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Symmetries, invariances, and boundary value problems for the Hamilton-Jacobi equation. (English) Zbl 1098.35011

Initial boundary value problems for the perturbed Hamilton-Jacobi equation \(u_t+H(\nabla _x u)+G(\cdot,w)=0,\; x\in \mathbb{R}^N_+,\; t\geq 0\) with Dirichlet, Neumann and periodic boundary conditions are considered. It is shown that these problems are governed by nonexpansive semigroups on certain closed subsets of the space of bounded uniformly continuous functions on the first quadrant. The existence of even solutions, odd and periodic solutions corresponding to indicated problems is proved. Lipschitz regularity of solutions is explored.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35F30 Boundary value problems for nonlinear first-order PDEs
47H20 Semigroups of nonlinear operators
35D10 Regularity of generalized solutions of PDE (MSC2000)
58J70 Invariance and symmetry properties for PDEs on manifolds
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