arXiv subjects

Junli Zhang

Publications and source records attributed to Junli Zhang.

9 recordsLinked to original sources

arXiv2026

Higher Weak Differentiability to Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

In this paper, we investigate the higher weak differentiability of solutions to a class of mixed local and nonlocal degenerate elliptic equations in the Heisenberg group $\mathbb{H}^n$. Owing to the non-commutative property and two-step nilpotent Lie algebra structure of $\mathbb{H}^n$, we first employ an iterative scheme involving fractional difference quotients to establish the weak differentiability of solutions in the vertical direction. This is subsequently extended to the horizontal and vertical gradients. Then, by coupling a truncation argument with the difference quotient method, we prove the higher weak differentiability of the gradients of solutions.

math.AP↗

arXiv2026

$C^{1,\alpha}$-regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

The regularity theory for equations combining both local and nonlocal operators in sub-Riemannian geometries is a huge challenge. In this paper, we investigate the $C^{1,\alpha}$-regularity of weak solutions to mixed local and nonlocal degenerate elliptic equations on the Heisenberg group. We first derive a sophisticated iteration scheme of Morrey-type by leveraging horizontal difference combined with the fractional Sobolev-type inequality on the Heisenberg group. Then, the H\"{o}lder continuity of the weak solutions is established by applying the local boundedness, the iteration scheme of Morrey-type, an iterative method and the Morrey inequality. Finally, we use the H\"{o}lder continuity in conjunction with Theorem 1.2 from Mukherjee and Zhong\cite{MZ21} to prove the $C^{1,\alpha}$-regularity of weak solutions.

math.AP↗

arXiv2025

Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group

In this paper, we investigate the regularity for mixed local and nonlocal degenerate elliptic equations in the Heisenberg group. Inspired by the De Giorgi-Nash-Moser theory, the local boundedness of weak subsolutions and the H\"{o}lder continuity of weak solutions to mixed local and nonlocal degenerate elliptic equations are established by deriving the Caccioppoli type inequality for weak subsolutions and the logarithmic estimates for weak supersolutions. Furthermore, the Harnack inequality for weak solutions and the weak Harnack inequality for weak supersolutions are proved by using the estimates involving a Tail term and expansion of positivity.

math.AP↗

arXiv2023

Local regularity for nonlocal double phase equations in the Heisenberg group

We prove interior boundedness and H\"{o}lder continuity for the weak solutions of nonlocal double phase equations in the Heisenberg group $\mathbb{H}^n$. This solves a problem raised by Palatucci and Piccinini et. al. in 2022 and 2023 for nonlinear integro-differential problems in the Heisenberg group $\mathbb{H}^n$. Our proof of the a priori estiamtes bases on the spirit of De Giorgi-Nash-Moser theory, where the important ingredients are Caccioppoli-type inequality and Logarithmic estimate. To achieve this goal, we establish a new and crucial Sobolev-Poincar\'{e} type inequality in local domain, which may be of independent interest and potential applications.

math.AP↗

arXiv2023

Validation of the plasma-wall self-organization model for density limit in ECRH-assisted start-up of Ohmic discharges on J-TEXT

A recently developed plasma-wall self-organization (PWSO) model predicts a significantly enhanced density limit, which may be attainable in tokamaks with ECRH-assisted ohmic startup and sufficiently high initial neutral density. Experiments have been conducted on J-TEXT to validate such a density limit scenario based on this model. Experimental results demonstrate that increasing the pre-filled gas pressure or ECRH power during the startup phase can effectively enhance plasma purity and raise the density limit at the flat-top. Despite the dominant carbon fraction in the wall material, some discharges approach the edge of the density-free regime of the 1D model of PWSO.

physics.plasm-ph↗

arXiv2023

On a Conjecture of Cai-Zhang-Shen for Figurate Primes

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints about the conjecture in the cases of odd and even integers and using the method of Lagrange multipliers, Cardano formula for cubic equations and the contradiction, we prove the conjecture.

math.NT↗

arXiv2023

Weak Differentiability to Nonuniform Nonlinear Degenerate Elliptic Systems under $p,q$-growth Condition on the Heisenberg Group

The paper concerns the weak differentiability of weak solutions to two kinds of nonuniform nonlinear degenerate elliptic systems under the $p,q$-growth condition on the Heisenberg Group. We use the iteration to fractional difference quotients on the Heisenberg Group to get the weak differentiability of weak solution $u$ in the vertical direction (i.e., ${L^p}$($1<p<4$) integrability of $Tu$) and then the second order weak differentiability of weak solution in the horizontal directions (i.e., ${L^2}$ integrability of $\nabla_H^2u$) and weak differentiability of gradient of weak solution in the vertical direction (i.e., ${L^2}$ integrability of $T{\nabla_H}u$)).

math.AP↗

arXiv2017

A proof of Riemann Hypothesis

Let $\Xi(t)$ be a function relating to the Riemann zeta function $\zeta (s)$ with $s = \frac{1} {2} + it$. In this paper, we construct a function $v$ containing $t$ and $\Xi(t)$, and prove that $v$ satisfies a nonadjoint boundary value problem to a nonsingular differential equation if $t$ is any nontrivial zero of $\Xi(t)$. Inspecting properties of $v$ and using known results of nontrivial zeros of $\zeta (s)$, we derive that nontrivial zeros of $\zeta (s)$ all have real part equal to $\frac{1} {2}$, which concludes that Riemann Hypothesis is true.

math.GM↗

arXiv2011

Microstructure and magnetic anisotropy of electrospun Cu$_{1-x}$Zn$_x$Fe$_2$O$_4$ nanofibers: A local probe study

Understanding the phenomena at the nanometer scale is of fundamental importance for future improvements of desired properties of nanomaterials. We report a detailed investigation of the microstructure and the resulting magnetic anisotropy by magnetic, transmission electron microscope (TEM) and Mössbauer measurements of the electrospun Cu$_{1-x}$Zn$_x$Fe$_2$O$_4$ nanofibers. Our results show that the electrospun Cu$_{1-x}$Zn$_x$Fe$_2$O$_4$ nanofibers exhibit nearly isotropic magnetic anisotropy. TEM measurements indicate that the nanofibers are composed of loosely connected and randomly aligned nanograins. As revealed by the Henkel plot, these nanofibers and the nanograins within the nanofibers are dipolar coupled, which reduces the effective shape anisotropy leading to a nearly random configuration of the magnetic moments inside the nanofibers, hence, the observed nearly isotropic magnetic anisotropy can be easily understood.

cond-mat.mes-hall↗