Univalent σ-Harmonic Mappings

Abstract

We study mappings from R 2 into R 2 whose components are weak solutions to the elliptic equation in divergence form, Div(σ ∇u) = 0, which we call σ -harmonic mappings. We prove sufficient conditions for the univalence, i.e., injectivity, of such mappings. Moreover we prove local bounds in BMO on the logarithm of the Jacobian determinant of such univalent mappings, thus obtaining the a.e. nonvanishing of their Jacobian. In particular, our results apply to σ -harmonic mapping associated with any periodic structure and therefore they play an important role in homogenization.

Experimental Evidence for Sign Reversal of the Hall Coefficient in Three-Dimensional Metamaterials

“…In the simplest case, the Hall coefficient A H is equal to the inverse of the charge density, i.e., A H ¼ ρ −1 . A few years ago, building upon earlier work [17][18][19], Marc Briane and Graeme W. Milton predicted theoretically that the sign of the isotropic Hall coefficient can be reversed in chainmail-like three-dimensional metamaterials [20]. Notably, art inspired science: Chainmail artist Dylon Whyte suggested to them the three-dimensional structure [21].…”

Some elementary inequalities in gas dynamics equation

“…For different values γ it can be a flow of gas, fluid, plastic, electric or chemical field in different mediums, and so forth (see, e.g., [ In general, a solution of (1.1) with a function σ of variables (x 1 ,...,x n ) is called σ-harmonic function. Such functions were studied in many works (see., e.g., [3,4] and literature quoted therein).…”

“…1)) and it is continuous at the point ε γ ; (c) If γ ∈ (2,3], then the function 1)) and it has the continuous derivative at the point ε γ ; 1)) and it has the second continuous derivative at the point ε γ . 18 Some elementary inequalities in gas dynamics equation…”

“…We rewrite the equality Therefore the function x γ (ε) is not doubly differentiable at the point ε γ for γ ∈ (2,3] and it has second continuous derivative at the point ε γ for γ ∈ (3,+∞). Thus property (5) is proved.…”

On Quasi-conformal Harmonic Maps Between Surfaces

“…The spherical metric ρ(w) = 2 1 + |w| 2 is approximately analytic, but the hyperbolic metric (1.5) λ(w) = 2 1 − |w| 2 is not. Let us mention the following important fact.…”