6,364 research outputs found
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II
We continue the development, by reduction to a first order system for the
conormal gradient, of \textit{a priori} estimates and solvability for
boundary value problems of Dirichlet, regularity, Neumann type for divergence
form second order, complex, elliptic systems. We work here on the unit ball and
more generally its bi-Lipschitz images, assuming a Carleson condition as
introduced by Dahlberg which measures the discrepancy of the coefficients to
their boundary trace near the boundary. We sharpen our estimates by proving a
general result concerning \textit{a priori} almost everywhere non-tangential
convergence at the boundary. Also, compactness of the boundary yields more
solvability results using Fredholm theory. Comparison between classes of
solutions and uniqueness issues are discussed. As a consequence, we are able to
solve a long standing regularity problem for real equations, which may not be
true on the upper half-space, justifying \textit{a posteriori} a separate work
on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has
changed nam
Determining the anisotropic traction state in a membrane by boundary measurements
We prove uniqueness and stability for an inverse boundary problem associated
to an anisotropic elliptic equation arising in the modeling of prestressed
elastic membranes.Comment: 6 page
Discrete Geometric Structures in Homogenization and Inverse Homogenization with application to EIT
We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions over the domain . Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to -convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem
Recent progress in elliptic equations and systems of arbitrary order with rough coefficients in Lipschitz domains
This is a survey of results mostly relating elliptic equations and systems of
arbitrary even order with rough coefficients in Lipschitz graph domains.
Asymptotic properties of solutions at a point of a Lipschitz boundary are also
discussed
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
We develop new solvability methods for divergence form second order, real and
complex, elliptic systems above Lipschitz graphs, with boundary data. The
coefficients may depend on all variables, but are assumed to be close to
coefficients that are independent of the coordinate transversal to the
boundary, in the Carleson sense defined by Dahlberg. We obtain a
number of {\em a priori} estimates and boundary behaviour results under
finiteness of . Our methods yield full characterization of weak
solutions, whose gradients have estimates of a non-tangential maximal
function or of the square function, via an integral representation acting on
the conormal gradient, with a singular operator-valued kernel. Also, the
non-tangential maximal function of a weak solution is controlled in by
the square function of its gradient. This estimate is new for systems in such
generality, and even for real non-symmetric equations in dimension 3 or higher.
The existence of a proof {\em a priori} to well-posedness, is also a new fact.
As corollaries, we obtain well-posedness of the Dirichlet, Neumann and
Dirichlet regularity problems under smallness of and
well-posedness for , improving earlier results for real symmetric
equations. Our methods build on an algebraic reduction to a first order system
first made for coefficients by the two authors and A. McIntosh in order
to use functional calculus related to the Kato conjecture solution, and the
main analytic tool for coefficients is an operational calculus to prove
weighted maximal regularity estimates.Comment: This is an extended version of the paper, containing some new
material and a road map to proofs on suggestion from the referee
Hardy spaces of the conjugate Beltrami equation
We study Hardy spaces of solutions to the conjugate Beltrami equation with
Lipschitz coefficient on Dini-smooth simply connected planar domains, in the
range of exponents . We analyse their boundary behaviour and certain
density properties of their traces. We derive on the way an analog of the Fatou
theorem for the Dirichlet and Neumann problems associated with the equation
with -boundary data
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