A priori estimates in weighted spaces for solutions of the Poisson and heat equations
Adam Kubica, Wojciech M. Zajączkowski · Applicationes Mathematicae · 2007
We prove a priori estimates for solutions of the Poisson and heat equations in weighted spaces of Kondrat'ev type.The weight is a power of the distance from a distinguished axis.1. Introduction.In this paper we obtain a priori estimates for solutions of the Poisson and heat equations in weighted spaces of Kondrat'ev type in R 3 .Here the weight is some power of the distance from a distinguished axis in R 3 and the power depends on the order of the derivative of the function.We want to stress that the weights we are concerned with are not A 2 weights (i.e.Muckenhoupt's weights, see [Mc72℄).This is the main diculty in deriving a priori estimates.Solutions which belong to weighted spaces of Kondrat'ev type can be found in e.g.[Na94℄, [Ko97℄, where the authors examine boundary value problems in dihedral domains.In our investigations we replace the dihedral domain by R 3 with the z-axis removed.We deduce a priori estimates for solutions of the Poisson (resp.heat) equation in the space) in three steps.First, utilizing Kondrat'ev's method [Ko67℄ we examine the Poisson equation in R 2 and we show that, if µ is noninteger, then for each f ∈ L 2,µ (R 2 ) there exists a unique solution u ∈ H 2 µ (R 2 ) of ∆u = f .Next we obtain estimates in weighted spaces for solutions of a related two-dimensional elliptic problem with a parameter.This is the main step of the proof.Finally, we obtain a priori estimates in the weighted spaces2000 Mathematics Subject Classication: