Potential theory for elliptic systems

Potential theory for elliptic systems

Z. Q. Chen and Z. Zhao

Source: Ann. Probab. Volume 24, Number 1 (1996), 293-319.

Abstract

The existence and uniqueness theorem is proved for solutions of the Dirichlet boundary value problems for weakly coupled elliptic systems on bounded domains. The elliptic systems are only assumed to have measurable coefficients and have singular coefficients for the lower-order terms. A probabilistic representation theorem for solutions of the Dirichlet boundary value problems is obtained by using the switched diffusion process associated with the system. A strong positivity result for solutions of the Dirichlet boundary value problems is proved. Formulas expressing resolvents and kernel functions for the system by those of the component elliptic operators are also obtained.

First Page: Show

Primary Subjects: 60H30, 35J45

Secondary Subjects: 60J60

Keywords: Weakly coupled elliptic system; weak solution; Dirichlet boundary value problem; Dirichlet space; switched diffusion; irreducibility; resolvent; kernel function

Full-text: Open access

PDF File (793 KB)

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1042644718
Mathematical Reviews number (MathSciNet): MR1387637
Digital Object Identifier: doi:10.1214/aop/1042644718
Zentralblatt MATH identifier: 0854.60062

back to Table of Contents

References

2 CHEN, Z. Q. and ZHAO, Z. 1994. Switched diffusion processes and sy stems of elliptic equations a Dirichlet space approach. Proc. Roy. Soc. Edinburgh Sect. A 124 673 701.

Mathematical Reviews (MathSciNet): MR1298586

Zentralblatt MATH: 0807.47056

3 CHEN, Z. Q. and ZHAO, Z. 1995. Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math. Ann. 302 323 357.

Zentralblatt MATH: 0876.35029

Mathematical Reviews (MathSciNet): MR1336339

Digital Object Identifier: doi:10.1007/BF01444498

4 CHUNG, K. L. and ZHAO, Z. 1995. From Brownian Motion to Schrodinger Equations. ¨ Springer, Berlin.

5 EIZENBERG, A. and FREIDLIN, M. 1990. On the Dirichlet problem for a class of second order PDE sy stems with small parameter. Stochastics Stochastics Rep. 33 111 148.

Mathematical Reviews (MathSciNet): MR91m:35061

Zentralblatt MATH: 0723.60095

6 FUKUSHIMA, M. 1980. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam.

Mathematical Reviews (MathSciNet): MR81f:60105

7 GILBERG, D. and TRUDINGER, N. S. 1977. Elliptic Partial Differential Equations of Second Order. Springer, New York.

Mathematical Reviews (MathSciNet): MR473443

8 HE, S.-W., WANG, J.-G. and YAN, J.-A. 1992. Semimartingale Theory and Stochastic Calculus. Science Press CRC Press, Beijing.

Mathematical Reviews (MathSciNet): MR1219534

Zentralblatt MATH: 0781.60002

9 IKEDA, N., NAGASAWA, M. and WATANABE, S. 1966. A construction of Markov process by piecing out. Proc. Japan Acad. Ser. A Math. Sci. 42 370 375.

Mathematical Reviews (MathSciNet): MR202197

Digital Object Identifier: doi:10.3792/pja/1195522037

Project Euclid: euclid.pja/1195522037

10 KATO, T. 1966. Perturbation Theory for Linear Operators. Springer, New York.

Mathematical Reviews (MathSciNet): MR203473

11 KIFER, Y. 1992. Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic sy stems of PDE. Preprint.

Mathematical Reviews (MathSciNet): MR94m:35221a

Zentralblatt MATH: 0803.35065

Digital Object Identifier: doi:10.1007/BF02790219

12 KUNITA, H. 1969. Sub-Markov semi-groups in Banach lattices. Functional Analy sis and Related Topics. Lecture Notes in Math. 1540. Springer, Berlin 332 343.

Mathematical Reviews (MathSciNet): MR267412

Zentralblatt MATH: 0193.42402

13 MEy ER, P. A. 1975. Renaissance, recollements, melanges, relentissement de processus de ´ () Markov. Ann. Inst. Fourier Grenoble 25 465 497.

14 PROTTER, M. and WEINBERGER, H. 1976. Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ.

Mathematical Reviews (MathSciNet): MR219861

Zentralblatt MATH: 0549.35002

15 SHARPE, M. 1986. General Theory of Markov Processes. Academic, New York.

16 SIMON, B. 1982. Schrodinger semigroups. Bull. Amer. Math. Soc. 7 447 526. ¨

17 SKOROKHOD, A. V. 1989. Asy mptotic Methods of the Theory of Stochastic Differential Equations. Trans. Amer. Math. Soc., Providence, RI.

Mathematical Reviews (MathSciNet): MR1020057

18 SWEERS, G. 1992. Strong positivity in C for elliptic sy stems. Math. Z. 209 251 271.

Mathematical Reviews (MathSciNet): MR92j:35046

Zentralblatt MATH: 0727.35045

Digital Object Identifier: doi:10.1007/BF02570833

19 TRUDINGER, N. S. 1973. Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 27 255 308.

Mathematical Reviews (MathSciNet): MR51:6113

ITHACA, NEW YORK COLUMBIA, MISSOURI 65211 E-mail: zchen@math.cornell.edu E-mail: mathzz@mizzou1.missouri.edu

previous :: next