In this paper, we prove coercive a priori estimates of solutions of a Dirichlet-type boundaryvalue problem in a strip for a certain higher-order degenerate elliptic equation containing weighted derivatives of a special form up to the order 2m and ordinary partial derivatives up to the order 2k − 1 under the condition 2m > 2k − 1. At the boundary of the strip, Dirichlet-type conditions are imposed. A coercive a priori estimate for solutions of the problem considered in special weighted Sobolev-type spaces is obtained.
In this paper, a new class of degenerate pseudo-differential operators is investigated, with a variable symbol depending on the complex parameter. Pseudodifferential operators are constructed by a special integral transform. Theorems on the composition and boundedness of these operators in special weighted spaces are proved. The behavior of these operators on hyperplanes of degeneration is investigated. The theorems on the commutation of these operators with differentiation operators are established. A adjoint operator is constructed and an analogue of Goring inequality for degenerate pseudodifferential operators is proved.
Доказаны коэрцитивные априорные оценки решений краевой задачи типа задачи Дирихле в полосе для одного вырождающегося эллиптического уравнения высокого порядка, содержащего весовые производные специального вида до порядка $2m$ и обычные частные производные до порядка $2k-1$ при условии $2m>2k-1$. На границе полосы наложены условия типа Дирихле. Получена коэрцитивная априорная оценка решения рассматриваемой задачи. Оценка получена в специальных весовых пространствах типа пространств Соболева.
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