Integral inequalities for infimal convolution and Hamilton-Jacobi equations

Original · EN

Let f,g:Rⁿ→ (-∞,∞] be Borel measurable, bounded below and such that f+ g≥ 0. We prove that with mf,g:=(f- g)/2, the inequality ||(f-mf,g)⁻¹||ϕ+||(g+mf,g)⁻¹||ϕ≤ 4||(f g)⁻¹||ϕ holds in every Orlicz space Lϕ, where f g denotes the infimal convolution of f and g and where ||· ||ϕ is the Luxemburg norm (i.e., the Lᵖ norm when Lϕ=Lᵖ). Although no genuine reverse inequality can hold in any generality, we also prove that such reverse inequalities do exist in the form ||(f g)⁻¹||ϕ≤ 2ⁿ⁻¹(||(f-mf,g)⁻¹||ϕ+||(g+mf,g)⁻¹||ϕ), where f and g are suitable transforms of f and g introduced in the paper and reminiscent of, yet very different from, nondecreasing rearrangement. Similar inequalities are proved for other extremal operations and applications are given to the long-time behavior of the solutions of the Hamilton-Jacobi and related equations.