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Yifei Pan

Publications and source records attributed to Yifei Pan.

50 records · Page 3Linked to original sources
arXiv2013

On local solvability of nonlinear elliptic partial differential systems of principle type: the second order

We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As applications to geometry, we prove the exsitence of local harmonic maps with given tangent plane at a point between any Riemannan manifolds. More generally geometric objects defined by Beltrami-Laplace always exist locally.

math.AP↗
arXiv2013

A variant of Hörmander's $L^2$ theorem for Dirac operator in Clifford analysis

In this paper, we give the Hörmander's $L^2$ theorem for Dirac operator over an open subset $Ω\in\R^{n+1}$ with Clifford algebra. Some sufficient condition on the existence of the weak solutions for Dirac operator has been found in the sense of Clifford analysis. In particular, if $Ω$ is bounded, then we prove that for any $f$ in $L^2$ space with value in Clifford algebra, there exists a weak solution of Dirac operator such that $$\bar{D}u=f$$ with $u$ in the $L^2$ space as well. The method is based on Hörmander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted space is utilised.

math.AP↗
arXiv2013

A Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces

In this paper we prove a Schwarz lemma for harmonic mappings between the unit balls in real Euclidean spaces. Roughly speaking, our result says that under a harmonic mapping between the unit balls in real Euclidean spaces, the image of a smaller ball centered at origin can be controlled. This extends the related result proved by Chen in complex plane.

math.CV↗
arXiv2013

On the existence of solutions to nonlinear systems of higher order Poisson type

In this paper, we study the existence of higher order Poisson type systems. In detail, we prove a Residue type phenomenon for the fundamental solution of Laplacian in $\RR^n, n\ge 3$. This is analogous to the Residue theorem for the Cauchy kernel in $\CC$. With the aid of the Residue type formula for the fundamental solution, we derive the higher order derivative formula for the Newtonian potential and obtain its appropriate $\s C^{k, α}$ estimates. The existence of solutions to higher order Poisson type nonlinear systems is concluded as an application of the fixed point theorem.

math.AP↗
arXiv2012

High-order Green Operators on the Disk and the Polydisc

In this paper, we give the explicit expressions of high-order Green operators on the disk and the polydisc, and hence the kernel functions of high-order Green operators are also presented. As applications, we present the explicit integral expressions of all the solutions for linear high-order partial differential equations in the disk.

math.CV↗
arXiv2011

On solvability of nonlinear partial differential systems of any order in the complex plane

We prove a general existence theorem for nonlinear partial differential systems of any order in one complex variable. A special case of first order contains a well-known theorem of Nijenhuis and Woolf concerning local existence of J-holomorphic curves on almost complex manifolds. As an application to differential geometry, we prove that non constant harmonic map always exists locally from a Riemann surface to a Riemannian manifold with a prescribed tangent vector at a given point.

math.CV↗
arXiv2011

The Schwarz-Pick lemma of high order in several variables

We prove a high order Schwarz-Pick lemma for mappings between unit balls in complex spaces in terms of the Bergman metric. From this lemma, Schwarz-Pick estimates for partial derivatives of arbitrary order of mappings are deduced.

math.CV↗
arXiv2009

Some nonlinear differential inequalities and an application to Hölder continuous almost complex structures

We consider some second order quasilinear partial differential inequalities for real valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex valued functions $f(z)$ satisfying $\partial f/\partial\bar z=|f|^α$, $0<α<1$, and $f(0)\ne0$, there is also a lower bound for $\sup|f|$ on the unit disk. For each $α$, we construct a manifold with an $α$-Hölder continuous almost complex structure where the Kobayashi-Royden pseudonorm is not upper semicontinuous.

math.CV↗