2018
Том 70
№ 6

# Existence of weak solutions of certain boundary value problems for equations of mixed type

Berezansky Yu. M.

Abstract

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.

Citation Example: Berezansky Yu. M. Existence of weak solutions of certain boundary value problems for equations of mixed type // Ukr. Mat. Zh. - 1963. - 15, № 4. - pp. 347-364.

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