Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.15245

Optimal boundary gradient estimates for the insulated conductivity problem

Haigang Li·Yan Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study the boundary gradient estimate of the solution to the insulated conductivity problem with the Neumann boundary data when a convex insulating inclusion approaches the boundary of the matrix domain. The gradient of solutions may blow up as the distance between the inclusion and the boundary, denoted as $\varepsilon$, approaches to zero. The blow up rate was previously known to be sharp in dimension $n=2$ (see Ammari et al.\cite{AKLLL}). However, the sharp rates in dimensions $n\geq3$ are still unknown. In this paper, we solve this problem by establishing upper and lower bounds on the gradient and prove that the optimal blow up rates of the gradient are always of order $ε^{-1/2}$ for general strictly convex inclusions in dimensions $n\geq3$. Several new difficulties are overcome and the impact of the boundary data on the gradient is specified. This result highlights a significant difference in blow-up rates compared to the interior estimates in recent works (\cites{LY,Weinkove,DLY,DLY2,LZ}), where the optimal rate is $ε^{-1/2+β(n)}$, with $β(n)\in(0,1/2)$ varying with dimension $n$. Furthermore, we demonstrate that the gradient does not blow up for the corresponding Dirichlet boundary problem.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Haigang Li, Yan Zhao. 2024-09-26. Optimal boundary gradient estimates for the insulated conductivity problem. https://arxiv.org/abs/2409.15245

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-$W$ branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed $b_i$ equation on $\mathbb T^3$. Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.

math.AP↗
arXiv2026

Unconditional uniqueness for the derivative nonlinear Schrödinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

math.AP↗
arXiv2026

Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

We study an elliptic operator $L:=-\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume a mixed \(L^1-L^\infty\) condition on only \(|\partial_t A|\), the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this this mixed \(L^1-L^\infty\) condition.

math.AP↗