On \(C_ 0\)-semigroups generated by elliptic second order differential expressions on \(L^ p\)-spaces. (English) Zbl 0852.47018

Summary: We study well-posedness in \(L^p\) of the Cauchy problem for second-order parabolic equations with time-independent measurable coefficients by means of constructing corresponding \(C_0\)-semigroups. Lower order terms are considered as form-bounded perturbations of the generator of the symmetric sub-Markovian semigroup associated with the Dirichlet form. It is shown that the \(C_0\)-semigroup corresponding to the Cauchy problem exists in a certain interval in the scale of \(L^p\)-spaces which depends only on form-bounds of perturbations. We establish also analyticity and \(L^p\)-smoothness of the semigroup constructed.


47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators
47D09 Operator sine and cosine functions and higher-order Cauchy problems
47D07 Markov semigroups and applications to diffusion processes