Pyatkov S.G. Shilenkov D.V.
Inverse problems of recovering surface fluxes from pointwise measurements
Reporter: Pyatkov S.G.
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\begin{center}
\title{}{\bf Inverse problem of
recovering surface fluxes from
pointwise measurements}
\author{}{Pyatkov$^{1}$ S.G., Shilenkov$^2$ D.V.}
$^{1}$ {\it Yugra State University, Khanty-Mansiysk; \\
Sobolev Institute of Mathematics, Novosibirsk \\ pyatkov@math.nsc.ru }
$^{2}$ {\it Yugra State University, Khanty-Mansiysk\\
deoltesh.projects@yandex.ru}
\end{center}
Under
consideration is the parabolic equation
\begin{equation}\label{e1}
Mu= u_t+Lu=f(t,x), \ \ (t,x)\in Q = (0,T)\times G,\ T\leq \infty,
\end{equation}
where $Lu=-\Delta u + \sum_{i=1}^n a_{i}(x)u_{x_{i}} +
a_{0}(x)u$, $G$ is a domain in ${\mathbb R}^n$ with boundary
$\Gamma\in C^{2}$, and $n=2,3$. The equation \eqref{e1} is
furnished with the initial-boundary conditions
\begin{equation}\label{e2}
Bu|_{S}=g(t,x)\ \ (S=(0,T) \times \Gamma),\ \ u|_{t=0}=u_{0}(x),
\end{equation}
where $Bu=\frac{\partial u}{\partial \nu} +\sigma(x)u$, with
$\nu$ the outward unit normal to $\Gamma$, and, respectively,
with the overdetermination conditions
\begin{equation}\label{e33}
u(t,b_{i})=\psi_{i}(t)\ (i=1,2,\ldots,r),
\end{equation}
where $\{b_{i}\}_{i=1}^{r}$ is a collection of points lying in
$G$. The problem is to find a solution to the equation
\eqref{e1} satisfying \eqref{e2}-\eqref{e33} and an unknown function
$g(t,x)=\sum_{j=1}^{r}\alpha_{i}(t)\Phi_{i}(x)$,
where the functions $\Phi_{i}(x)$ are given and $\alpha_{i}$
are unknowns.
Inverse problems of recovering the above type are
classical. The arise in many different problems of mathematical
physics, in particular, in the heat and mass transfer theory,
diffusion, filtration, and ecology.
In this article we describe a new approach to the existence
theory of solutions to this problem and establish the
corresponding existence and uniqueness theorems.
Sharp conditions on the data
ensuring existence and uniqueness in Sobolev
classes are exposed. They are smoothness conditions on the
data, geometric conditions on the location of measurement
points, and the boundary of a domain. The proof relies on
asymptotics of fundamental solutions to the corresponding
elliptic problems and the Laplace transform. The problem is reduced to a linear algebraic system
with a nondegerate matrix.
We hope that
these results can be used in developing new numerical
algorithms for solving the problem.
\end{document}
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