Premium
Fundamental solutions of homogeneous fully nonlinear elliptic equations
Author(s) -
Armstrong Scott N.,
Smart Charles K.,
Sirakov Boyan
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20360
Subject(s) - mathematics , bounded function , scaling , exponent , uniqueness , homogeneous , infinity , gravitational singularity , type (biology) , nonlinear system , mathematical analysis , homogeneity (statistics) , pure mathematics , mathematical physics , combinatorics , geometry , physics , quantum mechanics , ecology , philosophy , linguistics , statistics , biology
We prove the existence of two fundamental solutions Φ and $\tilde \Phi$ of the PDE \input amssym $$F(D^2\Phi) = 0 \quad {\rm in} \ {\Bbb{R}}^n \setminus \{ 0 \}$$ for any positively homogeneous, uniformly elliptic operator F . Corresponding to F are two unique scaling exponents α*, $\tilde\alpha^*$ > −1 that describe the homogeneity of Φ 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 2 u ) = 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 2 u ) = 0 in \input amssym ${\Bbb{R}}^n \setminus \{ 0 \}$ that 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. © 2010 Wiley Periodicals, Inc.