Integral inequalities for infimal convolution and Hamilton-Jacobi equations (Q2821977)

The author studies new properties of the infimal convolution of \(f\) and \(g\), which is defined as NEWLINE\[NEWLINE (f\square g)(x)=\inf_{y\in\mathbb R^N}(f(x-y)+g(y)). NEWLINE\]NEWLINE Firstly, he proves some measurability conditions of \(f\square g\). Then, denoting \(m_f\) and \(m_g\) the infimum values of \(f\) and \(g\), respectively, and \(m_{f,g}=(m_f-m_g)/2\), if \(\phi\) is a Young function and \(m_f+m_g\geq0\), then NEWLINE\[NEWLINE \|(f-m_{f,g})^{-1}\|_\phi+\|(g+m_{f,g})^{-1}\|_\phi\leq 4\|(f\square g)^{-1}\|_\phi, NEWLINE\]NEWLINE where \(f^{-1}=1/f\). The proof is based on the Brunn-Minkowski inequality. Reverse estimates are also considered. For this purpose, a converse Brunn-Minkowski inequality for Euclidean balls is used. In particular, introducing the radial transform of \(f\) as NEWLINE\[NEWLINE \hat f(x)=\gamma_f^+(|x|), NEWLINE\]NEWLINE where \(\gamma_f^+(t)=\inf\{\xi:\rho_f^+(\xi)\leq t\}\) and \(\rho_f^+(\xi)\) is the radius of the smallest ball containing the level set \(\{x\in\mathbb R^n:f(x)>\xi\}\) (\(\hat f\) is reminiscent of the nondecreasing rearrangement \(f^*\)), the result proved, under some assumptions on \(f\) and \(g\), says that NEWLINE\[NEWLINE \|(f\square g)^{-1}\|_\phi\leq 2^{N-1}(\|(\check{f}-m_{f,g})^{-1}\|_\phi+\|(\check{g}+m_{f,g})^{-1}\|_\phi), NEWLINE\]NEWLINE where \(\check{f}=-(\widehat{-f})\). Similar inequalities are also considered for other classical extremal operators like \(f \barwedge g\) and \(f\veebar g\). Applications are given to the long-time behavior of the solutions of the Hamilton-Jacobi and related equations.