Formula for a Solution of 𝑢𝑡 + 𝐻(𝑢, D𝑢) = 𝑔

Proceedings Mathematical Sciences

G D Veerappa Gowda¹  Adimurthi² 

[1] $$;TIFR Centre, P.B. , Indian Institute of Science Campus, Bangalore 0 0, India$$

关键词: Hamilton-Jacobi equation;    dynamic programming principle;    viscosity sub and super solutions.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We study the continuous as well as the discontinuous solutions of Hamilton–Jacobi equation 𝑢𝑡 + 𝐻(𝑢, D𝑢) = 𝑔 in $mathbb{R}^n × mathbb{R}_+$ with 𝑢(𝑥, 0)=𝑢0(𝑥). The Hamiltonian 𝐻(𝑠, 𝑝) is assumed to be convex and positively homogeneous of degree one in 𝑝 for each 𝑠 in $mathbb{R}$. If 𝐻 is non increasing in 𝑠, in general, this problem need not admit a continuous viscosity solution. Even in this case we obtain a formula for discontinuous viscosity solutions.

【 授权许可】

Unknown