Nonlinear diffusions, hypercontractivity and the optimal \(L^{p}\)-Euclidean logarithmic Sobolev inequality
The authors deal with the semigroup generated by the nonlinear diffusion equation \[ u_t= \Delta_p(u^{1/p-1})\tag{1} \] with \(\Delta_p w:= \text{div}(|\nabla w|^{p-2}\nabla w)\) for some \(p> 1\). They prove that associated to (1) semigroup is hyper- and ultra-contractive. To this end the authors use the \(L^p\)-Euclidean logarithmic Sobolev inequality. The last section of the paper is devoted to connections with large deviations and the Hamilton-Jacobi equation \[ v_t+{1\over p}|\nabla v|^p= 0 \] for which the optimal \(L^p\)-Euclidean logarithmic Sobolev inequality also plays an important role.
scientific article; zbMATH DE number 2082150