Abstract

We devote the integro-extremization method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations {F(Du)=0inΩu(x)=φ(x)forx∈∂Ω,\begin{cases} F(Du)=0 & \quad \textrm{in} \quad\O\cr u(x)=\varphi(x) & \quad \textrm{for} \quad x\in \partial \O, \end{cases} with a particular interest for non coercive hamiltonians FF, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations {∂u∂t+F(∇u)=0in]0,T[×Ωu(0,x)=η(x)forx∈Ωu(t,x)=ψ(x)for(t,x)∈[0,T]×∂Ω.\begin{cases} \frac{\partial u}{\partial t}+ F(\nabla u)=0 & \quad \textrm{in} \quad ]0,T[\times \O\cr u(0,x)=\eta(x) & \quad \textrm{for} \quad x\in\O \cr u(t,x)=\psi(x) & \quad \textrm{for} \quad (t,x)\in[0,T]\times \partial \O. \end{cases} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian FF, on the domain Ω\O and on the boundary datum, without any growth assumptions on FF

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Sandro Zagatti

SISSA, Via Bonomea 265, 34136 Trieste, Italy

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Suggested citation

S. Zagatti. “An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations.” Journal of Convex Analysis 18 (2011), No. 4, 1141–1170. https://doi.org/10.68381/jca18069

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.