1,721,072 research outputs found
Wave equation with Robin condition, quantitative estimates of strong unique continuation at the boundary
Get PDFThe main result of the present paper consists in a quanti- tative estimate of unique continuation at the boundary for solutions to the wave equation. Such estimate is the sharp quantitative counterpart of the following strong unique continuation property: let u be a solution to the wave equation that satisfies an homogeneous Robin condition on a portion S of the boundary and the restriction of u on S is flat on a segment \{0\} \times J with 0 \in S then u|S vanishes in a neighbourhood of \{0\} \times J
Size estimates for fat inclusions in an isotropic Reissner-Mindlin plate
Get PDFIn this paper we consider the inverse
problem of determining, within an elastic isotropic thick plate
modelled by the Reissner-Mindlin theory, the possible presence of
an inclusion made of a different elastic material. Under some a
priori assumptions on the inclusion, we deduce constructive upper
and lower estimates of the area of the inclusion in terms of a
scalar quantity related to the work developed in deforming the
plate by applying simultaneously a couple field and a transverse
force field at the boundary of the plate. The approach allows to
consider plates with boundary of Lipschitz class
Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-love plate
Get PDFIn this paper we consider the inverse problem of determining a
rigid inclusion inside a thin plate by applying a couple field at
the boundary and by measuring the induced transversal displacement
and its normal derivative at the boundary of the plate. The plate
is made by non-homogeneous, linearly elastic and isotropic
material. Under suitable a priori regularity assumptions on the
boundary of the inclusion, we prove a constructive stability
estimate of log type. Key mathematical tool is a recently proved optimal three
spheres inequality at the boundary for solutions to the
Kirchhoff-Love plate's equation
Optimal Three Spheres Inequality at the Boundary for the Kirchhoff–Love Plate’s Equation with Dirichlet Conditions
Get PDFWe prove a three spheres inequality with optimal exponent at the boundary for
solutions to the Kirchhoff–Love plate’s equation satisfying homogeneous Dirichlet
conditions. This result implies the Strong Unique Continuation Property at the
Boundary (SUCPB). Our approach is based on the method of Carleman estimates,
and involves the construction of an ad hoc conformal mapping preserving the structure
of the operator and the employment of a suitable reflection of the solution with
respect to the flattened boundary which ensures the needed regularity of the extended
solution. To the authors’ knowledge, this is the first (nontrivial) SUCPB
result for fourth-order equations with a bi-Laplacian principal part
Stability for quantitative photoacoustic tomography with well-chosen illuminations
Get PDFWe treat the stability issue for the three-dimensional inverse imaging modality called quantitative photoacoustic tomography. We provide universal choices of the illuminations which enable to recover, in a Hölder stable fashion, the diffusion and absorption coefficients from the interior pressure data. With such choices of illuminations we do not need the nondegeneracy conditions commonly used in previous studies, which are difficult to be verified a priori
Differentiability of the Dirichlet to Neumann map under movements of polygonal inclusions with an application to shape optimization.
Get PDFIn this paper we derive rigorously the derivative of the Dirichlet
to Neumann map and of the Neumann to Dirichlet map of the conductivity
equation with respect to movements of vertices of triangular conductivity in-
clusions. We apply this result to formulate an optimization problem based on
a shape derivative approach
Size Estimates of Unknown Boundaries with Robin Type Condition
Get PDFWe deal with the problem of determining an unknown part of the boundary of an electrical conductor that is not accessible from an exter- nal observation and where a corrosion process is going on. We obtain estimates from above and below of the size of this damaged region
A generalized Korn inequality and strong unique continuation for the Reissner–Mindlin plate system
Get PDFWe prove constructive estimates for elastic plates modelled by the
Reissner-Mindlin theory and made by general anisotropic material.
Namely, we obtain a generalized Korn inequality which allows to
derive quantitative stability and global H^2 regularity for the
Neumann problem. Moreover, in case of isotropic material, we
derive an interior three spheres inequality with optimal exponent
from which the strong unique continuation property follows
Doubling inequality at the boundary for the Kirchhoff - Love plate\u27s equation with Dirichlet conditions
Get PDFThe main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate\u27s equation satisfying homogeneous Dirichlet conditions. This result, like the three sphere inequality with optimal exponent at the boundary proved in Alessandrini, Rosset, Vessella, Arch. Ration. Mech. Anal. (2019), implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on a suitable Carleman estimate, and involves an ad hoc reflection of the solution. We also give a simple application of our main result, by weakening the standard hypotheses ensuring uniqueness for the Cauchy problem for the plate equation
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