The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton–Jacobi equations

We prove a general optimal Lp-Euclidean logarithmic Sobolev inequality by using Prékopa–Leindler inequality and a special Hamilton–Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.).