Integral Inequalities for Infimal Convolution and Hamilton-Jacobi Equations

Let

be Borel measurable, bounded below and such that

\inf f+\inf g\geq 0. inf ⁡ f + inf ⁡ g ≥ 0.

We prove that with

m_{f,g}:=(\inf f-\inf g)/2, m f , g : = ( inf ⁡ f − inf ⁡ g ) / 2 ,

the inequality

||(f-m_{f,g})^{-1}||_{\phi }+||(g+m_{f,g})^{-1}||_{\phi }\leq 4||(f\Box g)^{-1}||_{\phi } ∣ ∣ ( f − m f , g ) − 1 ∣ ∣ ϕ + ∣ ∣ ( g + m f , g ) − 1 ∣ ∣ ϕ ≤ 4 ∣ ∣ ( f □ g ) − 1 ∣ ∣ ϕ

holds in every Orlicz space

L_{\phi }, L ϕ ,

where

f\Box g f □ g

denotes the infimal convolution of

f f

and

g g

and where

||\cdot ||_{\phi } ∣ ∣ ⋅ ∣ ∣ ϕ

is the Luxemburg norm (i.e., the

L^{p} L p

norm when

L_{\phi }=L^{p} L ϕ = L p

). Although no genuine reverse inequality can hold in any generality, we also prove that such reverse inequalities do exist in the form

||(f\Box g)^{-1}||_{\phi }\leq 2^{N-1}(||(\check{f}-m_{f,g})^{-1}||_{\phi }+||(\check{ g}+m_{f,g})^{-1}||_{\phi }), ∣ ∣ ( f □ g ) − 1 ∣ ∣ ϕ ≤ 2 N − 1 ( ∣ ∣ ( f ˇ − m f , g ) − 1 ∣ ∣ ϕ + ∣ ∣ ( g ˇ + m f , g ) − 1 ∣ ∣ ϕ ) ,

where

\check{f} f ˇ

and

\check{g} g ˇ

are suitable transforms of

f f

and

g 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.