3,227 research outputs found
Laplacian transfer across a rough interface: Numerical resolution in the conformal plane
We use a conformal mapping technique to study the Laplacian transfer across a
rough interface. Natural Dirichlet or Von Neumann boundary condition are simply
read by the conformal map. Mixed boundary condition, albeit being more complex
can be efficiently treated in the conformal plane. We show in particular that
an expansion of the potential on a basis of evanescent waves in the conformal
plane allows to write a well-conditioned 1D linear system. These general
principle are illustrated by numerical results on rough interfaces
Single-logarithmic stability for the Calder\'on problem with local data
We prove an optimal stability estimate for Electrical Impedance Tomography
with local data, in the case when the conductivity is precisely known on a
neighborhood of the boundary. The main novelty here is that we provide a rather
general method which enables to obtain the H\"older dependence of a global
Dirichlet to Neumann map from a local one on a larger domain when, in the layer
between the two boundaries, the coefficient is known.Comment: 12 page
Cracks with impedance, stable determination from boundary data
We discuss the inverse problem of determining the possible presence of an
(n-1)-dimensional crack \Sigma in an n-dimensional body \Omega with n > 2 when
the so-called Dirichlet-to-Neumann map is given on the boundary of \Omega. In
combination with quantitative unique continuation techniques, an optimal
single-logarithm stability estimate is proven by using the singular solutions
method. Our arguments also apply when the Neumann-to-Dirichlet map or the local
versions of the D-N and the N-D map are available.Comment: 40 pages, submitte
Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations
When considering fractional diffusion equation as model equation in analyzing
anomalous diffusion processes, some important parameters in the model, for
example, the orders of the fractional derivative or the source term, are often
unknown, which requires one to discuss inverse problems to identify these
physical quantities from some additional information that can be observed or
measured practically. This chapter investigates several kinds of inverse
coefficient problems for the fractional diffusion equation
On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms
We consider the identification of nonlinear diffusion coefficients of the
form or in quasi-linear parabolic and elliptic equations.
Uniqueness for this inverse problem is established under very general
assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof
of our main result relies on the construction of a series of appropriate
Dirichlet data and test functions with a particular singular behavior at the
boundary. This allows us to localize the analysis and to separate the principal
part of the equation from the remaining terms. We therefore do not require
specific knowledge of lower order terms or initial data which allows to apply
our results to a variety of applications. This is illustrated by discussing
some typical examples in detail
Nodal and spectral minimal partitions -- The state of the art in 2015 --
In this article, we propose a state of the art concerning the nodal and
spectral minimal partitions. First we focus on the nodal partitions and give
some examples of Courant sharp cases. Then we are interested in minimal
spectral partitions. Using the link with the Courant sharp situation, we can
determine the minimal k-partitions for some particular domains. We also recall
some results about the topology of regular partitions and Aharonov-Bohm
approach. The last section deals with the asymptotic behavior of minimal
k-partition
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