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First Critical Field in the pinned three-dimensional Ginzburg--Landau Model: A matching upper bound

Carlos Román

Abstract

We continue our study of the first critical field $H_{c_1}$ for extreme type-II superconductors governed by the three-dimensional magnetic Ginzburg--Landau functional with a pinning term $a_\varepsilon$, as introduced in our previous work [arXiv:2507.10915]. Building upon the lower bound for $H_{c_1}$ and the characterization of the Meissner solution, we now establish a matching upper bound for $H_{c_1}$, thereby identifying its leading-order behavior. This result confirms the sharpness of the previously derived lower bound and further elucidates the connection between the onset of vorticity and a weighted variant of the \emph{isoflux problem}. Our argument is prompted by the upper bound construction we developed in [arXiv:2510.14910], based on the Biot--Savart law.

First Critical Field in the pinned three-dimensional Ginzburg--Landau Model: A matching upper bound

Abstract

We continue our study of the first critical field for extreme type-II superconductors governed by the three-dimensional magnetic Ginzburg--Landau functional with a pinning term , as introduced in our previous work [arXiv:2507.10915]. Building upon the lower bound for and the characterization of the Meissner solution, we now establish a matching upper bound for , thereby identifying its leading-order behavior. This result confirms the sharpness of the previously derived lower bound and further elucidates the connection between the onset of vorticity and a weighted variant of the \emph{isoflux problem}. Our argument is prompted by the upper bound construction we developed in [arXiv:2510.14910], based on the Biot--Savart law.
Paper Structure (7 sections, 4 theorems, 56 equations)

This paper contains 7 sections, 4 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. Tubular neighborhood of a curve
  4. Biot--Savart vector field
  5. Energy splitting
  6. Upper bound for the free energy
  7. Upper bound for the first critical field

Key Result

Theorem 1.1

Let $\Gamma$ be a $C^2$ simple open curve in $\Omega$, parametrized by arc length, that intersects $\partial\Omega$ transversally. Assume hypothesispmin holds. Then, for any $\varepsilon>0$ sufficiently small, there exists $(u_\varepsilon,A)\in H^1(\Omega,\mathbb{C})\times H^1(\mathbb{R}^3,\mathbb{R where $C_{\Omega,\Gamma}$ is the constant defined in comega. In addition, it holds that, for any $\

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Definition 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:upperbound']}
  • proof : Proof of Theorem \ref{['thm:firstcriticalfield']}