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<p>For a family <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P_{\gamma }, \gamma \in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mi>γ</mi> </msub> <mo>,</mo> <mi>γ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, of parabolic singular minimal surface equations we prove existence and uniqueness results by employing a fixed point argument. Assuming that the data satisfy suitable conditions, we show convergence of a subsequence <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(u(x,t_k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>t</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to a solution <i>u</i>(<i>x</i>) of the stationary problem. On the other hand, it will be shown that singularities of the flow must occur after finite time, in case these conditions are violated.</p>

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Evolutionary Singular Minimal Surfaces: Existence, Uniqueness and Singularities

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • Ulrich Dierkes,
  • Sebastian Holthausen

摘要

For a family \(P_{\gamma }, \gamma \in \mathbb {R}\) P γ , γ ∈ R , of parabolic singular minimal surface equations we prove existence and uniqueness results by employing a fixed point argument. Assuming that the data satisfy suitable conditions, we show convergence of a subsequence \(u(x,t_k)\) u ( x , t k ) to a solution u(x) of the stationary problem. On the other hand, it will be shown that singularities of the flow must occur after finite time, in case these conditions are violated.