대한수학회지 (Journal of the Korean Mathematical Society)

  • 제54권5호
  • /
  • Pages.1357-1377
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  • 2017
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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GLOBAL W1,2p ESTIMATES FOR NONDIVERGENCE PARABOLIC OPERATORS WITH POTENTIALS SATISFYING A REVERSE HÖLDER CONDITION

  • Pan, Guixia (School of Public Health Anhui Medical University) ;
  • Tang, Lin (LMAM, School of Mathematical Sciences Peking University)
  • 투고 : 2014.12.11
  • 심사 : 2017.06.19
  • 발행 : 2017.09.01
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초록

In this article, we first give the $L^p$ boundedness of the operator $D^2L^{-1}$ with BMO coefficients and a potential V satisfying an appropriate reverse $H{\ddot{o}}lder$ condition, then obtain global $W^{1,2}_p$ estimates for the nondivergence parabolic operator L with VMO coefficients and a potential V satisfying an appropriate reverse $H{\ddot{o}}lder$ condition.

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과제정보

연구 과제 주관 기관 : NNSF of China

참고문헌

  1. M. Bramanti, L. Brandolini, E. Harboure, and B. Viviani, Global $W^{2,p}$ estimates for nondivergence elliptic operators with potentials satisfying a reverse Holder condition, Ann. Mat. Pura Appl. (4) 191 (2012), no. 2, 339-362. https://doi.org/10.1007/s10231-011-0186-1
  2. M. Bramanti and M. C. Cerutti, $W^{1,2}_p$ solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (1993), no. 9-10, 1735-1763. https://doi.org/10.1080/03605309308820991
  3. N. Burger, Espace des fonctions a variation moyenne bournee sur un espace de nature homogene, C. R. Acad. Sci. Paris Ser. A-B 286 (1978), no. 3, 139-142.
  4. A. Carbonaro, G. Metafune, and C. Spina, Parabolic Schrodinger operators, J. Math. Anal. Appl. 343 (2008), no. 2, 965-974. https://doi.org/10.1016/j.jmaa.2008.02.010
  5. F. Chiarenza, M. Frasca, and P. Longo, Interior $W^{2,p}$ estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149-168.
  6. F. Chiarenza, M. Frasca, and P. Longo, $W^{2,p}$ solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficents, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841-853. https://doi.org/10.1090/S0002-9947-1993-1088476-1
  7. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nded., Springer-Verlag, Berlin, 1983.
  8. W. Gao and Y. Jiang, $L^p$ estimate for parabolic Schrodinger operator with certain potentials, J. Math. Anal. Appl. 310 (2005), no. 1, 128-143. https://doi.org/10.1016/j.jmaa.2005.01.049
  9. K. Kurata, An estimate on the heat kernel of magnetic Schrodinger operators and uniformly elliptic operators with non-negative potentials, J. London Math. Soc. 62 (2000), no. 3, 885-903. https://doi.org/10.1112/S002461070000137X
  10. G. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  11. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. https://doi.org/10.1090/S0002-9947-1972-0293384-6
  12. G. Pan and L. Tang, Solvability for Schrodinger equations with discontinuous coefficients, J. Funct. Anal. 270 (2016), no. 1, 88-133. https://doi.org/10.1016/j.jfa.2015.10.004
  13. Z. W. Shen, On the Neumann problem for Schrodinger operators in Lipschitz domains, Indiana Univ. Math. J. 453 (1994), no. 1, 143-176.
  14. Z. W. Shen, $L^p$ estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513-546. https://doi.org/10.5802/aif.1463
  15. E. M. Stein, Harmonic analysis: real variable methods orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, 1993.
  16. L. Tang, Weighted $L^p$ solvability for parabolic equations with partially BMO coefficients and its applications, J. Aust. Math. Soc. 96 (2014), no. 4, 396-428. https://doi.org/10.1017/S1446788714000020
  17. L. Tang and J. Han, $L^p$ boundedness for parabolic Schrodinger type operators with certain nonnegative potentials, Forum Math. 23 (2011), no. 1, 161-179. https://doi.org/10.1515/FORM.2011.007
  18. C. Vitanza, A new contribution to the $W^{2,p}$ regularity for a class of elliptic second order equations with discontinuous coefficients, Matematiche (Catania) 48 (1993), no. 2 287-296.