Regularity for variational problems in the Heisenberg group
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group.
math.AP↗
arXiv subjects
Xiao Zhong
Publications and source records attributed to Xiao Zhong.
$C^{1,α}$-Regularity for variational problems in the Heisenberg group
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the Hölder continuity of horizontal gradient of minima.
math.AP↗
Maximal function estimates and self-improvement results for Poincaré inequalities
Our main result is an estimate for a sharp maximal function, which implies a Keith-Zhong type self-improvement property of Poincaré inequalities related to differentiable structures on metric measure spaces. As an application, we give structure independent representation for Sobolev norms and universality results for Sobolev spaces.
math.CA↗
Guiding microscale swimmers using teardrop-shaped posts
The swimming direction of biological or artificial microscale swimmers tends to be randomised over long time-scales by thermal fluctuations. Bacteria use various strategies to bias swimming behaviour and achieve directed motion against a flow, maintain alignment with gravity or travel up a chemical gradient. Herein, we explore a purely geometric means of biasing the motion of artificial nanorod swimmers. These artificial swimmers are bimetallic rods, powered by a chemical fuel, which swim on a substrate printed with teardrop-shaped posts. The artificial swimmers are hydrodynamically attracted to the posts, swimming alongside the post perimeter for long times before leaving. The rods experience a higher rate of departure from the higher curvature end of the teardrop shape, thereby introducing a bias into their motion. This bias increases with swimming speed and can be translated into a macroscopic directional motion over long times by using arrays of teardrop-shaped posts aligned along a single direction. This method provides a protocol for concentrating swimmers, sorting swimmers according to different speeds, and could enable artificial swimmers to transport cargo to desired locations.
cond-mat.soft↗
Dynamic self-assembly of microscale rotors and swimmers
Biological systems often involve the self-assembly of basic components into complex and function- ing structures. Artificial systems that mimic such processes can provide a well-controlled setting to explore the principles involved and also synthesize useful micromachines. Our experiments show that immotile, but active, components self-assemble into two types of structure that exhibit the fundamental forms of motility: translation and rotation. Specifically, micron-scale metallic rods are designed to induce extensile surface flows in the presence of a chemical fuel; these rods interact with each other and pair up to form either a swimmer or a rotor. Such pairs can transition reversibly be- tween these two configurations, leading to kinetics reminiscent of bacterial run-and-tumble motion.
cond-mat.soft↗
An inverse problem for the p-Laplacian: boundary determination
We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.
math.AP↗
Gradient regularity for elliptic equations in the Heisenberg Group
We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised by Manfredi & Mingione (Math. Ann. 2007) where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the horizontal p-Laplacean operator, extending some regularity proven by Domokos & Manfredi (Cont. Math. 2005). In turn, the a priori estimates found are shown to imply the suitable local Calderon-Zygmund theory for the related class of non-homogeneous, possibly degenerate equations involving discontinuous coefficients. These last results extend to the sub-elliptic setting a few classical non-linear Euclidean results of Iwaniec and Dibenedetto & Manfredi, and to the non-linear case estimates of the same nature that were available in the sub-elliptic setting only for solutions to linear equations.
math.AP↗
Beltrami equations with coefficient in the Sobolev space $W^{1,p}$
We study the removable singularities for solutions to the Beltrami equation $\bar\partial f=μ\partial f$, assuming that the coefficient $μ$ lies on some Sobolev space $W^{1,p}$, $p\leq 2$. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient $μ\in W^{1,2}$ preserve compact sets of $σ$-finite length and vanishing analytic capacity, even though they need not be bilipschitz.
math.AP↗