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

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Marginals of Geometric Inequalities (2007, unknown): “The proof in [ 22 ] relies on the Pr´ekopa-Leindler inequality (see also the proof of the logarithmic Sobolev inequality in [8]).”
Sharp affine Sobolev type inequalities via the $\Lp$ Busemann-Petty centroid inequality (2015, unknown): “p Lp version of this latter one as well as its extremal functions. For the complete list of references concerning the Lp log-Sobolev inequality (or Lp-entropy inequality), we refer to [4], [9], [10], [15], [17] and [24]. For some applications on the study of optimal estimates for solutions of certai”
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences (2013, methodology): “In the case p = 2 with the Euclidean norm | · |, Proposition 1.1 reproduces the inequality (1.1) of Bakry, Gentil and Ledoux.”
New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality (2015, methodology): “garithmic-Sobolev inequality The classical sharp Lp logarithmic-Sobolev inequality was proved by Del Pino and Dolbeault [15, 16] for p∈ (1,n) and was extended in more general for all p>1 by Gentil [29]. This inequality states that for n≥ 1 and p>1, then for any functions f∈ W1,p(Rn) such that ”