A new inequality for superdiffusions and its applications to nonlinear differential equations
Author:
E. B. Dynkin
Journal:
Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 68-77
MSC (2000):
Primary 60H30; Secondary 35J60, 60J60
DOI:
https://doi.org/10.1090/S1079-6762-04-00131-3
Published electronically:
August 2, 2004
Comment(s):
Additional information about this paper
MathSciNet review:
2075898
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: 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.
- E. B. Dynkin, A probabilistic approach to one class of nonlinear differential equations, Probab. Theory Related Fields 89 (1991), no. 1, 89–115. MR 1109476, DOI https://doi.org/10.1007/BF01225827
- E. B. Dynkin, Diffusions, superdiffusions and partial differential equations, American Mathematical Society Colloquium Publications, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1883198
- E. B. Dynkin, On upper bounds for positive solutions of semilinear equations, J. Funct. Anal. 210 (2004), no. 1, 73–100. MR 2051633, DOI https://doi.org/10.1016/S0022-1236%2803%2900147-2
[Dy04b]Dy04b ---, Superdiffusions and positive solutions of nonlinear partial differential equations, Uspekhi Matem. Nauk 59 (2004), to appear.
- Eugene B. Dynkin, Absolute continuity results for superdiffusions with applications to differential equations, C. R. Math. Acad. Sci. Paris 338 (2004), no. 8, 605–610 (English, with English and French summaries). MR 2056468, DOI https://doi.org/10.1016/j.crma.2004.01.028
[Dy04d]Dy04d ---, Superdiffusions and positive solutions of nonlinear partial differential equations, American Mathematical Society, Providence, RI, 2004, to appear.
- E. B. Dynkin and S. E. Kuznetsov, Poisson capacities, Math. Res. Lett. 10 (2003), no. 1, 85–95. MR 1960126, DOI https://doi.org/10.4310/MRL.2003.v10.n1.a9
[DK04]DK04 ---, $\mathbb {N}$-measures for branching exit Markov systems and their applications to differential equations, Probab. Theory and Related Fields, to appear.
[Ku04]Ku04 S. E. Kuznetsov, An upper bound for positive solutions of the equation $\Delta u=u^\alpha$, Amer. Math. Soc., Electronic Research Announcements, to appear.
[MV04]MV04 M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion, J. European Math. Soc., to appear.
[Ms02]Ms02 B. Mselati, Classification et représentation probabiliste des solutions positives de $\Delta u=u^2$ dans un domaine, Thése de Doctorat de l’Université Paris 6, 2002.
[Ms04]Ms04 B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society 168 (2004), no. 798, to appear.
[Dy91]Dy91 E. B. Dynkin, A probabilistic approach to one class of nonlinear differential equations, Probab. Th. Rel. Fields 89 (1991), 89–115.
[Dy02]Dy02 ---, Diffusions, superdiffusions and partial differential equations, American Mathematical Society, Providence, RI, 2002.
[Dy04a]Dy04a ---, On upper bounds for positive solutions of semilinear equations, J. Functional Analysis 210 (2004), 73–100.
[Dy04b]Dy04b ---, Superdiffusions and positive solutions of nonlinear partial differential equations, Uspekhi Matem. Nauk 59 (2004), to appear.
[Dy04c]Dy04c ---, Absolute continuity results for superdiffusions with applications to differential equations, C. R. Acad. Sc. Paris, Série I, 338 (2004), 605–610.
[Dy04d]Dy04d ---, Superdiffusions and positive solutions of nonlinear partial differential equations, American Mathematical Society, Providence, RI, 2004, to appear.
[DK03]DK03 E. B. Dynkin and S. E. Kuznetsov, Poisson capacities, Math. Research Letters 10 (2003), 85–95.
[DK04]DK04 ---, $\mathbb {N}$-measures for branching exit Markov systems and their applications to differential equations, Probab. Theory and Related Fields, to appear.
[Ku04]Ku04 S. E. Kuznetsov, An upper bound for positive solutions of the equation $\Delta u=u^\alpha$, Amer. Math. Soc., Electronic Research Announcements, to appear.
[MV04]MV04 M. Marcus and L. Véron, Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion, J. European Math. Soc., to appear.
[Ms02]Ms02 B. Mselati, Classification et représentation probabiliste des solutions positives de $\Delta u=u^2$ dans un domaine, Thése de Doctorat de l’Université Paris 6, 2002.
[Ms04]Ms04 B. Mselati, Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation, Memoirs of the American Mathematical Society 168 (2004), no. 798, to appear.
Similar Articles
Retrieve articles in Electronic Research Announcements of the American Mathematical Society
with MSC (2000):
60H30,
35J60,
60J60
Retrieve articles in all journals
with MSC (2000):
60H30,
35J60,
60J60
Additional Information
E. B. Dynkin
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853
Email:
ebd1@cornell.edu
Keywords:
Positive solutions of semilinear elliptic PDEs,
superdiffusions,
conditional diffusions,
$\mathbb {N}$-measures
Received by editor(s):
April 23, 2004
Published electronically:
August 2, 2004
Additional Notes:
Partially supported by the National Science Foundation Grant DMS-0204237
Communicated by:
Mark Freidlin
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
