Given a sequence of approximate H-surfaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{u_n\}_{n\in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> from a Riemann surface M to a compact Riemannian manifold N with free boundaries <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$u_n(\partial M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> lying on a given supporting submanifold <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {K}\subset N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">K</mi> <mo>⊂</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, admitting uniform energy bound and tension fields uniformly bounded in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p(M, TN)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>T</mi> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p \ge {6}/{5}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>6</mn> <mo stretchy="false">/</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, we employ the doubling annulus Pohozaev-type identity to establish energy identity and no-neck property during bubbling process near free boundaries for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\{u_n\}_{n\in \mathbb {N}}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Bubbling Analysis at Free Boundary for Approximate H-surfaces with Tension Fields Bounded in $L^p$

期刊论文
发表日期：

2026年7月21日

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Rui Gao,
Zijun Wang,
Miaomiao Zhu

摘要

Given a sequence of approximate H-surfaces $\{u_n\}_{n\in \mathbb {N}}$ ${u_{n}}_{n \in N}$ from a Riemann surface M to a compact Riemannian manifold N with free boundaries $u_n(\partial M)$ $u_{n} (\partial M)$ lying on a given supporting submanifold $\mathcal {K}\subset N$ $K \subset N$ , admitting uniform energy bound and tension fields uniformly bounded in $L^p(M, TN)$ $L^{p} (M, T N)$ for $p \ge {6}/{5}$ $p \geq 6 / 5$ , we employ the doubling annulus Pohozaev-type identity to establish energy identity and no-neck property during bubbling process near free boundaries for $\{u_n\}_{n\in \mathbb {N}}.$ ${u_{n}}_{n \in N} .$

Bubbling Analysis at Free Boundary for Approximate H-surfaces with Tension Fields Bounded in \(L^p\)

摘要