Sobolev Type Equations in Spaces of Differential Forms on Riemannian Manifolds without Boundary

Статья содержит обзор результатов, полученных автором как самостоятельно, так и в соавторстве с другими представителями Челябинской научной школы Г.А. Свиридюка по "Уравнениям соболевского типа" в специфических пространствах, а именно пространствах дифференциальных форм, заданных на каком-либо римановом многообразии без края. Уравнения соболевского типа относятся к неклассическим уравнениям математической физики и характеризуются необратимым оператором при старшей производной. При рассмотрении в наших пространствах пришлось использовать специальные обобщения операторов на пространство дифференциальных форм, в частности, оператор Лапласа заменили на его обобщение - оператор Лапласа - Бельтрами. Рассмотрены конкретные интерпритации уравнений с относительно ограниченными операторами: линейное Баренблатта - Желтова - Кочиной, линейное и полулинейное Хоффа, линейное Осколкова. Для этих уравнений исследованы в различных случаях разрешимость задач Коши, Шоуолтера - Сидорова и начально-конечной. В зависимости от выбора типа уравнения (линейное или полулинейное) применялась соответствующая модификация метода фазового пространства. Для использования этого метода, основанного на расщеплении области определения и действия соответствующих операторов, в пространствах дифференциальных форм базой служит теорема Ходжа - Кодаиры о расщеплении области определения оператора Лапласа - Бельтрами.

Ключевые слова

уравнения соболевского типа; метод фазового пространства; дифференциальные формы; риманово многообразие без края.

Литература

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