditions ($u=g\in C(\partial B_R), \Delta u =0 \textrm{ on } \partial B_R$). We also prove that there exists a solution in $C^{1,\alpha}(\mathbb{R}^n)$ for all $\alpha>0$ to the eigenvalue problem on $\mathbb{R}^n$: $$\Delta_\infty^2 u =-\lambda u+f(x)$$ whenever $n\geq 3, \lambda<0,$ and $f(x)$ is continuous, bounded, and supported on an annulus.