ЗАДАЧА З ДВОМА КРАТНИМИ ВУЗЛАМИ ДЛЯ ЛIНIЙНИХ СИСТЕМ РIВНЯНЬ IЗ ЧАСТИННИМИ ПОХIДНИМИ

M. M. Symotiuk

doi:10.31861/bmj2019.02.086

M. M. Symotiuk Iнститут прикладних проблем механiки i математики iм. Я.С. Пiдстригача НАН України

DOI: https://doi.org/10.31861/bmj2019.02.086

Ключові слова: двоточкова задача, системи рiвнянь iз частинними похiдними, малi знаменники, метричний пiдхiд

Анотація

Запроваджено класи $H, H_{δ}, S_{δ}$ лiнiйних систем рiвнянь iз частинними похiдними. Цi
класи характеризуються нетривiальнiстю та наявнiстю степеневих оцiнок знизу для певних симетричних многочленiв вiд коренiв характеристичних рiвнянь даних систем. На
пiдставi метричного пiдходу та теорiї симетричних многочленiв показано, що до запроваджених класiв належать майже всi системи рiвнянь iз частинними похiдними зi сталими
коефiцiєнтами (стосовно мiри Лебега у просторi, натягнутому на коефiцiєнти системи).
Дослiджено задачу з двома кратними вузлами за видiленою змiнною t та умовами перiодичностi за iншими координатами x1, . . . , xp для лiнiйних систем рiвнянь iз частинними
похiдними, якi належать до описаних класiв. Встановлено умови розв’язностi задачi у просторах гладких вектор-функцiй iз експоненцiйним спаданням вектор-коефiцiєнтiв Фур’є.
Доведено, що оцiнки знизу для малих знаменникiв, достатнi для iснування розв’язку задачi, виконуються для майже всiх (стосовно мiри Лебега та фрактальної мiри Гаусдорфа)
значень другого вузла iнтерполяцiї для лiнiйних систем з класiв $H, H_{δ}, S_{δ}$

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