<p>We study extreme type-II superconductors described by the three-dimensional magnetic Ginzburg–Landau functional incorporating a pinning term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a_\varepsilon (x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which we assume to be a bounded measurable function satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b\le a_\varepsilon (x)\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≤</mo> <msub> <mi>a</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for some constant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. A hallmark of such materials is the formation of vortex filaments, which emerge when the applied magnetic field exceeds the first critical field <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H_{c_1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> </msub> </math></EquationSource> </InlineEquation>. In this work, we provide a lower bound for this critical field and provide a characterization of the Meissner solution, that is, the unique vortexless configuration that globally minimizes the energy below <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H_{c_1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <msub> <mi>c</mi> <mn>1</mn> </msub> </msub> </math></EquationSource> </InlineEquation>. Moreover, we show that the onset of vorticity is intrinsically linked to a weighted variant of the <i>isoflux</i> problem studied in [<CitationRef CitationID="CR33">33</CitationRef>, <CitationRef CitationID="CR36">36</CitationRef>]. A crucial role is played by the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ε</mi> </math></EquationSource> </InlineEquation>-level tools developed in [<CitationRef CitationID="CR34">34</CitationRef>], which we adapt to the weighted Ginzburg–Landau framework.</p>

First critical field in the pinned three-dimensional Ginzburg–Landau model of superconductivity

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发表日期：

2026年6月23日

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Matías Díaz-Vera,
Carlos Román

摘要

We study extreme type-II superconductors described by the three-dimensional magnetic Ginzburg–Landau functional incorporating a pinning term $a_\varepsilon (x)$ $a_{ε} (x)$ , which we assume to be a bounded measurable function satisfying $b\le a_\varepsilon (x)\le 1$ $b \leq a_{ε} (x) \leq 1$ for some constant $b>0$ $b > 0$ . A hallmark of such materials is the formation of vortex filaments, which emerge when the applied magnetic field exceeds the first critical field $H_{c_1}$ $H_{c_{1}}$ . In this work, we provide a lower bound for this critical field and provide a characterization of the Meissner solution, that is, the unique vortexless configuration that globally minimizes the energy below $H_{c_1}$ $H_{c_{1}}$ . Moreover, we show that the onset of vorticity is intrinsically linked to a weighted variant of the isoflux problem studied in [33, 36]. A crucial role is played by the $\varepsilon $ $ε$ -level tools developed in [34], which we adapt to the weighted Ginzburg–Landau framework.