Fundamental solutions of homogeneous fully nonlinear elliptic equations

Scott Armstrong, Charles K. Smart, Boyan Sirakov

Research output: Contribution to journalArticle

Abstract

We prove the existence of two fundamental solutions Φ and Φ̃ of the PDE F(D2Phi) = 0 R{double-struck}n\{0} for any positively homogeneous, uniformly elliptic operator F. Corresponding to F are two unique scaling exponents α*, α̃* > -1 that describe the homogeneity of Φ and Φ̃. 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(D2Phi) = 0 R{double-struck}n\{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.

Original languageEnglish (US)
Pages (from-to)737-777
Number of pages41
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number6
DOIs
StatePublished - Jun 2011

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Fully Nonlinear Elliptic Equations
Scaling Exponent
Fundamental Solution
Random processes
Infinity
Transience
Isolated Singularity
Differential Games
Nontrivial Solution
Elliptic Operator
Homogeneity
Recurrence
Stochastic Processes
Demonstrate

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Fundamental solutions of homogeneous fully nonlinear elliptic equations. / Armstrong, Scott; Smart, Charles K.; Sirakov, Boyan.

In: Communications on Pure and Applied Mathematics, Vol. 64, No. 6, 06.2011, p. 737-777.

Research output: Contribution to journalArticle

Armstrong, Scott ; Smart, Charles K. ; Sirakov, Boyan. / Fundamental solutions of homogeneous fully nonlinear elliptic equations. In: Communications on Pure and Applied Mathematics. 2011 ; Vol. 64, No. 6. pp. 737-777.
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