Nonuniqueness for second-order elliptic equations with measurable coefficients

SIAM Journal on Mathematical Analysis

Volumn 30, Issue 4, 1999, Pages 879-895

  Safonov, Mikhail V ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Elliptic PDE; Measurable coefficients; Nonuniqueness

Indexed keywords

EID: 0033243615     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141096309046     Document Type: Article

References (40)

1. A. D. ALEKSANDROV, Uniqueness conditions and estimates for the solutions of the Dirichlet problem, Vestnik Leningrad Univ., 18 (1963), pp. 5-29 (in Russian); Amer. Math. Soc. Trans., 68 (1968), pp. 89-119 (in English).
2. A. D. ALEKSANDROV, Uniqueness conditions and estimates for the solutions of the Dirichlet problem, Vestnik Leningrad Univ., 18 (1963), pp. 5-29 (in Russian); Amer. Math. Soc. Trans., 68 (1968), pp. 89-119 (in English).
3. n, Vestnik Leningrad Univ., 18 (1966), pp. 5-29 (in Russian); Amer. Math. Soc. Trans., 68 (1968), pp. 89-119 (in English).
4. n, Vestnik Leningrad Univ., 18 (1966), pp. 5-29 (in Russian); Amer. Math. Soc. Trans., 68 (1968), pp. 89-119 (in English).
5. 3 in non-divergent form with measurable coefficients, Le Matematiche, 48 (1993), pp. 163-177.
6. R. F. BASS, The Dirichlet problem for radially homogeneous elliptic operators, Trans. Amer. Math. Soc., 320 (1990), pp. 593-614.
7. R. F. BASS AND E. PARDOUX, Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields, 76 (1987), pp. 557-572.
8. A. BENSOUSSAN, J. L. LIONS, AND G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, New York, Oxford, 1978.
9. L. BERS, F. JOHN, AND M. SCHECTER, Partial Differential Equations, Interscience, New York, London, Sydney, 1964.
10. L. CAFFARELLI, M. G. CRANDALL, M. KOCAN, AND A. SWIECH, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math., 49 (1996), pp. 365-397.
11. M. C. CERUTTI, L. ESCAURIAZA, AND E. B. FABES, Uniqueness for some diffusions with discontinuous coefficients, Ann. Probab., 19 (1991), pp. 525-537.
12. M. C. CERUTTI, L. ESCAURIAZA, AND E. B. FABES, Uniqueness in the Dirichlet problem for some elliptic operators with discontinuous coefficients, Ann. Mat. Pura Appl. (4) Ser. A, 163 (1993), pp. 161-180.
13. M. C. CERUTTI, E. B. FABES, AND P. MANSELLI, Uniqueness for elliptic operators with time-independent coefficients, Pitman Res. Notes Math. Ser., 350, Longman, Harlow, 1996, pp. 112-135.
14. H. O. CORDES, Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen, Math. Ann., 131 (1956), pp. 278-312.
15. M. I. FREIDLIN, Dirichlet's problem for an equation with periodic coefficients depending on a small parameter, Teor. Veroyatnost i Primenen., 9 (1964), pp. 133-138 (in Russian); Theory Probab. Appl., 9 (1964), pp. 121-125 (in English).
16. M. I. FREIDLIN, Dirichlet's problem for an equation with periodic coefficients depending on a small parameter, Teor. Veroyatnost i Primenen., 9 (1964), pp. 133-138 (in Russian); Theory Probab. Appl., 9 (1964), pp. 121-125 (in English).
17. D. GILBARG AND J. SERRIN, On isolated singularities of solutions of second order elliptic differential equations, J. Anal. Math., 4 (1955/56), pp. 309-340.
18. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.
19. R. R. JENSEN, Uniformly elliptic PDEs with bounded, measurable coefficients, J. Fourier Anal. Appl., 2 (1996), pp. 237-259.
20. V. V. JIKOV, S. M. KOZLOV, AND O. A. OLEINIK, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, New York, 1994.
21. N. V. KRYLOV, Nonlinear Elliptic and Parabolic Equations of Second Order, Nauka, Moscow, 1985 (in Russian); Reidel, Dordrecht, 1987 (in English).
22. N. V. KRYLOV, On Itô's stochastic integral equations, Teor. Veroyatnost i Primenen., 14 (1969), pp. 340-348 (in Russian); Theory Probab. Appl., 14 (1969), pp. 330-336 (in English).
23. N. V. KRYLOV, On Itô's stochastic integral equations, Teor. Veroyatnost i Primenen., 14 (1969), pp. 340-348 (in Russian); Theory Probab. Appl., 14 (1969), pp. 330-336 (in English).
24. N. V. KRYLOV, On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes, Izv. Akad. Nauk SSSR, ser. Matem., 37 (1973), pp. 691-708 (in Russian); Math. USSR Izvestija, 7 (1973), pp. 691-709 (in English).
25. N. V. KRYLOV, On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes, Izv. Akad. Nauk SSSR, ser. Matem., 37 (1973), pp. 691-708 (in Russian); Math. USSR Izvestija, 7 (1973), pp. 691-709 (in English).
26. N. V. KRYLOV, Once more about the connection between elliptic operators and Itô's stochastic equations, in Statistics and Control of Stochastic Processes, Moscow, 1984, pp. 214-229 (in Russian); Optimization Software, New York, 1985 (in English).
27. N. V. KRYLOV, Once more about the connection between elliptic operators and Itô's stochastic equations, in Statistics and Control of Stochastic Processes, Moscow, 1984, pp. 214-229 (in Russian); Optimization Software, New York, 1985 (in English).
28. N. V. KRYLOV, On one-point uniqueness for elliptic equations, Comm. Partial Differential Equations, 17 (1992), pp. 1759-1784.
29. N. V. KRYLOV AND M. V. SAFONOV, A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR, Ser. Mat., 44 (1980), pp. 161-175 (in Russian); Math. USSR Izvestija, 16 (1981), pp. 151-164 (in English).
30. N. V. KRYLOV AND M. V. SAFONOV, A certain property of solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR, Ser. Mat., 44 (1980), pp. 161-175 (in Russian); Math. USSR Izvestija, 16 (1981), pp. 151-164 (in English).
31. O. A. LADYZHENSKAYA AND N. N. URAL'TSEVA, Linear and Quasilinear Elliptic Equations, Nauka, Moscow, 1964 (in Russian); Academic Press, New York, 1968 (in English); 2nd Russian ed. 1973.
32. N. S. NADIRASHVILI, Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), pp. 537-550.
33. C. PUCCI, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl., 74 (1966), pp. 15-30.
34. M. V. SAFONOV, Harnack inequality for elliptic equations and the Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 96 (1980), pp. 272-287 (in Russian); J. Soviet Math., 21 (1983), pp. 851-863 (in English).
35. M. V. SAFONOV, Harnack inequality for elliptic equations and the Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 96 (1980), pp. 272-287 (in Russian); J. Soviet Math., 21 (1983), pp. 851-863 (in English).
36. M. V. SAFONOV, Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients, Mat. Sb., 132 (1987), pp. 272-288 (in Russian); Math. USSR Sbornik, 60 (1988), pp. 269-281 (in English).
37. M. V. SAFONOV, Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients, Mat. Sb., 132 (1987), pp. 272-288 (in Russian); Math. USSR Sbornik, 60 (1988), pp. 269-281 (in English).
38. M. V. SAFONOV, On a weak uniqueness for some elliptic equations. Comm. Partial Differential Equations, 19 (1994), pp. 943-957.
39. D. W. STROOCK AND S. R. S. VARADHAN, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, Heidelberg, New York, 1979.
40. D. W. STROOCK AND S. R. S. VARADHAN, Diffusion processes with continuous coefficients, I and II, Comm. Pure Appl. Math., 22 (1969), pp. 345-400, 479-530.