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| Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity |
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Citation： |
Charles DAPOGNY,Michael S. VOGELIUS.Uniform Asymptotic Expansion of the Voltage Potential in the Presence of Thin Inhomogeneities with Arbitrary Conductivity[J].Chinese Annals of Mathematics B,2017,38(1):293~344 |
| Page view： 706
Net amount： 661 |
Authors： |
Charles DAPOGNY; Michael S. VOGELIUS |
Foundation： |
This work was partially supported by NSF grant DMS-12-11330
while CD was a postdoctoral visitor at Rutgers University and by the
NSF IR/D program while MSV served at the National Science
Foundation. |
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| Abstract： |
Asymptotic expansions of the voltage potential in terms of the
``radius'' of a diametrically small (or several diametrically small)
material inhomogeneity(ies) are by now quite well-known. Such
asymptotic expansions for diametrically small inhomogeneities are
uniform with respect to the conductivity of the inhomogeneities.
In contrast, thin inhomogeneities, whose limit set is a smooth,
codimension 1 manifold, $\sigma$, are examples of inhomogeneities
for which the convergence to the background potential, or the
standard expansion cannot be valid uniformly with respect to the
conductivity, $a$, of the inhomogeneity. Indeed, by taking $a$ close
to 0 or to infinity, one obtains either a nearly homogeneous Neumann
condition or nearly constant Dirichlet condition at the boundary of
the inhomogeneity, and this difference in boundary condition is
retained in the limit.
The purpose of this paper is to find a ``simple" replacement for the
background potential, with the following properties: (1) This
replacement may be (simply) calculated from the limiting domain
$\Omega \setminus \sigma$, the boundary data on the boundary of
$\Omega$, and the right-hand side. (2) This replacement depends on
the thickness of the inhomogeneity and the conductivity, $a$,
through its boundary conditions on $\sigma$. (3) The difference
between this replacement and the true voltage potential converges to
$0$ uniformly in $a$, as the inhomogeneity thickness tends to $0$. |
Keywords： |
Uniform asymptotic expansions, Conductivity problem, Thininhomogeneities |
Classification： |
35J25, 74G10, 74G75, 78M35 |
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