Title: Characterizations of non-radiating
sources in the elastic waves
Abstract:
In this talk, I would like to discuss the
characterization of non-radiating volume and surface
(faulting) sources for the elastic waves in anisotropic
inhomogeneous media. Each type of the source can be
decomposed into a radiating part and a non-radiating part.
The radiating part can be uniquely determined by an explicit
formula containing the near-field measurements. On the other
hand, the non-radiating part does not induce scattered waves
at a certain frequency. In other words, such non-radiating
sources can not be detected by measuring the field at one
single frequency in a region outside of the domain where the
source is located. This is a recent joint work with Pu-Zhao
Kow.
Thursday 4th March 2021,
4:00–5:00 pm JST (UTC+9), online on Zoom
Abstract: We
introduce a method for solving Calder ́on type inverse
problems forsemilinear equations. The method is based on
higher order linearizations, and it allows one to
solve inverse problems for certain nonlinear equations in
cases where the solution for a corresponding linear equation
is not known.
Thursday 18th March 2021,
4:00–5:00 pm JST (UTC+9), online on Zoom
Abstract: We will
start the talk with a stability result for the surface
diffusion flow in the three dimensional flat torus. Then we
shall consider the surface diffusion flow with a non local
elastic term. We shall present an existence result of a
short time smooth solution of this evolution equation. We
will also discuss long time existence and asymptotic
convergence of solutions starting close to strictly stable
stationary sets.
★DISTINGUISHED LECTURE
Wednesday 24th March 2021,
6:00–7:00 pm JST (UTC+9), online on Zoom
Title:New
perspectives on Sobolev norms Abstract: The
classical Sobolev spaces involve L^p norms of the gradient
of a function u. We present an original point of view where
derivatives are replaced by appropriate finite differences
and the Lebesgue space L^p is replaced by the slightly
larger Marcinkiewicz space M^p (aka weak L^p space) --- a
popular tool in Harmonic Analysis. Surprisingly, these
spaces coincide with the standard Sobolev spaces, a fact
which sheds a new light onto these classical objects and
should have numerous applications. It allows e.g., to
rectify some well-known “irregularities” occurring in the
theory of fractional Sobolev spaces. In particular, we may
derive alternative estimates in some exceptional cases
(involving W^{1,1}) where the anticipated fractional Sobolev
and Gagliardo-Nirenberg inequalities fail. Part of the
central argument relies on an innocuous looking new calculus
inequality which might be useful in other situations. The
current proof of this inequality is more complicated than
expected and it would be desirable to find a simpler one.
The lecture is based on a recent joint work with Jean Van
Schaftingen and Po-Lam Yung (PNAS Feb 2021).
Thursday 1st April 2021, 4:00–5:00
pm JST (UTC+9), online on Zoom
Title:Shape optimization of light structures and the vanishing
mass conjecture Abstract: It is a classical problem in the
theory of shape optimization to find a shape with minimal
(linear) elastic compliance (or, equivalently, maximal
stiffness) for a given amount of mass and prescribed
external forces. It is an intriguing question with a long
history, going back to Michell's seminal 1904 work on
trusses, to determine what happens in the limit of vanishing
mass. Contrary to all previous results, which utilize a soft
mass constraint by introducing a Lagrange multiplier, we
here consider the hard mass constraint. Our results
establish the convergence of approximately optimal shapes of
(exact) size tending to zero to a limit generalized shape
represented by a (possibly diffuse) probability measure.
This limit generalized shape is a minimizer of the limit
compliance, which involves a new integrand, namely the one
conjectured by Bouchitte in 2001 and predicted heuristically
before in works of Allaire-Kohn (80's) and Kohn-Strang
(90's). This integrand gives the energy of the limit
generalized shape understood as a fine oscillation of
(optimal) lower-dimensional structures. Its appearance is
surprising since the integrand in the original compliance is
just a quadratic form and the non-convexity of the problem
is not immediately obvious. I will also present connections
to the theory of Michell trusses and show how our results
can be interpreted as a rigorous justification of that
theory on the level of functionals in both two and three
dimensions. This is joint work with J.F. Babadjian
(Paris-Saclay) and F. Iurlano (Paris-Sorbonne).
★SPECIAL LECTURE Part 1/3
Wednesday
7th April 2021, 4:30–6:30 pm JST (UTC+9), online on Zoom
Abstract: We
consider a one-dimensional variational problem arising in
connection with a model for cholesteric liquid crystals. The
principal feature of our study is the assumption that the
twist deformation of the nematic director incurs much higher
energy penalty than other modes of deformation. The
appropriate ratio of the elastic constants then gives a
small parameter epsilon entering an Allen-Cahn-type energy
functional augmented by a twist term. We consider the
behavior of the energy as epsilon tends to zero. We
demonstrate existence of local energy minimizers classified
by their overall twist, find the Gamma-limit of
these energies and show that it consists of twist and jump
terms. This is joint work with Dmitry Golovaty (Akron) and
Michael Novack (UT Austin).
★SPECIAL
LECTURE Part 2/3
Wednesday
14th April 2021, 4:30–6:30 pm JST (UTC+9), online on Zoom
Abstract: So-called
Schauder estimates are a standard tool in the analysis of
linear elliptic and parabolic PDEs. They had been proved by
Hopf (1929 interior case), and by Scahuder and
Caccioppoli (1934, global estimates). Since then, several
proofs have been given (Campanato, Trudinger, Simon). The
nonlinear case is a more recent achievement from the 80s
(Giaquinta \& Giusti, Ivert, J. Manfredi, Lieberman). All
these classical results take place in the uniformly
elliptic case. I will discuss progress in the
nonuniformly elliptic one. From recent, joint work with
Cristiana De Filippis (Torino).
★SPECIAL LECTURE Part 3/3
Wednesday
21st April 2021, 4:30–6:30 pm JST (UTC+9), online on Zoom
Abstract: It is known that minimizers of
strongly polyconvex variational integrals need not be
regular nor unique. However, if a suitable Gårding type
inequality is assumed for the variational integral, then
both regularity and uniqueness of minimizers can be restored
under natural smallness conditions on the data. In turn, the
Gårding inequality turns out to always hold under an a
priori C1 regularity hypothesis on the minimizer, while its
validity is not known in the general case. In this talk, we
discuss these issues and how they are naturally connected to
convexity of the variational integral on the underlying
Dirichlet classes. Part of the talk is based on ongoing
joint work with Judith Campos Cordero, Bernd Kirchheim and
Jan Kolar.
