41 research outputs found
The linear constraints in Poincar\'{e} and Korn type inequalities
We investigate the character of the linear constraints which are needed for
Poincar\'e and Korn type inequalities to hold. We especially analyze
constraints which depend on restriction on subsets of positive measure and on
the trace on a portion of the boundary.Comment: Revised versio
On doubling inequalities for elliptic systems
We prove doubling inequalities for solutions of elliptic systems with an
iterated Laplacian as diagonal principal part and for solutions of the Lame'
system of isotropic linearized elasticity. These inequalities depend on global
properties of the solutions.Comment: 13 pages, submitted for publicatio
Symmetry of singular solutions of degenerate quasilinear elliptic equations
We prove radial symmetry of singular solutions to an overdetermined boundary
value problem for a class of degenerate quasilinear elliptic equations.Comment: 8 pages, to appear on Rendiconti dell'Istituto di Matematica
dell'Universita' di Triest
Stable determination of an inclusion in an elastic body by boundary measurements (unabridged)
We consider the inverse problem of identifying an unknown inclusion contained
in an elastic body by the Dirichlet-to-Neumann map. The body is made by
linearly elastic, homogeneous and isotropic material. The Lam\'e moduli of the
inclusion are constant and different from those of the surrounding material.
Under mild a-priori regularity assumptions on the unknown defect, we establish
a logarithmic stability estimate. For the proof, we extend the approach used
for electrical and thermal conductors in a novel way. Main tools are
propagation of smallness arguments based on three-spheres inequality for
solutions to the Lam\'e system and refined local approximation of the
fundamental solution of the Lam\'e system in presence of an inclusion.Comment: 58 pages, 4 figures. This is the extended, and revised, version of a
paper submitted for publication in abridged for
Stable determination of a rigid inclusion in an anisotropic elastic plate
In 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
material belonging to a general class of anisotropy. For this severely
ill-posed problem, under suitable a priori regularity assumptions on the
boundary of the inclusion, we prove a stability estimate of log-log type
A generalized Korn inequality and strong unique continuation for the Reissner-Mindlin plate system.
We 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's equation with Dirichlet conditions
The main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's 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