Fundamental solutions of homogeneous fully nonlinear elliptic equations

By S. Armstrong, Charles K. Smart, Boyan Sirakov. 2009

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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(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 \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.

Subfields: Applied Mathematics

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In Applied Mathematics

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Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle (2010, background): “In particular, the inequality F D2u ≥ u has no positive solutions in any exterior domain in , provided that ≤ ∗ + 2 / ∗ (or ∗ ≤ 0), where ∗ = ∗ F characterizes the homogeneity of the upward-pointing fundamental solution of the operator F (as found in [2]).”