Uniform estimates near the initial state for solutions of the two-phase parabolic problem
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- by D. E. Apushkinskaya and N. N. Uraltseva
- St. Petersburg Math. J. 25 (2014), 195-203
- DOI: https://doi.org/10.1090/S1061-0022-2014-01285-X
- Published electronically: March 12, 2014
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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 $C^{1,1}$.References
- D. E. Apushkinskaya and N. N. Uraltseva, Boundary estimates for solutions to the two-phase parabolic obstacle problem, J. Math. Sci. (N.Y.) 156 (2009), no. 4, 569–576. Problems in mathematical analysis. No. 38. MR 2493233, DOI 10.1007/s10958-009-9284-7
- Luis A. Caffarelli and Carlos E. Kenig, Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math. 120 (1998), no. 2, 391–439. MR 1613650
- Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics, vol. 68, American Mathematical Society, Providence, RI, 2005. MR 2145284, DOI 10.1090/gsm/068
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
- Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
- Kaj Nyström, Andrea Pascucci, and Sergio 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, DOI 10.1016/j.jde.2010.05.020
- 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 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 1, 93–115. MR 2760150, DOI 10.1090/S1061-0022-2011-01188-4
- Kaj Nyström, On the behaviour near expiry for multi-dimensional American options, J. Math. Anal. Appl. 339 (2008), no. 1, 644–654. MR 2370682, DOI 10.1016/j.jmaa.2007.06.068
- Henrik Shahgholian, Free boundary regularity close to initial state for parabolic obstacle problem, Trans. Amer. Math. Soc. 360 (2008), no. 4, 2077–2087. MR 2366975, DOI 10.1090/S0002-9947-07-04292-4
- Henrik Shahgholian, Nina Uraltseva, and Georg S. Weiss, A parabolic two-phase obstacle-like equation, Adv. Math. 221 (2009), no. 3, 861–881. MR 2511041, DOI 10.1016/j.aim.2009.01.011
- N. N. Uraltseva, Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 235–246. MR 2343613, DOI 10.1090/trans2/220/10
Bibliographic 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
- 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.
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 195-203
- MSC (2010): Primary 35K05
- DOI: https://doi.org/10.1090/S1061-0022-2014-01285-X
- MathSciNet review: 3114850
Dedicated: To the memory of V. S. Buslaev