A probabilistic approach to mixed boundary value problems for elliptic operators with singular coefficients

Chen, Zhen-Qing; Zhang, Tusheng

doi:10.1090/S0002-9939-2014-11907-1

A probabilistic approach to mixed boundary value problems for elliptic operators with singular coefficients
HTML articles powered by AMS MathViewer

by Zhen-Qing Chen and Tusheng Zhang PDF

Proc. Amer. Math. Soc. 142 (2014), 2135-2149

Abstract:

In this paper, we establish existence and uniqueness of solutions of a class of mixed boundary value problems for elliptic operators with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms plays an important role.

References

Alano Ancona, First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997), 45–92. MR 1482989, DOI 10.1007/BF02843153
Richard F. Bass and Pei Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), no. 2, 486–508. MR 1106272
Zhen Qing Chen, On reflecting diffusion processes and Skorokhod decompositions, Probab. Theory Related Fields 94 (1993), no. 3, 281–315. MR 1198650, DOI 10.1007/BF01199246
Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639–4679. MR 1926893, DOI 10.1090/S0002-9947-02-03059-3
Zhen-Qing Chen, Analytic characterization of conditional gaugeability for non-local Feynman-Kac transforms, J. Funct. Anal. 202 (2003), no. 1, 226–246. MR 1994771, DOI 10.1016/S0022-1236(02)00096-4
Z.-Q. Chen, P. J. Fitzsimmons, K. Kuwae, and T.-S. Zhang, Stochastic calculus for symmetric Markov processes, Ann. Probab. 36 (2008), no. 3, 931–970. MR 2408579, DOI 10.1214/07-AOP347
Z.-Q. Chen, P. J. Fitzsimmons, K. Kuwae, and T.-S. Zhang, Perturbation of symmetric Markov processes, Probab. Theory Related Fields 140 (2008), no. 1-2, 239–275. MR 2357677, DOI 10.1007/s00440-007-0065-2
Zhen-Qing Chen and Masatoshi Fukushima, Symmetric Markov processes, time change, and boundary theory, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR 2849840
Zhen-Qing Chen and Tusheng Zhang, Time-reversal and elliptic boundary value problems, Ann. Probab. 37 (2009), no. 3, 1008–1043. MR 2537548, DOI 10.1214/08-AOP427
Z. Q. Chen and Z. Zhao, Diffusion processes and second order elliptic operators with singular coefficients for lower order terms, Math. Ann. 302 (1995), no. 2, 323–357. MR 1336339, DOI 10.1007/BF01444498
Patrick J. Fitzsimmons, Even and odd continuous additive functionals, Dirichlet forms and stochastic processes (Beijing, 1993) de Gruyter, Berlin, 1995, pp. 139–154. MR 1366430
Mark Freidlin, Functional integration and partial differential equations, Annals of Mathematics Studies, vol. 109, Princeton University Press, Princeton, NJ, 1985. MR 833742, DOI 10.1515/9781400881598
Masatoshi Fukushima, On a decomposition of additive functionals in the strict sense for a symmetric Markov process, Dirichlet forms and stochastic processes (Beijing, 1993) de Gruyter, Berlin, 1995, pp. 155–169. MR 1366431
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
W. D. Gerhard, The probabilistic solution of the Dirichlet problem for $\frac 12\Delta +\langle a,\nabla \rangle +b$ with singular coefficients, J. Theoret. Probab. 5 (1992), no. 3, 503–520. MR 1176434, DOI 10.1007/BF01060432
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Konrad Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann. 283 (1989), no. 4, 679–687. MR 990595, DOI 10.1007/BF01442860
Pavel Gyrya and Laurent Saloff-Coste, Neumann and Dirichlet heat kernels in inner uniform domains, Astérisque 336 (2011), viii+144 (English, with English and French summaries). MR 2807275
Pei Hsu, Probabilistic approach to the Neumann problem, Comm. Pure Appl. Math. 38 (1985), no. 4, 445–472. MR 792399, DOI 10.1002/cpa.3160380406
Shizuo Kakutani, Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo 20 (1944), 706–714. MR 14647
Hiroshi Kunita, Sub-Markov semi-groups in Banach lattices, Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969) Univ. Tokyo Press, Tokyo, 1970, pp. 332–343. MR 0267412
John Lunt, T. J. Lyons, and T. S. Zhang, Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels, J. Funct. Anal. 153 (1998), no. 2, 320–342. MR 1614598, DOI 10.1006/jfan.1997.3182
Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
Zhi Ming Ma and Ren Ming Song, Probabilistic methods in Schrödinger equations, Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989) Progr. Probab., vol. 18, Birkhäuser Boston, Boston, MA, 1990, pp. 135–164. MR 1042345, DOI 10.1007/978-1-4612-3458-6_{8}
Shintaro Nakao, Stochastic calculus for continuous additive functionals of zero energy, Z. Wahrsch. Verw. Gebiete 68 (1985), no. 4, 557–578. MR 772199, DOI 10.1007/BF00535345
Daniel W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 316–347. MR 960535, DOI 10.1007/BFb0084145
Neil S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 265–308. MR 369884
Xue Yang and Tusheng Zhang, Estimates of heat kernels with Neumann boundary conditions, Potential Anal. 38 (2013), no. 2, 549–572. MR 3015364, DOI 10.1007/s11118-012-9286-9