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<p>We investigate the Dirichlet problem of the two-dimensional Lagrangian mean curvature equation in bounded domains. Infinitely many <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({C^{1,\alpha}}({\alpha \in ({0,{1 \over 5}})})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </mrow> <mo stretchy="false">(</mo> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> very weak solutions are built through Nash-Kuiper construction. Moreover, we note that there are infinitely many <i>C</i><sup>1,<i>α</i></sup> very weak solutions that cannot be improved to be <i>C</i><sup>2,<i>α</i></sup>.</p>

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Ill-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equations

  • 期刊论文

  • 发表日期：

    2026年4月21日

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  • Wentao Cao,
  • Zhehui Wang

摘要

We investigate the Dirichlet problem of the two-dimensional Lagrangian mean curvature equation in bounded domains. Infinitely many \({C^{1,\alpha}}({\alpha \in ({0,{1 \over 5}})})\) C 1 , α ( α ∈ ( 0 , 1 5 ) ) very weak solutions are built through Nash-Kuiper construction. Moreover, we note that there are infinitely many C1,α very weak solutions that cannot be improved to be C2,α.