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

TL;DR

This paper extends the application of viscosity solutions to higher-order PDEs, proving existence of solutions for the inhomogeneous infinity-Bilaplacian and related eigenvalue problems in specific function spaces.

Contribution

It introduces a novel application of viscosity solutions to fourth-order PDEs, establishing existence results for complex boundary value and eigenvalue problems.

Findings

Existence of solutions in $C^{2,eta}$ for the inhomogeneous infinity-Bilaplacian.

Existence of solutions in $C^{1,eta}$ for the eigenvalue problem on $ eal^n$.

Application of viscosity solution theory to higher-order PDEs.

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.