Linear parabolic problems involving measures

Amann, H (2001). Linear parabolic problems involving measures. Real Academia de Ciencias Exactas, Fisicas y Naturales. Revista. Serie A, Matematicas, 95(1):85-119.

Abstract

We develop a general solvability theory for linear evolution equations of the form u˙+Au=μ on R+, where −A is the infinitesimal generator of a strongly continuous analytic semigroup, and μ is a bounded Banach-space-valued Radon measure. It is based on the theory of interpolation-extrapolation spaces and the Riesz representation theorem for such measures.
"The abstract results are illustrated by applications to second-order parabolic initial value problems. In particular, the case where Radon measures occur on the Dirichlet boundary can be handled, which is important in control theory and has not been treated so far.
"We also give sharp estimates under various regularity assumptions. They form the basis for the study of semilinear parabolic evolution equations with measures, which is to be studied in a forthcoming joint paper of the author and P. Quittner.

Abstract

We develop a general solvability theory for linear evolution equations of the form u˙+Au=μ on R+, where −A is the infinitesimal generator of a strongly continuous analytic semigroup, and μ is a bounded Banach-space-valued Radon measure. It is based on the theory of interpolation-extrapolation spaces and the Riesz representation theorem for such measures.
"The abstract results are illustrated by applications to second-order parabolic initial value problems. In particular, the case where Radon measures occur on the Dirichlet boundary can be handled, which is important in control theory and has not been treated so far.
"We also give sharp estimates under various regularity assumptions. They form the basis for the study of semilinear parabolic evolution equations with measures, which is to be studied in a forthcoming joint paper of the author and P. Quittner.