ESTIMATES FOR THE SINGULAR SOLUTIONS OF THE 3-D PROTTER’S PROBLEM

ЗаглавиеESTIMATES FOR THE SINGULAR SOLUTIONS OF THE 3-D PROTTER’S PROBLEM
Вид публикацияJournal Article
Година на публикуване2004
АвториPopivanov N, Popov T
СписаниеAnnuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique
Том96
ключови думиboundary value problems, generalized solution, propagation of singularities, singular solutions, special functions, wave equation
Резюме

For the wave equation we study boundary value problems, stated by Protter in 1952, as some three-dimensional analogues of Darboux problems on the plane. It is known that Protter's problems are not well posed and the solution may have singularity at the vertex $O$ of a characteristic cone, which is a part of the domain's boundary $\partial \Omega $. It is shown that for $n$ in $\mathbb{N}$ there exists a right-hand side smooth function from $C^{n}(\bar{\Omega})$, for which the corresponding unique generalized solution belongs to $C^{n}(\bar{\Omega}\backslash O)$, but it has a strong power-type singularity. It is isolated at the vertex $O$ and does not propagate along the cone. The present article gives some necessary and sufficient conditions for the existence of a fixed order singularity. It states some exact a priori estimates for the solution.

2000 MSC

main 35L05, 35L20, 35D05, 35A20, secondary 33C05, 33C90

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