Nonremovable sets for H\"older continuous quasiregular mappings in the plane

We show that for any dimension t>2(1+alpha K)/(1+K) there exists a compact set E of dimension t and a function alpha-Holder continuous on the plane, which is K-quasiregular only outside of E. To do this, we construct an explicit K-quasiconformal mapping that gives, by one side, extremal dimension distortion on a Cantor-type set, and by the other, more Holder continuity than the usual 1/K.Comment: 17 pages, 3 figure

Removable singularities for H\"older continuous quasiregular mappings in the plane

We give necessary conditions for a set E to be removable for Holder continuous quasiregular mappings in the plane. We also obtain some removability results for Holder continuous mappings of finite distortion.Comment: 8 page

Weighted estimates for Beltrami equations

We obtain a priori estimates in $L^p(\omega)$ for the generalized Beltrami equation, provided that the coefficients are compactly supported $VMO$ functions with the expected ellipticity condition, and the weight $\omega$ lies in the Muckenhoupt class $A_p$. As an application, we obtain improved regularity for the jacobian of certain quasiconformal mappings.Comment: 25 page

Sharp nonremovability examples for H\"older continuous quasiregular mappings in the plane

Let $\alpha\in(0,1)$, $K\geq 1$, and $d=2\frac{1+\alpha K}{1+K}$. Given a compact set E\subset\C, it is known that if \H^d(E)=0 then $E$ is removable for $\alpha$-H\"older continuous $K$-quasiregular mappings in the plane. The sharpness of the index $d$ is shown with the construction, for any $t>d$, of a set $E$ of Hausdorff dimension $\dim(E)=t$ which is not removable. In this paper, we improve this result and construct compact nonremovable sets $E$ such that 0<\H^d(E)<\infty. For the proof, we give a precise planar $K$-quasiconformal mapping whose H\"older exponent is strictly bigger than $\frac{1}{K}$, and that exhibits extremal distortion properties.Comment: 19 pages, 1 figur

Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $\gamma>1$, we construct an example of a bounded autonomous velocity field $b$ with $$\operatorname{div} b\in \operatorname{Exp}\left(\frac{L}{\log^\gamma L}\right) ,$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the $BV$ setting are also addressed

A Note on Transport Equation in Quasiconformally Invariant Spaces

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel-Lizorkin spaces.Comment: 14 page

Manifolds of quasiconformal mappings and the nonlinear Beltrami equation

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.Comment: 26 pages, Proposition 2.2 is improved and the proofs of Schauder estimates and the lower bound for the Jacobian are moved to arXiv:1511.0837

Flows for non-smooth vector fields with subexponentially integrable divergence

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.Comment: 24 pages, revise

Nonlinear transport equations and quasiconformal maps

We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy Kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings

Pointwise descriptions of nearly incompressible vector fields with bounded curl

Among those nearly incompressible vector fields ${\bf{v}}:{\mathbb{R}}^n\to{\mathbb{R}}^n$ with $|x|\log|x|$ growth at infinity, we give a pointwise characterization of the ones for which $\operatorname{curl}{\bf{v}}= D{\bf{v}}-D^t{\bf{v}}$ belongs to $L^\infty$. When $n=2$ we can go further and describe, still in pointwise terms, the vector fields ${\bf{v}}:{\mathbb{R}}^2\to{\mathbb{R}}^2$ for which $|\operatorname{div}{\bf{v}}|+|\operatorname{curl}{\bf{v}}|\in L^\infty$