Fundamental solutions of homogeneous fully nonlinear elliptic equations |
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Authors: | Scott N Armstrong Charles K Smart Boyan Sirakov |
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Institution: | 1. The University of Chicago, Department of Mathematics, 5734 S. University Avenue, Chicago, IL 60637;2. University of California, Berkeley, Department of Mathematics, Berkeley, CA 94720, Courant Institute, 251 Mercer Street, New York, NY 10012;3. Université Paris 10, UFR SEGMI, 92001 Nanterre Cedex, FRANCE;4. CAMS, EHESS, 54 bd Raspail, 75270 Paris Cedex 06, FRANCE |
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Abstract: | We prove the existence of two fundamental solutions Φ and 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*, > −1 that describe the homogeneity of Φ and . 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. |
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