Abstract
This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part.
We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions.
It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality.
The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point.
A little more details about this are as follows.
Factorize this operator with constant coefficients into two first order differential operators.
Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.
AbstractWe prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.
Abstract
For a parabolic equation in the spatial variable
x
=
(
x
1
,
…
,
x
n
)
{x=(x_{1},\ldots,x_{n})}
and time t,
we consider an inverse problem of determining a coefficient which is independent
of one spatial component
x
n
{x_{n}}
by lateral boundary data.
We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem.
Also, we prove similar results for the corresponding inverse source problem.
This paper deals with the global uniqueness of an inverse problem for
the stochastic plate with structural damping. The key point is the
Carleman estimate for the fourth order stochastic plate operators dyt −
ρ∆ytdt + ∆2ydt. To this aim, a weighted point- wise identity for a
fourth order stochastic plate operator is established, via which we
obtained the desired Carleman estimate for the corresponding stochastic
plate equation with structural damping.
Abstract
Under a priori boundedness conditions of solutions and coefficients, we prove
a Hölder stability estimate for an inverse problem of
determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix
in a one-dimensional fractional diffusion system of
half order in time.
The proof relies on the conversion of the fractional diffusion
system to a system of order 4
in the space variable and the Carleman estimate.
Unique continuation property of solutions to general second order elliptic systems
