On the Smoothness of a Single Layer Parabolic Potential with Dini Continuous Density
DOI:
https://doi.org/10.24160/1993-6982-2026-4-195-200Keywords:
parabolic equation, single layer parabolic potential, modulus of continuity, Dini conditionAbstract
One of the classical methods for solving initial boundary value problems for parabolic equations is the potential method. In this approach, the solution is sought in the form of parabolic potentials generated by the fundamental solution of the equation. In the case of noncylindrical domains with nonsmooth lateral boundaries, the single layer parabolic potential is of most interest, which makes it possible to obtain accurate results from a smoothness point of view with minimal assumptions for the equation coefficients, domain lateral boundary, and functions from boundary conditions. Therefore, it is important to investigate the properties of this potential. The article investigates the smoothness of a single layer parabolic potential whose kernel is the fundamental solution of a multidimensional uniformly parabolic second-order equation with real coefficients. It is assumed that the equation coefficients are bounded and uniformly continuous, with the continuity modulus satisfying the double Dini condition. This potential is considered in a semibounded spatial domain with a generally nonsmooth, with respect to the time variable, lateral boundary belonging to the Dini-Hölder class. The density of the potential is bounded and Dini continuous on the domain lateral boundary (the density carrier surface) with a modulus of continuity satisfying the Dini condition. Estimates in the Dini class for this potential and its first order spatial derivatives, as well as for their increments with respect to time and spatial variables have been determined. For the proof, the representation of the fundamental solution of the parabolic equation constructed by the Levy method is used. The obtained estimates can be used in studying the solvability of initial boundary value problems for second order parabolic equations with Dini continuous coefficients, as well as to study the smoothness of solutions.
References
1. Бадерко Е.А. О решении методом граничных интегральных уравнений краевых задач для линейных параболических уравнений произвольного порядка в нецилиндрических областях: автореф. дис. … доктора физ.-мат. наук. М.: МГУ им. М.В. Ломоносова, 1992.
2. Бадерко Е.А. О гладкости 2m-параболического потенциала простого слоя // Дифференциальные уравнения. 1990. Т. 26. № 1. C. 3—10.
3. Камынин Л.И. О гладкости тепловых потенциалов в пространстве Дини-Гельдера // Сибирский математический журнал. 1970. Т. 11. № 5. C. 1017—1045.
4. Камынин Л.И. О решении методом потенциалов основных краевых задач для одномерного параболического уравнения второго порядка // Сибирский математический журнал. 1974. Т. 15. № 4. C. 806—834.
5. Бадерко Е.А. О потенциалах для 2p-параболических уравнений // Дифференциальные уравнения. 1983. Т. 19. № 1. С. 9—18.
6. Зейнеддин М. Гладкость потенциала простого слоя для параболической системы второго порядка в классах Дини // Деп. в ВИНИТИ РАН. № 1294-В92.
7. Черепова М.Ф. О гладкости параболического потенциала простого слоя // Деп. в ВИНИТИ АН СССР. № 5194-В88.
8. Дзядык В.К. Введение в теорию равномерного приближения функций полиномами. М.: Наука, 1977.
9. Zhenyakova I.V., Cherepova M.F. The Cauchy Problem for a Multi-dimensional Parabolic Equation with Dini-continuous Coefficients // J. Math. Sci. 2022. V. 264(5). Pp. 581—602.
10. Friedman A. Partial Differential Equations of Parabolic Type. Mineola: Dover Publ., Inc., 2008.
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Для цитирования: Михайличенко И.В. О гладкости параболического потенциала простого слоя с Дини-непрерывной плотностью // Вестник МЭИ. 2026. № 4. С. 195—200. DOI: 10.24160/1993-6982-2026-4-195-200
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Результаты работы получены в рамках выполнения государственного задания Министерства науки и высшего образования Российской Федерации (проект № FSWF-2026-0010)
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1. Baderko E.A. O Reshenii Metodom Granichnyh Integral'nyh Uravneniy Kraevyh Zadach dlya Lineynyh Parabolicheskih Uravneniy Proizvol'nogo Poryadka v Netsilindricheskih Oblastyah: Avtoref. Dis. … Doktora Fiz.-mat. Nauk. M.: MGU im. M.V. Lomonosova, 1992. (in Russian).
2. Baderko E.A. O Gladkosti 2m-parabolicheskogo Potentsiala Prostogo Sloya. Differentsial'nye Uravneniya. 1990;26(1):3—10. (in Russian).
3. Kamynin L.I. O Gladkosti Teplovyh Potentsialov v Prostranstve Dini-Gel'dera. Sibirskiy Matematicheskiy Zhurnal. 1970;11(5):1017—1045. (in Russian).
4. Kamynin L.I. O Reshenii Metodom Potentsialov Osnovnyh Kraevyh Zadach dlya Odnomernogo Parabolicheskogo Uravneniya Vtorogo Poryadka. Sibirskiy Matematicheskiy Zhurnal. 1974;15(4):806—834. (in Russian).
5. Baderko E.A. O Potentsialah dlya 2p-parabolicheskih Uravneniy. Differentsial'nye Uravneniya. 1983;19(1):9—18. (in Russian).
6. Zeyneddin M. Gladkost' Potentsiala Prostogo Sloya dlya Parabolicheskoy Sistemy Vtorogo Poryadka v Klassah Dini. Dep. v VINITI RAN. № 1294-V92. (in Russian).
7. Cherepova M.F. O Gladkosti Parabolicheskogo Potentsiala Prostogo Sloya. Dep. v VINITI AN SSSR. № 5194-V88. (in Russian).
8. Dzyadyk V.K. Vvedenie v Teoriyu Ravnomernogo Priblizheniya Funktsiy Polinomami. M.: Nauka, 1977. (in Russian).
9. Zhenyakova I.V., Cherepova M.F. The Cauchy Problem for a Multi-dimensional Parabolic Equation with Dini-continuous Coefficients. J. Math. Sci. 2022;264(5):581—602.
10. Friedman A. Partial Differential Equations of Parabolic Type. Mineola: Dover Publ., Inc., 2008
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For citation: Mikhailichenko I.V. On the Smoothness of a Single Layer Parabolic Potential with Dini Continuous Density. Bulletin of MPEI. 2026;4:195—200. (in Russian). DOI: 10.24160/1993-6982-2026-4-195-200
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The Results were Obtained as Part of the State Assignment of the Ministry of Science and Higher Education of the Russian Federation (Project No. FSWF-2026-0010)
