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Potential theory for elliptic systems. (English) Zbl 0854.60062
Summary: 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.

MSC:
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J60 Diffusion processes
35J45 Systems of elliptic equations, general (MSC2000)
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[1] 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. · Zbl 0807.47056
[2] 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. · Zbl 0876.35029
[3] CHUNG, K. L. and ZHAO, Z. 1995. From Brownian Motion to Schrodinger Equations. \" Springer, Berlin.
[4] 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. · Zbl 0723.60095
[5] FUKUSHIMA, M. 1980. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam. · Zbl 0422.31007
[6] GILBERG, D. and TRUDINGER, N. S. 1977. Elliptic Partial Differential Equations of Second Order. Springer, New York. · Zbl 0361.35003
[7] HE, S.-W., WANG, J.-G. and YAN, J.-A. 1992. Semimartingale Theory and Stochastic Calculus. Science Press CRC Press, Beijing. · Zbl 0781.60002
[8] 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. · Zbl 0178.53401
[9] KATO, T. 1966. Perturbation Theory for Linear Operators. Springer, New York. · Zbl 0148.12601
[10] KIFER, Y. 1992. Principal eigenvalues and equilibrium states corresponding to weakly coupled parabolic sy stems of PDE. · Zbl 0803.35065
[11] 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. · Zbl 0193.42402
[12] MEy ER, P. A. 1975. Renaissance, recollements, melanges, relentissement de processus de \' () Markov. Ann. Inst. Fourier Grenoble 25 465 497. · Zbl 0304.60041
[13] PROTTER, M. and WEINBERGER, H. 1976. Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0549.35002
[14] SHARPE, M. 1986. General Theory of Markov Processes. Academic, New York.
[15] SIMON, B. 1982. Schrodinger semigroups. Bull. Amer. Math. Soc. 7 447 526. \" · Zbl 0524.35002
[16] SKOROKHOD, A. V. 1989. Asy mptotic Methods of the Theory of Stochastic Differential Equations. Trans. Amer. Math. Soc., Providence, RI.
[17] SWEERS, G. 1992. Strong positivity in C for elliptic sy stems. Math. Z. 209 251 271. · Zbl 0727.35045
[18] TRUDINGER, N. S. 1973. Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 27 255 308. · Zbl 0279.35025
[19] ITHACA, NEW YORK COLUMBIA, MISSOURI 65211 E-mail: zchen@math.cornell.edu E-mail: mathzz@mizzou1.missouri.edu
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