Preprint No.
A-00-06
S. R. Umarov, Yu. F. Luchko, R. Gorenflo
On boundary value problems for elliptic equations with boundary operators of frational order
Abstract: We deal with boundary value problems for pseudo-differential
equations of elliptic type with boundary operators depending on a
positive real parameter $\alpha$. In particular, boundary conditions
can be given through the one-sided Marchaud, Gr\"unwald-Letnikov
or Liouville-Weyl fractional derivatives of order
$\alpha$. We prove solvability theorems in spaces of
different type including some subspaces of the Sobolev space
$H^s(\re^n)$, the Lizorkin type spaces and some modifications
of the space $\Psi_{G,p}(\re^n),\ 1<p<\infty,$ of functions
in $L_p$ whose Fourier transforms are compactly supported
in a domain $G\subset \re^n$. The distributions defined
on the last space may have singularities of infinite order
and can be considered as hyper-functions. We show that
they can be treated also as boundary limits in a
special weak topology of the one-sided Marchaud,
Gr\"unwald-Letnikov or Liouville-Weyl fractional derivatives of
harmonic functions.
Keywords: pseudo-differential equations, boundary value problems, fractional derivatives
Mathematics Subject Classification (MSC91): 26A33 Fractional derivatives and integrals
, 45K05 Integro-partial differential equations
, 35A05 General existence and uniqueness theorems
, 35S15 Boundary value problems for $\Psi$DO
Language: ENG
Available: Pr-A-00-06.ps
Contact: Gorenflo, R.; Luchko, Yu. F., Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (gorenflo@math.fu-berlin.de)
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