On some PDEs involving homogeneous operators. Spectral analysis, semigroups and Hardy inequalities
Continuing their earlier work on homogeneous operators and semigroups, the authors of the paper under review present a spectral analysis of homogeneous operators and give simple characterisations of generators of homogeneous semigroups and of homogeneous sectorial operators. Here an operator \(L\) with domain \(D(L)\) in a Banach space \(X\) of dilation-invariant functions or distributions on \(\mathbb{R}^N\) is said to be \emph{homogeneous} (of degree \(\sigma\in\mathbb{R}\)) if \[L\phi_R=R^\sigma(L\phi)_R,\qquad \phi\in D(L),\ R>0,\] where \(\phi_R(x)=\phi(Rx)\) for all \(x\in\mathbb{R}^N\) and \(R>0.\) Examples of homogeneous operators include positive fractional powers of the Laplacian \(-\Delta\) and, more generally, pseudodifferential operators with homogeneous symbol on \(\mathbb{R}^N\). Among other things the authors analyse homogeneous perturbations of homogeneous operators and semigroups, and they apply these results to study certain parabolic PDEs involving homogeneous operators, including fractional diffusion problems. The authors moreover investigate the connection between their new results and some Gagliardo-Nirenberg and Hardy type inequalities.
scientific article; zbMATH DE number 7471769
scientific article