arXiv subjects

Hanye Zhu

Publications and source records attributed to Hanye Zhu.

6 recordsLinked to original sources

arXiv2025

Gradient estimates for the insulated conductivity problem with partially flat inclusions

We study the insulated conductivity problem with inclusions embedded in a bounded domain in $\mathbb{R}^n$. It was known that in the setting of strictly convex inclusions, the gradient of solutions may blow up as the distance between inclusions approaches 0. The optimal blow-up rate was proved in [10] and was achieved in the presence of a uniform background gradient field. In this paper, we demonstrate that when the inclusions are partially flat, the gradient of solutions does not blow up under any uniform background fields.

math.AP↗

arXiv2024

Gradient estimates for the conductivity problem with imperfect bonding interfaces

We study the field concentration phenomenon between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type. The boundary condition on these interfaces is given by a Robin-type boundary condition. We discover a \textit{new} dichotomy for the field concentration depending on the bonding parameter $γ$. Specifically, we show that the gradient of solution is uniformly bounded independent of $\varepsilon$ (the distance between two inclusions) when $γ$ is sufficiently small. However, the gradient may blow up when $γ$ is large. Moreover, we identify the threshold of $γ$ and the optimal blow-up rates under certain symmetry assumptions. The proof relies on a crucial anisotropic gradient estimate in the thin neck between two inclusions. We develop a general framework for establishing such estimate, which is applicable to a wide range of elliptic equations and boundary conditions.

math.AP↗

arXiv2023

Asymptotics of the solution to the perfect conductivity problem with $p$-Laplacian

We study the perfect conductivity problem with closely spaced perfect conductors embedded in a homogeneous matrix where the current-electric field relation is the power law $J=σ|E|^{p-2}E$. The gradient of solutions may be arbitrarily large as $\varepsilon$, the distance between inclusions, approaches to 0. To characterize this singular behavior of the gradient in the narrow region between two inclusions, we capture the leading order term of the gradient. This is the first gradient asymptotics result on the nonlinear perfect conductivity problem.

math.AP↗

arXiv2023

The insulated conductivity problem with $p$-Laplacian

We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order $\varepsilon^{-α}$, where $α= 1/2$ when $p \in(1,3]$ and any $α> 1/(p-1)$ when $p > 3$. We provide examples to show that this exponent is almost optimal. In dimensions $n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + β}$ for some $β> 0$, and show that $β\nearrow 1/2$ as $n \to \infty$.

math.AP↗

arXiv2021

Gradient estimates for singular parabolic $p$-Laplace type equations with measure data

We are concerned with gradient estimates for solutions to a class of singular quasilinear parabolic equations with measure data, whose prototype is given by the parabolic $p$-Laplace equation $u_t-Δ_p u=μ$ with $p\in (1,2)$. The case when $p\in \big(2-\frac{1}{n+1},2\big)$ were studied in [15]. In this paper, we extend the results in [15] to the open case when $p\in \big(\frac{2n}{n+1},2-\frac{1}{n+1}\big]$ if $n\geq 2$ and $p\in(\frac{5}{4}, \frac{3}{2}]$ if $n=1$. More specifically, in a more singular range of $p$ as above, we establish pointwise gradient estimates via linear parabolic Riesz potential and gradient continuity results via certain assumptions on parabolic Riesz potential.

math.AP↗

arXiv2021

Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the $p$-Laplace equation $-Δ_p u=μ$ with $p\in (1,2)$. The cases when $p\in \big(2-\frac 1 n,2\big)$ and $p\in \big(\frac{3n-2}{2n-1},2-\frac{1}{n}\big]$ were studied in [9] and [22], respectively. In this paper, we improve the results in [22] and address the open case when $p\in \big(1,\frac{3n-2}{2n-1}\big]$. Interior and global modulus of continuity estimates of the gradients of solutions are also established.

math.AP↗