Analytic aspects of quasiconformality.
On Peano's theorem in locally convex spaces
Calderón's problem for Lipschitz classes and the dimension of quasicircles.
In the last years the mapping properties of the Cauchy integral CΓf(z) = 1/(2πi) ∫Γ [f(ξ) / ξ - z] dξ have been widely studied. The most important question in this area was Calderón's problem, to determine those rectifiable Jordan curves Γ for which CΓ defines a bounded operator on L2(Γ). The question was solved by Guy David [Da] who proved that CΓ is bounded on...
Teichmüller spaces and BMOA.
Rectifiability in Teichmüller theory
A boundary integral equation for Calderón's inverse conductivity problem.
Towards a constructive method to determine an L-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L-conductivity problem can be determined in a stable manner at the boundary and outside of the body.
Quasiconformal mappings with Sobolev boundary values
We consider quasiconformal mappings in the upper half space of , , whose almost everywhere defined trace in has distributional differential in . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space . More generally, we consider certain positive functions defined on , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them....
Convex integration and the theory of elliptic equations
This paper deals with the theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions are based...
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