Abstract
Studying PDEs on the sets of complicated structure and, in particular, stratified sets has been gaining popularity since recently. This article establishes some local Aleksandrov–Bakelman–Krylov type maximum estimates for solutions to linear elliptic and parabolic second-order equations on a book-type stratified set.
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References
Alexandrov A.D., “Certain estimates for the Dirichlet problem,” Soviet Math. Dokl., vol. 1, 1151–1154 (1961).
Bakelman I.Ya., “To the theory of quasilinear elliptic equations,” Sib. Math. J., vol. 2, no. 2, 179–186 (1961).
Krylov N.V., “Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation,” Sib. Math. J., vol. 17, no. 2, 226–236 (1976).
Krylov N.V., “On the maximum principle for nonlinear parabolic and elliptic equations,” Math. USSR-Izv., vol. 13, no. 2, 335–347 (1979).
Nazarov A.I., “The A.D. Aleksandrov maximum principle,” J. Math. Sci., vol. 142, 2154–2171 (2007).
Apushkinskaya D.E. and Nazarov A.I., “The normal derivative lemma and surrounding issues,” Russian Math. Surveys, vol. 77, no. 2, 189–249 (2022).
Oshchepkova S.N. and Penkin O.M., “The mean-value theorem for elliptic operators on stratified sets,” Math. Notes, vol. 81, no. 3, 365–372 (2007).
Luo Y., “An Aleksandrov–Bakelman type maximum principle and applications,” J. Differ. Equ., vol. 101, no. 2, 213–231 (1993).
Apushkinskaya D.E. and Nazarov A.I., “Hölder estimates of solutions to initial-boundary value problems for parabolic equations of nondivergent form with Venttsel boundary condition,” Amer. Math. Soc. Transl., vol. 64, 1–13 (1995).
Apushkinskaya D.E., “An estimate for the maximum of solutions of parabolic equations with Venttsel boundary conditions,” Vestn. Leningrad. Univ., Mat. Mekh. Astron., Ser. I, no. 2, 3–12 (1991).
Apushkinskaya D.E. and Nazarov A.I., “Linear two-phase Venttsel problems,” Ark. Mat., vol. 39, no. 2, 201–222 (2001).
Mironenko F.D. and Nazarov A.I., “Local Aleksandrov–Bakelman type maximum estimate for solutions to elliptic equations on a book-type stratified set,” Zap. Nauchn. Sem. POMI, vol. 519, 105–113 (2022).
Rockafellar R.T., Convex Analysis, Princeton University, Princeton (1997).
Nazarov A.I. and Uraltseva N.N., “Convex-monotonous bulls and an estimate of the maxima of solutions of a parabolic equation,” Zap. Nauchn. Sem. POMI, vol. 147, 95–109 (1985).
Acknowledgment
The author is grateful to A.I. Nazarov for stating the problem and paying constant attention to his work.
Funding
Supported by the Russian Science Foundation (Grant no. 22–21–00027).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 6, pp. 1263–1278. https://doi.org/10.33048/smzh.2023.64.612
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Mironenko, F.D. Maximum Estimates for Solutions to Elliptic and Parabolic Equations on a Book-Type Stratified Set. Sib Math J 64, 1385–1398 (2023). https://doi.org/10.1134/S0037446623060125
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DOI: https://doi.org/10.1134/S0037446623060125