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<p>In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\le 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, for all values of the Lagrangian phase, any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^{1,1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> solution is smooth and derive a <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(C^{k,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> estimate for it, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Regularity for Hamiltonian stationary equations in \(\mathbb {R}_{n\le 4}^{n}\)

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Arunima Bhattacharya

摘要

In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions \(n\le 4\) n ≤ 4 , for all values of the Lagrangian phase, any \(C^{1,1}\) C 1 , 1 solution is smooth and derive a \(C^{k,\alpha }\) C k , α estimate for it, where \(k \ge 2\) k ≥ 2 .