司靈得 (Daniel Spector)

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.

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.

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).  

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).

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). 

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.