99 research outputs found
Local boundedness property for parabolic BVP's and the gaussian upper bound for their Green functions
In the present note, we give a concise proof for the equivalence between the
local boundedness property for parabolic Dirichlet BVP's and the gaussian upper
bound for their Green functions. The parabolic equations we consider are of
general divergence form and our proof is essentially based on the gaussian
upper bound by Daners \cite{Da} and a Caccioppoli's type inequality. We also
show how the same analysis enables us to get a weaker version of the local
boundedness property for parabolic Neumann BVP's assuming that the
corresponding Green functions satisfy a gaussian upper bound
Gaussian lower bound for the Neumann Green function of ageneral parabolic operator
Based on the fact that the Neumann Green function can be constructed as a
perturbation of the fundamental solution by a single-layer potential, we
establish gaussian two-sided bounds for the Neumann Green function for a
general parabolic operator. We build our analysis on classical tools coming
from the construction of a fundamental solution of a general parabolic operator
by means of the so-called parametrix method. At the same time we provide a
simple proof for the gaussian two-sided bounds for the fundamental solution. We
also indicate how our method can be adapted to get a gaussian lower bound for
the Neumann heat kernel of a compact Riemannian manifold with boundary having
non negative Ricci curvature
Stability of the determination of a time-dependent coefficient in parabolic equations
We establish a Lipschitz stability estimate for the inverse problem
consisting in the determination of the coefficient , appearing in a
Dirichlet initial-boundary value problem for the parabolic equation
, from Neumann boundary data. We
extend this result to the same inverse problem when the previous linear
parabolic equation in changed to the semi-linear parabolic equation
Determining a boundary coefficient in a dissipative wave equation: Uniqueness and directional lipschitz stability
We are concerned with the problem of determining the damping boundary
coefficient appearing in a dissipative wave equation from a single boundary
measurement. We prove that the uniqueness holds at the origin provided that the
initial condition is appropriately chosen. We show that the choice of the
initial condition leading to uniqueness is related to a fine version of unique
continuation property for elliptic operators. We also establish a Lipschitz
directional stability estimate at the origin, which is obtained by a
linearization process
New Stability Estimates for the Inverse Medium Problem with Internal Data
A major problem in solving multi-waves inverse problems is the presence of
critical points where the collected data completely vanishes. The set of these
critical points depend on the choice of the boundary conditions, and can be
directly determined from the data itself. To our knowledge, in the most
existing stability results, the boundary conditions are assumed to be close to
a set of CGO solutions where the critical points can be avoided. We establish
in the present work new weighted stability estimates for an electro-acoustic
inverse problem without assumptions on the presence of critical points. These
results show that the Lipschitz stability far from the critical points
deteriorates near these points to a logarithmic stability
The problem of detecting corrosion by an electric measurement revisited
We establish a logarithmic stability estimate for the problem of detecting
corrosion by a single electric measurement. We give a proof based on an
adaptation of the method initiated in \cite{BCJ} for solving the inverse
problem of recovering the surface impedance of an obstacle from the scattering
amplitude. The key idea consists in estimating accurately a lower bound of the
-norm, locally at the boundary, of the solution of the boundary value
problem used in modeling the problem of detection corrosion by an electric
measuremen
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