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Communications on Pure and Applied Mathematics 64, 6 (2011) 737-777

Versions disponibles :

v1 (24-10-2009)

v2 (18-12-2009)

v3 (11-02-2010)

Fundamental solutions of fully nonlinear elliptic equations

Scott Armstrong 1, Boyan Sirakov 2, Charles Smart 3

(2011)

We prove the existence of two fundamental solutions $\Phi$ and $\tilde \Phi$ of the PDE \begin{equation*} F(D^2\Phi) = 0 \quad \mbox{in} \ \R^n \setminus \{ 0 \} \end{equation*} for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $\alpha^*, \tilde\alpha^* > -1$ which describe the homogeneity of $\Phi$ and $\tilde \Phi$. 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 $\R^n \setminus \{ 0 \}$ which are bounded on one side in both 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.

1 :	Department of Mathematics
	Louisiana State University
2 :	Modélisation aléatoire de Paris X (MODAL'X)
	Université Paris X - Paris Ouest Nanterre La Défense
3 :	Courant Institute of Mathematical Sciences
	New York University

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Mathématiques/Equations aux dérivées partielles

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Contributeur : Boyan Sirakov <>

Soumis le : Jeudi 11 Février 2010, 19:31:36

Dernière modification le : Vendredi 13 Avril 2012, 20:42:46