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Uniform estimates near the initial state for solutions of the two-phase parabolic problem


Authors: D. E. Apushkinskaya and N. N. Uraltseva
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 195-203
MSC (2010): Primary 35K05
DOI: https://doi.org/10.1090/S1061-0022-2014-01285-X
Published electronically: March 12, 2014
MathSciNet review: 3114850
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Abstract: Optimal regularity near the initial state is established for weak solutions of the two-phase parabolic obstacle problem. The approach is sufficiently general to allow the initial data to belong to the class $ C^{1,1}$.


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  • [AU09] D. E. Apushkinskaya and N. N. Uraltseva, Boundary estimates for solutions to the two-phase parabolic obstacle problem, Problems in mathematical analysis. No. 38. J. Math. Sci. (N.Y.) 156 (2009), no. 4, 569-576. MR 2493233 (2010d:35412)
  • [CK98] L. A. Caffarelli and C. E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math. 120 (1998), no. 2, 391-439. MR 1613650 (99b:35081)
  • [CS05] L. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Grad. Stud. in Math., vol. 68, Amer. Math. Soc., Providence, RI, 2005. MR 2145284 (2006k:35310)
  • [GT01] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin, 2001, pp. xiv+517. MR 1814364 (2001k:35004)
  • [LSU67] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Izdat. Nauka, Moskow, 1967; English. transl., Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, RI, 1967. MR 0241822 (39;3159a)
  • [LU68] O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and quasilinear elliptic equations, Acad. Press, New York, 1968. MR 0244627 (39:5941)
  • [NPP10] K. Nyström, A. Pascucci, and S. Polidoro, Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators, J. Differential Equations 249 (2010), no. 8, 2044-2060. MR 2679015 (2011i:35266)
  • [NU11] A. I. Nazarov and N. N. Ural'tseva, The Harnack inequality and related properties of solutions of elliptic and parabolic equations with divergence-free lower-order coefficients, Algebra i Analiz 23 (2011), no. 1, 136-168; English transl., St. Petersburg Math. J. 23 (2012), no. 1, 93-115. MR 2760150 (2012d:35026)
  • [Nys08] K. Nyström, On the behaviour near expiry for multi-dimensional American options, J. Math. Anal. Appl. 339 (2008) no. 1, 644-654. MR 2370682 (2008m:91127)
  • [Sha08] H. Shahgholian, Free boundary regularity close to initial state for parabolic obstacle problem, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2077-2087. MR 2366975 (2009g:35350)
  • [SUW09] H. Shahgholian, N. Uraltseva, and G. S. Weiss, A parabolic two-phase obstacle-like equation, Adv. Math. 221 (2009), no. 3, 861-881. MR 2511041 (2010f:35441)
  • [Ura07] N. N. Uraltseva, Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities, Nonlinear equations and spectral theory, pp. 235-246, Amer. Math. Soc. Transl. Ser. 2, Vol. 220, Amer. Math. Soc., Providence, RI, 2007. MR 2343613 (2008k:35040)

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Additional Information

D. E. Apushkinskaya
Affiliation: Department of mathematics and mechanics, St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: darya@math.uni-sb.de

N. N. Uraltseva
Affiliation: Department of mathematics and mechanics, St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: uraltsev@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01285-X
Keywords: Two-phase parabolic obstacle problem, free boundary, optimal regularity
Received by editor(s): September 27, 2012
Published electronically: March 12, 2014
Additional Notes: Supported by RFBR (grant no. 11-01-00825) and by the St. Petersburg State University Grant. The second author thanks the Alexander von Humboldt Foundation and Saarland University, where this work was done, for hospitality and support.
Dedicated: To the memory of V. S. Buslaev
Article copyright: © Copyright 2014 American Mathematical Society

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