Estimates for the second-order derivatives of a solution to the two-phase parabolic obstacle problem are established. Similar results in the elliptic case were obtained by the authors in 2006. Bibliography: 4 titles.
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D. E. Apushkinskaya, N. N. Uraltseva, “Boundary estimates for solutions of two-phase obstacle problems” [in Russian], Probl. Math. Anal. 34 (2006), 3–11; English transl.: J. Math. Sci. (New York) 142 (2007), 1723–1732.
L. A. Caffarelli, C. Kenig, “Gradient estimates for variable coefficient parabolic equations and singular perturbation problems,” Amer. J. Math. 120 (1998), 391–439.
H. Shahgholian, N. Uraltseva, G. Weiss, The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions, Preprint, http://www.citebase.org/abstract?id=oai:arXiv.org:0712.3411, 2007.
N. N. Uraltseva, “Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities,” In: Nonlinear Equations and Spectral Theory, Am. Math. Soc. Transl. (2) 220 (2007), 235–246.
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Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 3–10.
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Apushkinskaya, D.E., Uraltseva, N.N. Boundary estimates for solutions to the two-phase parabolic obstacle problem. J Math Sci 156, 569–576 (2009). https://doi.org/10.1007/s10958-009-9284-7
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DOI: https://doi.org/10.1007/s10958-009-9284-7