On the Uniqueness for One-Dimensional Constrained Hamilton-Jacobi Equations.
On the Uniqueness for One-Dimensional Constrained Hamilton-Jacobi Equations.
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关于一维约束 Hamilton-Jacobi 方程的唯一性。
DOI：
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发表时间：
2018
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通讯作者：
Yeoneung Kim
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文献类型：
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作者：
Yeoneung Kim
本文的目的是研究一维Hamilton-Jacobi方程egin{equation*}的唯一性。
The goal of this paper is to study uniqueness of a one-dimensional Hamilton-Jacobi equation egin{equation*}
egin{cases}
u_t=|u_x|^2+R(x,I(t)) & ext{in }mathbb{R} imes (0,infty),
max_{mathbb{R}} u(cdot,t)=0 & ext{on }[0,infty),
end{cases} end{equation*} with an initial condition $u_0(x,0)=u_0(x)$ on $mathbb{R}$. A reaction term $R(x,I(t))$ is given while $I(t)$ is an unknown constraint (Lagrange multiplier) that forces maximum of $u$ to be always zero. In the paper, we prove uniqueness of a pair of unknowns (u,I) using dynamic programming principle in one dimensional space for some particular class of nonseparable reaction $R(x,I(t))$.