The Dirichlet problem for radially homogeneous elliptic operators
Author:
Richard F. Bass
Journal:
Trans. Amer. Math. Soc. 320 (1990), 593614
MSC:
Primary 35J25; Secondary 35B50, 60J60
MathSciNet review:
968415
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: The Dirichlet problem in the unit ball is considered for the strictly elliptic operator , where the , are smooth away from the origin and radially homogeneous: . 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.
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L. Caffarelli, private communication.
 [1]
 R. F. Bass and E. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probability Theory and Related Fields 76 (1987), 557572. MR 917679 (89b:60183)
 [2]
 J. R. Baxter and G. A. Brosamler, Energy and the law of the iterated logarithm, Math. Scand. 38 (1976), 115136. 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), 185201. 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), 309340. 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), 199325. MR 0038008 (12:341b)
 [7]
 N. V. Krylov, An inequality in the theory of stochastic integrals, Theory Probab. Appl. 16 (1971), 438448. 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), 253255.
 [9]
 , A certain property of solutions of parabolic equation with measurable coefficients, Math. USSRIzv. 16 (1981), 151235.
 [10]
 L. Lamberti and P. Manselli, Existenceuniqueness theorems and counterexamples for an axially symmetric elliptic operator, Boll. Un. Ital. B 2 (1983), 431443. MR 716741 (85f:35068)
 [11]
 C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 1530. 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), 275288. (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), 225246. MR 805961 (87b:60122)
 [15]
 L. Caffarelli, private communication.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199009684159
PII:
S 00029947(1990)09684159
Keywords:
Dirichlet problem,
maximum principle,
elliptic operator,
radially homogeneous,
diffusion processes,
martingale problem
Article copyright:
© Copyright 1990
American Mathematical Society
