Fundamental solutions of homogeneous fully nonlinear elliptic equations

Abstract

We prove the existence of two fundamental solutions $Φ$ and $\tilde Φ$ of the PDE \[ F(D^2Φ) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \} \] for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $α^*, \tildeα^* > -1$ which describe the homogeneity of $Φ$ and $\tilde Φ$. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of $F(D^2u) = 0$ in $\mathbb{R}^n \setminus \{0 \}$ which are bounded on one side in a neighborhood of the origin as well as at infinity. Finally, we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two-player differential game.