67 research outputs found
Composites and quasiconformal mappings: new optimal bounds in two dimensions
No full textWe prove new bounds for the homogenized tensor of two dimensional multiphase conducting composites. The bounds are optimal for a large class of composites. In physical terms these are mixtures of one polycrystal and several isotropic phases, with prescribed volume fractions. Optimality is understood in the strongest possible sense of exact microgeometries. The techniques to prove the bounds for composites are based on variational methods and results from quasiconformal mappings. We need to refine the quasiconformal area distortion theorem due to the first author and prove new distortion results with weigths. These distortion theorems are of independent interest for PDE's and quasiconformal mappings. They imply e.g. the following surprising theorem on integrability of derivatives at the borderline case: For K > 1, if f is an element of W-loc(1,2) (R-2, R-2) is K-quasiregular, if E subset of R-2 is measurable and bounded and if (&PARTIAL;) over barf(x) = 0 a.e. in E, then integral(E) D f(x)(p) dx < infinity for p = 2K/K-1
Lower semicontinuity, Stoilow factorization and principal maps
No full textWe consider a refinement of the usual quasiconvexity condition of Morrey in two dimensions that allows us to prove lower semicontinuity and existence of minimizers for a class of functionals which are unbounded as the determinant vanishes and are non-polyconvex in general. This notion, that we call principal quasiconvexity, arose from the planar theory of quasiconformal mappings and mappings of finite distortion. We compare it with other quasiconvexity conditions that have appeared in the literature and provide a number of concrete examples of principally quasiconvex functionals that are not polyconvex. We also describe local conditions which combined with quasiconvexity yield principal quasiconvexity. The Stoilow factorization, that in the context of maps of integrable distortion was developed by Iwaniec and Šverák, plays a prominent role in our approach
Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale
No full textWe study quasiconformal mappings in planar domains and their
regularity properties described in terms of Sobolev, Bessel potential or
Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the
geometry of the boundary and of the smoothness of the
Beltrami coefficient, that guarantee the global regularity of the mappings in
these classes. In the Triebel-Lizorkin class with smoothness below , the
same conditions give global regularity in for the principal solutions
with Beltrami coefficient supported in .Comment: 59 pages, 4 figure
Homogenization of iterated singular integrals with applications to random quasiconformal maps
No full textWe study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let.(F-j) / (j >= 1) be a sequence of normalized homeomorphic solutions to the planar Beltrami equation partial derivative((Z) over bar)Fj(z) = / mu(j)((Z) over bar,omega) partial derivative F-z(j)(z), where the random dilatation satisfies vertical bar mu(j vertical bar) (0.1) mu(j) (z,omega) =sic(z) Sigma (n is an element of Z2) g(2(j) z-n,Xn.(omega)), where g(z;omega) decays rapidly in z, the random variables Xn are i.i.d., and sic is an element of C (infinity) (0). We establish the almost sure and local uniform convergence as j ->infinity of the maps Fj to a deterministic quasiconformal limit F infinity. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let T1,...,Tm be translation and dilation invariant singular integrals on R-d, and consider a d-dimensional version of mu (j), e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function f such that almost surely, mu (j) T-m mu (j) : : : T-1 mu (j) -> f as j ->infinity weakly in Lp, for 1 < p < infinity.Peer reviewe
Dimer models and conformal structures
No full textDimer models have been the focus of intense research efforts over the last years. Our paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal domains, much studied in numerical simulations and elsewhere. In particular, we show that the frozen boundaries are always algebraic curves. Our classification also implies that the Pokrovsky-Talapov law holds for all dimer models at a generic point on the frozen boundary and, in addition, shows a very strong local rigidity of dimer models, which can be interpreted as a geometric universality result. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realized already by the lozenge model. To achieve these goals we develop a new study on the boundary regularity for a class of Monge-Amp & egrave;re equations in non-strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers.Peer reviewe
Numerical computation of complex geometrical optics solutions to the conductivity equation
No full textAbstractA numerical method is introduced for the evaluation of complex geometrical optics (cgo) solutions to the conductivity equation ∇⋅σ∇u(⋅,k)=0 in R2 for piecewise smooth conductivities σ. Here k is a complex parameter. The algorithm is based on the solution by Astala and Päivärinta (2006) [1] of Calderón's inverse conductivity problem and involves the solution of a Beltrami equation in the plane with an exponential asymptotic condition. The numerical strategy is to solve a related periodic problem using fft and gmres and show that the solutions agree on the unit disc. The cgo solver is applied to the problem of computing nonlinear Fourier transforms corresponding to nonsmooth conductivities. These computations give new insight into the D-bar method for the medical imaging technique of electric impedance tomography. Furthermore, the asymptotic behavior of the cgo solutions as k→∞ is studied numerically. The evidence so gained raises interesting questions about the best possible decay rates for the subexponential growth argument in the uniqueness proof for Calderón's problem with L∞ conductivities
- …
