Fundamental solutions of homogeneous fully nonlinear elliptic equations

2009/10/21 by Armstrong, Scott N., Sirakov, Boyan, Smart, Charles K. · 2 citations
#35A08 #35J60 #35P30 #49N70 #91A15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0910.4002

Abstract

We prove the existence of two fundamental solutions Φ and Φ of the PDE F(D²Φ) = 0 in ℝⁿ ∖ \0 \ for any positively homogeneous, uniformly elliptic operator F. Corresponding to F are two unique scaling exponents α^*, α^* > -1 which describe the homogeneity of Φ and Φ. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation F(D²u) = 0, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of F(D²u) = 0 in ℝⁿ ∖ \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.

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