ABSTRACT
This chapter discusses the isoperimetric inequalities and other related inequalities of the spectral geometry for integral operators of several types, appearing as solutions to different boundary value problems for elliptic and parabolic partial differential equations. It examines different versions of the isoperimetric inequalities for the singular numbers, for the heat operators of different types: higher-order heat operators, as well as the heat operators with the Cauchy-Dirichlet, Cauchy-Robin, Cauchy-Neumann and Cauchy-Dirichlet-Neumann boundary conditions. The chapter proposes the spectral properties and several spectral geometric inequalities for the Riesz potential operators. It considers the spectral geometric properties of the Riesz transforms in non-flat geometries, namely, in the cases of the spheres and of the hyperbolic spaces. The chapter describes some inequalities of the spectral geometry of non-self-adjoint operators.
