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<p>We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>D</mi> <mi>u</mi> <mo>,</mo> <mi>λ</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$H(x, Du, \lambda u) = 0 $</EquationSource> </InlineEquation> in <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{n} $</EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>λ</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$\lambda \to 0^{+} $</EquationSource> </InlineEquation>. Assuming that the Aubry set is localized, we use a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.</p>

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Vanishing discount limits for fully nonlinear Hamilton-Jacobi equations on noncompact domains

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  • 发表日期：

    2026年5月4日

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  • Son N. T. Tu,
  • Jianlu Zhang

摘要

We study the asymptotic behavior of solutions to the fully nonlinear Hamilton-Jacobi equation H ( x , D u , λ u ) = 0 $H(x, Du, \lambda u) = 0 $ in R n $\mathbb{R}^{n} $ as λ → 0 + $\lambda \to 0^{+} $ . Assuming that the Aubry set is localized, we use a variational approach to derive limiting Mather-type measures and formulate a selection principle. Central to our analysis is a modified variational formula that bridges global and local state-constraint solutions, thereby extending localization techniques to the nonlinear framework.