An Application of Viscosity Theory to Higher Order Differential Equations

We directly apply the viscosity theory for fully nonlinear second-order differential equations to higher order differential operators. We prove existence and uniqueness theorems for equations of the general form: $$\epsilon \Delta v+G(D(\Delta v), D^2(\Delta v)) = f(x)$$ when we have appropriate conditions on the functional $G$. As an explicit application, we show that 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 ($u=g\in C(\partial B_R)$, $\Delta u=0$ on $\partial B_R$) admits solutions in $W^{2,\infty}(B_R)$ under some mild conditions on $f(x)$ (e.g. H\"older continuity).Comment: 20 page