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Gao-Feng Zheng

Publications and source records attributed to Gao-Feng Zheng.

6 recordsLinked to original sources
arXiv2024

Optimal higher regularity for biharmonic maps via quantitative stratification

This little note is devoted to refining the almost optimal regularity results of Breiner and Lamm \cite{Breiner-Lamm-2015} on minimizing and stationary biharmonic maps via the powerful quantitative stratification method introduced by Cheeger and Naber \cite{Cheeger-Naber-2013} and further developed by Naber and Valtorta \cite{Naber-V-2017,Naber-V-2018} for harmonic maps. In particular, we obtain an optimal regularity results for minimizing biharmonic maps.

math.AP↗
arXiv2023

Sharp Morrey regularity theory for a fourth order geometrical equation

This paper is a continuation of the recent work of Guo-Xiang-Zheng \cite{Guo-Xiang-Zheng-2021-CV}. We deduce sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation \begin{equation*} Δ^{2}u=Δ(V\nabla u)+div(w\nabla u)+(\nablaω+F)\cdot\nabla u+f\qquad\text{in }B^{4},\end{equation*} under smallest regularity assumptions of $V,w,ω, F$ and that $f$ belongs to some Morrey spaces, which was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the $L^p$ type regularity theory of \cite{Guo-Xiang-Zheng-2021-CV}, and generalizes the work of Du, Kang and Wang \cite{Du-Kang-Wang-2022} on a second order problem to our fourth order problems.

math.AP↗
arXiv2021

$L^p$ regularity theory for even order elliptic systems with antisymmetric first order potentials

Motivated by a challenging expectation of Rivière (2011), in the recent interesting work of deLongueville-Gastel (2019), de Longueville and Gastel proposed the following geometrical even order elliptic system \begin{equation*} Δ^{m}u=\sum_{l=0}^{m-1}Δ^{l}\left\langle V_{l},du\right\rangle +\sum_{l=0}^{m-2}Δ^{l}δ\left(w_{l}du\right)\qquad \text{ in } B^{2m}\label{eq: Longue-Gastel system} \end{equation*} which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order potential, they successfully established a conservation law for this system, from which everywhere continuity of weak solutions follows. This beautiful result amounts to a significant advance in the expectation of Rivière. In this paper, we seek for the optimal interior regularity of the above system, aiming at a more complete solution to the aforementioned expectation of Rivière. Combining their conservation law and some new ideas together, we obtain optimal Hölder continuity and sharp $L^p$ regularity theory, similar to that of Sharp and Topping \cite{Sharp-Topping-2013-TAMS}, for weak solutions to a related inhomogeneous system. Our results can be applied to study heat flow and bubbling analysis for polyharmonic mappings.

math.AP↗
arXiv2020

The Lamm-Riviere system I: $L^p$ regularity theory

Motived by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm-Riviere system $$Δ^{2}u=Δ(V\cdot\nabla u)+{\rm div}(w\nabla u)+(\nablaω+F)\cdot\nabla u+f$$ in dimension four, with an inhomogeneous term $f$ which belongs to some natural function space. We obtain optimal higher order regularity and sharp Holder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mappings.

math.AP↗
arXiv2007

Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=Δu+V(x) |u|^{p-1}u$ in $Ω\times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.

math.AP↗