Dirichlet problem for degenerate elliptic equations

Authors:
Avner Friedman and Mark A. Pinsky

Journal:
Trans. Amer. Math. Soc. **186** (1973), 359-383

MSC:
Primary 35J70; Secondary 60H15

MathSciNet review:
0328345

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Abstract: Let be a degenerate second order elliptic operator with no zeroth order term in an *m*-dimensional domain *G*, and let . One divides the boundary of *G* into disjoint sets is the noncharacteristic part, and on the ``drift'' is outward. When *c* is negative, the following Dirichlet problem has been considered in the literature: in *G, u* is prescribed on . In the present work it is assume that . Assuming additional boundary conditions on a certain finite number of points of , a unique solution of the Dirichlet problem is established.

**[1]**M. I. Freĭdlin,*The smoothness of the solutions of degenerate elliptic equations*, Izv. Akad. Nauk SSSR Ser. Mat.**32**(1968), 1391–1413 (Russian). MR**0237944****[2]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[3]**Avner Friedman and Mark A. Pinsky,*Asymptotic behavior of solutions of linear stochastic differential systems*, Trans. Amer. Math. Soc.**181**(1973), 1–22. MR**0319268**, 10.1090/S0002-9947-1973-0319268-3**[4]**Avner Friedman and Mark A. Pinsky,*Asymptotic stability and spiraling properties for solutions of stochastic equations*, Trans. Amer. Math. Soc.**186**(1973), 331–358 (1974). MR**0329031**, 10.1090/S0002-9947-1973-0329031-5**[5]**J. J. Kohn and L. Nirenberg,*Degenerate elliptic-parabolic equations of second order*, Comm. Pure Appl. Math.**20**(1967), 797–872. MR**0234118****[6]**M. Pinsky,*A note on degenerate diffusion processes*, Teor. Verojatnost. i Primenen**14**(1969), 522–527 (English, with Russian summary). MR**0263174****[7]**D. Stroock and S. R. S. Varadhan,*On degenerate elliptic-parabolic operators of second order and their associated diffusions*, Comm. Pure Appl. Math.**25**(1972), 651–713. MR**0387812**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0328345-2

Keywords:
Dirichlet problem,
degenerate elliptic operator,
weak solution,
stochastic differential equations,
stochastic integrals,
exit time

Article copyright:
© Copyright 1973
American Mathematical Society