Advances in Differential Equations

Hopf-Lax type formula for sub- and supersolutions

Adimurthi and G. D. Veerappa Gowda

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Abstract

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

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

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651380

Mathematical Reviews number (MathSciNet)
MR1734538

Zentralblatt MATH identifier
0987.35034

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

Citation

Adimurthi; Veerappa Gowda, G. D. Hopf-Lax type formula for sub- and supersolutions. Adv. Differential Equations 5 (2000), no. 1-3, 97--119. https://projecteuclid.org/euclid.ade/1356651380


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