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Presentation av examensarbete

Mattias Enstedt presenterar sitt examensarbete tisdagen den 16 augusti kl 10.15.

Lokal: Glashuset.

Titel: On uniqueness of slutions to polyharmonic equations in the half-space with Cauchy data and imcomplete Cauchy data.

Abstract: In this thesis we study uniqueness of solutions to a biharmonic equations in the upper half-space with incomplete Cauchy data. That is, $u=0$ is the only function in $C^4 (\mathbb{R}^{n+1}_+) \cap C^2(\overline{\mathbb{R}^{n+1}_+})$ such that $u$ is biharmonic in $\mathbb{R}^{n+1}_+$, equal to 0 on $\partial \mathbb{R}^{n+1}_+$ and $\partial_y u=\partial_y^2 u = 0$ on a domain $G$ in $\partial \mathbb{R}^{n+1}_+$. This is evidently not the Dirichlet problem for the biharmonic operator in the upper half-space since we only have one boundary condition on $\partial \mathbb{R}^{n+1}_+ \setminus G$. Moreover, this is not a Cauchy problem since we don't have any third order Cauchy data on $G$. Hence, standard arguments (e.g. the argument involving Green's formula) won't work on this problem. Therefore, we introduce a new method to handle this problem. We show that this method can be used to show unique continuation properties for the Poisson integral.

We also show unique continuation properties for all polyharmonic operators (of finite degree) in the half-space. In this thesis we also present convergence results for a large class of approximate identities and we introduce theory on the Poisson integral.

Välkomna

Vladimir Kozlov

Material till Lite Mat lämnas till Bodil Stavklint senast torsdagar kl 08.00.
Linköpings universitet, 581 83 Linköping
Tel 013-281000, Fax 013-100746
E-mail: litemat@mai.liu.se