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

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Abstract | References | Similar Articles | 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.

**[1]**R. F. Bass and É. Pardoux,*Uniqueness for diffusions with piecewise constant coefficients*, Probab. Theory Related Fields**76**(1987), no. 4, 557–572. MR**917679**, 10.1007/BF00960074**[2]**J. R. Baxter and G. A. Brosamler,*Energy and the law of the iterated logarithm*, Math. Scand.**38**(1976), no. 1, 115–136. MR**0426178****[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), no. 2, 185–201. MR**663900**, 10.1007/BF00531822**[4]**D. Gilbarg and James Serrin,*On isolated singularities of solutions of second order elliptic differential equations*, J. Analyse Math.**4**(1955/56), 309–340. MR**0081416****[5]**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****[6]**M. G. Kreĭn and M. A. Rutman,*Linear operators leaving invariant a cone in a Banach space*, Amer. Math. Soc. Translation**1950**(1950), no. 26, 128. MR**0038008****[7]**N. V. Krylov,*A certain estimate from the theory of stochastic integrals*, Teor. Verojatnost. i Primenen.**16**(1971), 446–457 (Russian, with English summary). MR**0298792****[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. Mat. Ital. B (6)**2**(1983), no. 2, 431–443 (English, with Italian summary). MR**716741****[11]**Carlo Pucci,*Limitazioni per soluzioni di equazioni ellittiche*, Ann. Mat. Pura Appl. (4)**74**(1966), 15–30 (Italian, with English summary). MR**0214905****[12]**M. V. Safonov,*Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients*, Mat. Sb. (N.S.)**132(174)**(1987), no. 2, 275–288 (Russian); English transl., Math. USSR-Sb.**60**(1988), no. 1, 269–281. MR**882838****[13]**Daniel W. Stroock and S. R. Srinivasa Varadhan,*Multidimensional diffusion processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR**532498****[14]**R. J. Williams,*Brownian motion with polar drift*, Trans. Amer. Math. Soc.**292**(1985), no. 1, 225–246. MR**805961**, 10.1090/S0002-9947-1985-0805961-0**[15]**L. Caffarelli, private communication.

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0968415-9

Keywords:
Dirichlet problem,
maximum principle,
elliptic operator,
radially homogeneous,
diffusion processes,
martingale problem

Article copyright:
© Copyright 1990
American Mathematical Society