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We study the Cauchy problem of a Hamilton-Jacobi equation with the spatialvariable in a closed convex cone. A monotonicity assumption on the nonlinearityallows us to prescribe no condition on the boundary of the cone. We show thewell-posedness of the equation in the viscosity sense and prove severalproperties of the solution: monotonicity, Lipschitzness, and representations byvariational formulas.

Hamilton-Jacobi equations with monotone nonlinearities on convex cones