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In this paper we show that the maximal viscosity solution of a class ofquasi-convex Hamilton--Jacobi equations, coupled with inequality constraints onthe boundary, can be recovered by taking the limit as $p\to\infty$ in a familyof Finsler $p$-Laplace problems. The approach also enables us to provide anoptimal solution to a Beckmann-type problem in general Finslerian setting andallows recovering a bench of known results based on the Evans--Gangbotechnique.

Quasi-convex Hamilton--Jacobi equations via limits of Finsler $p$-Laplace problems as $p\to \infty$