DOI QR코드

DOI QR Code

NOTE ON LOCAL ESTIMATES FOR WEAK SOLUTION OF BOUNDARY VALUE PROBLEM FOR SECOND ORDER PARABOLIC EQUATION

  • Choi, Jongkeun (Department of Mathematical Sciences Seoul National University)
  • Received : 2015.07.15
  • Published : 2016.07.31

Abstract

The aim of this note is to provide detailed proofs for local estimates near the boundary for weak solutions of second order parabolic equations in divergence form with time-dependent measurable coefficients subject to Neumann boundary condition. The corresponding parabolic equations with Dirichlet boundary condition are also considered.

Acknowledgement

Supported by : SNU

References

  1. J. Choi and S. Kim, Green's function for second order parabolic systems with Neumann boundary condition, J. Differential Equations 254 (2013), no. 7, 2834-2860. https://doi.org/10.1016/j.jde.2013.01.003
  2. M. Choulli, Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions, Evol. Equ. Control Theory 4 (2015), no. 1, 61-67. https://doi.org/10.3934/eect.2015.4.61
  3. E. DiBenedetto, Partial Differential Equations, Second edition. Birkhauser Boston, Inc., Boston, MA, 2010.
  4. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.
  5. S. Kim, Note on local boundedness for weak solutions of Neumann problem for second- order elliptic equations, J. Korean Soc. Ind. Appl. Math. 19 (2015), no. 2, 189-195. https://doi.org/10.12941/jksiam.2015.19.189
  6. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1967.
  7. G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  8. L. G. Rogers, Degree-independent Sobolev extension on locally uniform domains, J. Funct. Anal. 235 (2006), no. 2, 619-665. https://doi.org/10.1016/j.jfa.2005.11.013