Advances in Differential Equations

Hopf-Lax type formula for sub- and supersolutions

Adimurthi and G. D. Veerappa Gowda

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We study the continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation, $u_t + H(u, D u)=0$ in $\mathbb R^{\,n} \times \mathbb R_+$ with $u(x,0)=u_0(x)$. We obtain explicit formulas for continuous as well as for the sub- and supersolutions. In the latter case, furthermore, if $ H(u,p) > 0 $ for $|p| \neq 0 $ then the supersolution becomes a solution.

Article information

Adv. Differential Equations, Volume 5, Number 1-3 (2000), 97-119.

First available in Project Euclid: 27 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 35K90: Abstract parabolic equations 35L55: Higher-order hyperbolic systems


Adimurthi; Veerappa Gowda, G. D. Hopf-Lax type formula for sub- and supersolutions. Adv. Differential Equations 5 (2000), no. 1-3, 97--119.

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