×

The Dirichlet problem for radially homogeneous elliptic operators. (English) Zbl 0743.60073

Summary: 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: \(a_{ij}(rx)=a_{ij}(x)\), \(r>0\), \(x\neq 0\). 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.

MSC:

60J60 Diffusion processes
35J15 Second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. F. Bass and É. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields 76 (1987), no. 4, 557 – 572. · Zbl 0617.60075 · doi: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. · Zbl 0346.60020 · doi:10.7146/math.scand.a-11622
[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. · Zbl 0468.60034 · doi: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. · Zbl 0071.09701 · doi:10.1007/BF02787726
[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. · Zbl 0562.35001
[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. · Zbl 0030.12902
[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).
[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. · Zbl 0459.60067
[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). · Zbl 0525.35029
[11] Carlo Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. (4) 74 (1966), 15 – 30 (Italian, with English summary). · Zbl 0144.35801 · doi:10.1007/BF02416445
[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.
[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. · Zbl 0426.60069
[14] R. J. Williams, Brownian motion with polar drift, Trans. Amer. Math. Soc. 292 (1985), no. 1, 225 – 246. · Zbl 0573.60072
[15] L. Caffarelli, private communication.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.