<p>We derive the Euler-Lagrange equations for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(f_1,f_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-harmonic maps with potential from the unit ball <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B=B(m)\subset \mathbb {R}^m$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$m\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) into standard stationary Lorentzian manifolds. By showing that the Euler-Lagrange equations can be rewritten as certain elliptic systems without an <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-antisymmetric structure, we establish the corresponding regularity results for weakly <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$(f_1,f_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-harmonic maps with potential.</p>

Regularity for $(f_1,f_2)$-harmonic maps with potential into stationary Lorentzian manifolds

期刊论文
发表日期：

2025年12月6日

查看全文

Tian Chong,
Miaomiao Zhu

摘要

We derive the Euler-Lagrange equations for $(f_1,f_2)$ $(f_{1}, f_{2})$ -harmonic maps with potential from the unit ball $B=B(m)\subset \mathbb {R}^m$ $B = B (m) \subset R^{m}$ ( $m\geqslant 2$ $m ⩾ 2$ ) into standard stationary Lorentzian manifolds. By showing that the Euler-Lagrange equations can be rewritten as certain elliptic systems without an $L^2$ $L^{2}$ -antisymmetric structure, we establish the corresponding regularity results for weakly $(f_1,f_2)$ $(f_{1}, f_{2})$ -harmonic maps with potential.

Regularity for \((f_1,f_2)\)-harmonic maps with potential into stationary Lorentzian manifolds

摘要