Potential Analysis

, Volume 26, Issue 4, pp 345–361 | Cite as

Parabolic Equations with Measurable Coefficients

Article

Abstract

We investigate the unique solvability of second order parabolic equations in non-divergence form in \(W_p^{1,2}((0,T) \times \mathbb{R}^d)\), p ≥ 2. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables.

Key words

second-order equations vanishing mean oscillation 

Mathematics Subject Classifications (2000)

35K10 35K20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bramanti, M., Cerutti, M.C.: \(W^{{1,2}}_{p} \) solvability for the Cauchy–Dirichlet problem for parabolic equations with VMO coefficients. Comm. Partial Differential Equations 18(9-10), 1735–1763 (1993)MATHMathSciNetGoogle Scholar
  2. 2.
    Chiarenza, F., Frasca, M., Longo, P.: Interior W 2, p estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat. 40(1), 149–168 (1991)MATHMathSciNetGoogle Scholar
  3. 3.
    Chiarenza, F., Frasca, M., Longo, P.: W 2, p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc. 336(2), 841–853 (1993)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chiti, G.: A W 2,2 bound for a class of elliptic equations in nondivergence form with rough coefficients. Invent. Math. 33, 55–60 (1976)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Haller-Dintelmann, R., Heck, H., Hieber, M.: L pL q-estimates for parabolic systems in non-divergence form with VMO coefficients. J. London Math. Soc. 74(3), 717–736 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kim, Doyoon: Second order elliptic equations in \(\mathbb{R}^{d}\) with piecewise continuous coefficients. Potential Anal. (2007) (in press)Google Scholar
  7. 7.
    Kim, Doyoon: Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 5(1), 1–22 (2006)Google Scholar
  8. 8.
    Kim, Doyoon, Krylov, N.V.: Elliptic differential equations with measurable coefficients. SIMA (http://arxiv.org/abs/math.AP/0512515) (2007) (in press)
  9. 9.
    Krylov, N.V.: Parabolic and elliptic equations with VMO coefficients. Comm. Partial Differential Equations (http://arxiv.org/pdf/math.AP/0511731) (2007) (in press)
  10. 10.
    Lorenzi, A.: On elliptic equations with piecewise constant coefficients. Appl. Anal. 2, 79–96 (1972)MATHMathSciNetGoogle Scholar
  11. 11.
    Lorenzi, A.: On elliptic equations with piecewise constant coefficients. II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3). 26, 839–870 (1972)MATHMathSciNetGoogle Scholar
  12. 12.
    Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and parabolic equations with discontinuous coefficients. In: Mathematical Research, vol. 109, Wiley, New York (2000)Google Scholar
  13. 13.
    Palagachev, D., Softova, L.: A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete Contin. Dynam. Systems 13(3), 721–742 (2005)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Softova, L.: Quasilinear parabolic operators with discontinuous ingredients. Nonlinear Anal. 52(4), 1079–1093 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Softova, L.: \(W^{{1,2}}_{p} \)-solvability for parabolic Poincaré problem. Comm. Partial Differential Equations 29(11, 12), 1783–1798 (2004)MATHMathSciNetGoogle Scholar
  16. 16.
    Salsa, S.: Un problema di Cauchy per un operatore parabolico con coefficienti costanti a tratti. Matematiche (Catania) 31 (1976) (1), 126–146 (1977)MATHMathSciNetGoogle Scholar
  17. 17.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin Heidelberg New York (1979)MATHGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.University of MinnesotaMinneapolisUSA

Personalised recommendations