Fully nonlinear elliptic equations

Problem's Description

Consider the Dirichlet problem $$\cases{F(D^2u)=0 &in $\Omega$\cr u=\phi &on $\partial\Omega\;,$\cr}(1)$$ where $\Omega \subset {\bf R}^n$ is a bounded domain with smooth boundary $\partial \Omega$, $\phi$ is a continuous function on $\partial\Omega$ and $F$ is a fully nonlinear uniformly elliptic operator, i.e., we assume that $F$ is a $C^1$ function defined on the space of ${n\times n}$ symmetric matrices satisfying the uniform ellipticity condition: there exists a constant $C=C(F)\ge 1$ (called an ellipticity constant) such that $${1\over C}|\xi|^2\le F_{u_{ij}}\xi_i\xi_j\le C |\xi |^2\;, \forall\xi\in {\bf R}^n\;.$$ Here, $u_{ij}$ denotes the partial derivative $\partial^2 u/\partial x_i\partial x_j$. A function $u$ is called a classical solution of (1) if $u\in C^2(\Omega)$ and $u$ satisfies (1). Actually, any classical solution of (1) is a smooth ($C^{\alpha +3}$) solution, provided that $F$ is a smooth $(C^\alpha )$ function of its arguments.

The problem of the existence of classical solutions for the Dirichlet problem (1) is a fundamental problem for fully nonlinear elliptic equations. For $n=2$ problem (1) has always a classical solutions, for $n\geq 4$ the problem of classical solutions was settled negatively. For $n=3$ the problem remains open.

1. Book Nonlinear elliptic equations and nonassociative algebras

   • N. Nadirashvili,
   • V. Tkachev,
   • S. Vladuts

   year of publication: 2014