Geodesic
A new inequality for superdiffusions and its applications to nonlinear differential equations
Dynkin, E.1
1 Department of Mathematics, Cornell University, Ithaca, NY 14853
Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 68-77.

Voir la notice de l'article provenant de la source American Mathematical Society

Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation $L u=u^\alpha$ with $\alpha >1$ in a bounded smooth domain $E\subset \mathbb {R}^d$. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called $\sigma$-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation $\Delta u=u^2$ in a domain of class $C^4$. His principal tool—the Brownian snake—is not applicable to the case $\alpha \neq 2$. In 2003 Dynkin and Kuznetsov modified most of Mselati’s arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.
DOI : 10.1090/S1079-6762-04-00131-3

Dynkin, E. 1

1 Department of Mathematics, Cornell University, Ithaca, NY 14853 
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Dynkin, E. A new inequality for superdiffusions and its applications to nonlinear differential equations. Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 68-77. doi : 10.1090/S1079-6762-04-00131-3. http://geodesic.mathdoc.fr/articles/10.1090/S1079-6762-04-00131-3/

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[5] Dynkin, E. B., Kuznetsov, S. E. Poisson capacities Math. Res. Lett. 2003 85 95

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