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<p>In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton–Jacobi equations with state constraints. For every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(p&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \mathcal {O}(\varepsilon ^{1/2}) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>ε</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and obtain an improved upper rate of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \mathcal {O}\left( \varepsilon ^{{p\over 2(p-1)}}\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mfenced close=")" open="("> <msup> <mi>ε</mi> <mfrac> <mi>p</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for semiconcave data.</p>

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On the rate of convergence in superquadratic Hamilton–Jacobi equations with state constraints

  • 期刊论文

  • 发表日期：

    2026年3月5日

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  • Prerona Dutta,
  • Khai T. Nguyen,
  • Son N. T. Tu

摘要

In this paper, we investigate the convergence rate in the vanishing viscosity limit for solutions to superquadratic Hamilton–Jacobi equations with state constraints. For every \(p>2\) p > 2 , we establish the rate of convergence for nonnegative Lipschitz data vanishing on the boundary to be of order \( \mathcal {O}(\varepsilon ^{1/2}) \) O ( ε 1 / 2 ) and obtain an improved upper rate of order \( \mathcal {O}\left( \varepsilon ^{{p\over 2(p-1)}}\right) \) O ε p 2 ( p - 1 ) for semiconcave data.