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)



The Dirichlet problem for radially homogeneous elliptic operators

Author: Richard F. Bass
Journal: Trans. Amer. Math. Soc. 320 (1990), 593-614
MSC: Primary 35J25; Secondary 35B50, 60J60
MathSciNet review: 968415
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Dirichlet problem in the unit ball is considered for the strictly elliptic operator $ L = \sum {{a_{ij}}{D_{ij}}} $, where the $ {a_{ij}}$, are smooth away from the origin and radially homogeneous: $ {a_{ij}}(rx) = {a_{ij}}(x),\;r > 0,\;x \ne 0$. Existence and uniqueness are proved for solutions in a certain space of functions. Necessary and sufficient conditions are given for an extended maximum principle to hold.

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

  • [1] R. F. Bass and E. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probability Theory and Related Fields 76 (1987), 557-572. MR 917679 (89b:60183)
  • [2] J. R. Baxter and G. A. Brosamler, Energy and the law of the iterated logarithm, Math. Scand. 38 (1976), 115-136. MR 0426178 (54:14124)
  • [3] R. N. Bhattacharya, On the functional central limit theorem and the law of the iterated logarithm for Markov processes, Z. Wahrsch. Verw. Gebiete 60 (1982), 185-201. MR 663900 (83h:60072)
  • [4] D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4 (1955/56), 309-340. MR 0081416 (18:399a)
  • [5] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Springer, New York, 1983. MR 737190 (86c:35035)
  • [6] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl. 10 (1962), 199-325. MR 0038008 (12:341b)
  • [7] N. V. Krylov, An inequality in the theory of stochastic integrals, Theory Probab. Appl. 16 (1971), 438-448. MR 0298792 (45:7841)
  • [8] N. V. Krylov and M. V. Safonov, An estimate of the probability that a diffusion process hits a set of positive measure, Soviet Math. Dokl. 20 (1979), 253-255.
  • [9] -, A certain property of solutions of parabolic equation with measurable coefficients, Math. USSR-Izv. 16 (1981), 151-235.
  • [10] L. Lamberti and P. Manselli, Existence-uniqueness theorems and counterexamples for an axially symmetric elliptic operator, Boll. Un. Ital. B 2 (1983), 431-443. MR 716741 (85f:35068)
  • [11] C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15-30. MR 0214905 (35:5752)
  • [12] M. V. Safonov, Unimprovability of estimates of Hölder continuity for solution to linear elliptic equations with measurable coefficients, Mat. Sb. 132 (174) (1987), 275-288. (Russian) MR 882838 (88e:35049)
  • [13] D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Springer, New York, 1979. MR 532498 (81f:60108)
  • [14] R. J. Williams, Brownian motion with polar drift, Trans. Amer. Math. Soc. 292 (1985), 225-246. MR 805961 (87b:60122)
  • [15] L. Caffarelli, private communication.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35J25, 35B50, 60J60

Retrieve articles in all journals with MSC: 35J25, 35B50, 60J60

Additional Information

Keywords: Dirichlet problem, maximum principle, elliptic operator, radially homogeneous, diffusion processes, martingale problem
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society