Hamilton-jacobi Equations with Partial Gradient and Application to Homogenization
نویسندگان
چکیده
The paper proves a new uniqueness result for viscosity solutions of the Dirichlet problem for Hamilton-Jacobi equations of the form H(x; u; D x 0 u) = 0 in ; u = g on @; where is an open subset or R n and D x 0 u is the partial gradient of the scalar function u with respect to the rst n 0 variables (n 0 n). The main theorem states that there is a viscosity solution of the equation which is unique a.e. This result is applied to the periodic homogenization of Hamilton-Jacobi equations in a general perforated set. It yields the a.e. convergence of the solutions of the problem at scale " as " ! 0 to the solution of the homogenized Hamilton-Jacobi equation with partial gradient.
منابع مشابه
Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment
We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends p...
متن کاملBreakdown of Homogenization for the Random Hamilton-jacobi Equations
We study the homogenization of Lagrangian functionals of Hamilton-Jacobi equations (HJ) with quadratic nonlinearity and unbounded stationary ergodic random potential in R, d≥1. We show that homogenization holds if and only if the potential is bounded from above. When the potential is unbounded from above, homogenization breaks down, due to the almost sure growth of the running maxima of the ran...
متن کاملThe vertex test function for Hamilton-Jacobi equations on networks
A general method for proving comparison principles for Hamilton-Jacobi equations on networks is introduced. It consists in constructing a vertex test function to be used in the doubling variable technique. The first important consequence is that it provides very general existence and uniqueness results for Hamilton-Jacobi equations on networks with Hamiltonians that are not convex with respect ...
متن کاملA note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable
for some δ > 0. Regularity of solutions of Hamilton-Jacobi equations with superlinear growth have been the object of several works (see in particular Lions [6], Barles [3], Rampazzo, Sartori [7]). Our aim is to show that u is locally Hölder continuous with Hölder exponent and constant depending only M , δ, q and T . What is new compared to the previous works is that the regularity does not depe...
متن کاملPeriodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations
We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, “viscous” Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.
متن کامل