Принцип Дирихле (математическая физика)

В математической физике при́нцип Дирихле́ относится к теории потенциала и формулируется следующим образом: если функция u(x) есть решение уравнения Пуассона:

в области с граничным условием на границе , то u может быть найдена как решение вариационной задачи: найти минимум

среди всех дважды дифференцируемых функций таких, что на границе .

Данное утверждение сформулировал (но не доказал) немецкий математик Дирихле. Карл Вейерштрасс показал, что в некоторых ситуациях принцип Дирихле неверен; позднее условия его применения уточнили Бернгард Риман, Анри Пуанкаре, Давид Гильберт и другие математики.

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