Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Trace on the boundary for solutions of nonlinear differential equations


Authors: E. B. Dynkin and S. E. Kuznetsov
Journal: Trans. Amer. Math. Soc. 350 (1998), 4499-4519
MSC (1991): Primary 60J60, 35J60; Secondary 60J80, 60J45, 35J65
DOI: https://doi.org/10.1090/S0002-9947-98-01952-7
MathSciNet review: 1422602
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $L$ be a second order elliptic differential operator in $\mathbb{R}^{d}$ with no zero order terms and let $E$ be a bounded domain in $\mathbb{R}^{d}$ with smooth boundary $\partial E$. We say that a function $h$ is $L$-harmonic if $Lh=0$ in $E$. Every positive $L$-harmonic function has a unique representation

\begin{equation*}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation*}

where $k$ is the Poisson kernel for $L$ and $\nu $ is a finite measure on $\partial E$. We call $\nu $ the trace of $h$ on $\partial E$. Our objective is to investigate positive solutions of a nonlinear equation

\begin{equation*}L u=u^{\alpha }\quad \text{in } E \end{equation*}

for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $\alpha $-superdiffusion which is not defined for $\alpha >2$]. We associate with every solution $u$ a pair $(\Gamma ,\nu )$, where $\Gamma $ is a closed subset of $\partial E$ and $\nu $ is a Radon measure on $O=\partial E\setminus \Gamma $. We call $(\Gamma ,\nu )$ the trace of $u$ on $\partial E$. $\Gamma $ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.


References [Enhancements On Off] (What's this?)

  • 1. E. B. Dynkin, Markov Processes, Springer-Verlag, Berlin, 1965. MR 33:1887
  • 2. -, A probabilistic approach to one class of nonlinear differential equations, Probab. Th. Rel. Fields 89 (1991), 89-115. MR 92d:35090
  • 3. -, Superdiffusions and parabolic nonlinear differential equations, Ann. Probab. 20 (1992), 942-962. MR 93d:60124
  • 4. -, Superprocesses and partial differential equations, Ann. Probab. 21 (1993), 1185-1262. MR 94j:60156
  • 5. -, An Introduction to Branching Measure-Valued Processes, American Mathematical Society, Providence, Rhode Island, 1994. MR 96f:60145
  • 6. E. B. Dynkin and S. E. Kuznetsov, Superdiffusions and removable singularities for quasilinear partial differential equations, Comm. Pure & Appl. Math. 49 (1996), 125-176. MR 97m:60144
  • 7. -, Solutions of $Lu = u^{\alpha }$ dominated by $L$-harmonic functions, Journal d'Analyse Mathématique 68 (1996), 15-37. MR 97f:35048
  • 8. -, Linear additive functionals of superdiffusions and related nonlinear P.D.E., Trans. Amer. Math. Soc. 348 (1996), 1959-1987. MR 97d:60135
  • 9. -, Nonlinear parabolic P.D.E. and additive functionals of superdiffusions, Ann. Probab. 25 (1997), 662-701. CMP 97:08
  • 10. -, Natural linear additive functionals of superprocesses, Ann. Probab. 25 (1997), 640-661. CMP 97:08
  • 11. -, Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary, Trans. Amer. Math. Soc., 350 (1998), 4521-4552. CMP 97:11
  • 12. A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math.J. 64 (1991), 271-324. MR 93a:35053
  • 13. J.F. Le Gall, Solutions positives de $\Delta u=u^{2}$ dans le disque unité, C.R. Acad. Sci. Paris, Série I 317 (1993), 873-878. MR 94h:35059
  • 14. -, The Brownian snake and solutions of $\Delta =u^{2}$ in a domain, Probab. Theory Relat. Fields 102 (1995), 393-402. MR 96c:60098
  • 15. -, A probabilistic Poisson representation for positive solutions of $\Delta u = u^{2}$ in a planar domain, Comm. Pure Appl. Math. 50 (1997), 69-103. MR 98c:60144
  • 16. -, A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation, J. Appl. Math. Stochastic Anal. 9 (1996), 399-414. MR 97m:35125
  • 17. M. Marcus and L. Véron, Trace au bord des solutions positives d'équations elliptiques non linéaires, C.R. Acad. Sci Paris Sér. I Math. 321 (1995), 179-184. MR 96f:35045
  • 18. -, Trace au bord des solutions positives d'équations elliptiques et paraboliques non linéaires. Résultats d'existence et d'unicité, C.R. Acad. Sci Paris Sér. I Math. 323 (1996), 603-608. MR 97f:35012
  • 19. V. G. Maz'ya, Beurling's theorem on a minimum principle for positive harmonic functions, [First published (in Russian) in Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 30 (1972), 76-90], J.Soviet Math. 4 (1975) 367-379. MR 48:8821

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 60J60, 35J60, 60J80, 60J45, 35J65

Retrieve articles in all journals with MSC (1991): 60J60, 35J60, 60J80, 60J45, 35J65


Additional Information

E. B. Dynkin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901; Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395
Email: ebd1@cornell.edu

S. E. Kuznetsov
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-7901
Email: sk47@cornell.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01952-7
Additional Notes: Partially supported by National Science Foundation Grant DMS-9301315.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society