2017
Том 69
№ 9

Всі номери

Существование слабых решений некоторых краевых задач для уравнений смешанного типа

Березанский Ю. М.

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Абстракт

The differential equation of mixed type $$Lu =\sum^2_{j, k=1}D_j (b_{jk} (x) D_ku) + \sum^2_{j=1}p_i(x)D_ju + p(x)u = f(x)$$ is considered in a bounded domain of the $(x_1, x_2)$-plane, the equation being for $x_2 > 0$ elliptic and for $x_2 < 0$ of the form $k(x_2) D^2_1u + D^2_2u = f(x)$. For boundary conditions of the Tricomi type, as well as for more general conditions, two energetic inequalities are proved (for the original and adjoint problems). The existence of the weak and the uniqueness of the strong solutions follows directly for the problems under consideration. Similar problems are investigated for certain unbounded domains.

Зразок цитування: Березанский Ю. М. Существование слабых решений некоторых краевых задач для уравнений смешанного типа // Укр. мат. журн. - 1963. - 15, № 4. - С. 347-364.

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