Remarks on bounded solutions of steady Hamilton–Jacobi equations

Abstract

We study here the equation H ( D u ) = H ( 0 ) , x ∈ R N . More precisely we investigate under which hypotheses the constant functions are the only bounded solutions. In arbitrary space dimension we prove that this happens when convexity and coercivity occur. In one space dimension we show that the above property holds true for Hamiltonians in a larger class. These results apply when studying the long time behaviour of solutions for time-dependent Hamilton–Jacobi equations. To cite this article: M. Bostan, G. Namah, C. R. Acad. Sci. Paris, Ser. I 347 (2009).