Differential Equations of Elliptic Type with Variable Operators and General Robin Boundary Condition in UMD Spaces

В данной работе изучается абстрактное дифференциальное уравнение второго порядка эллиптического типа с переменными операторными коэффициентами и общим граничным условием Робина, которое содержит неограниченный линейный оператор. Исследование проводится в случае, когда второй член принадлежит пространству Соболева и использует знаменитую теорему Доре - Венни. В исследовании не предполагается дифференцируемость резольвентных операторов. Приводятся необходимые и достаточные условия на данные, для того чтобы получить существование, единственность классического решения, которое удовлетворяет свойству максимальной регулярности, полученного в предположении Лаббаса - Террени. Используемые методы по существу основаны на теории полугрупп, дробных степенях линейных операторов, функциональном исчислении Данфорда и теории интерполяции. Работа является продолжением работ, изученных Р. Хауа в пространствах UMD и однородных случаях. Приведен пример, к которому применима данная теория.

Ключевые слова

абстрактные эллиптические дифференциальные уравнения второго порядка; граничные условия Робина; аналитическая полугруппа; максимальная регулярность.

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