Trace theorem and non-zero boundary value problem for parabolic equations in weighted Sobolev spaces

A Study on the Residential Environment Polarization and Characteristics in Seoul

“…T : H α,2 p,q,w → B 2θ p,q,w2 , where T u(t, x) := u(0, x) and prove that B 2θ p,q,w2 is the smallest space satisfying the above (extension theorem). As a key step in proving the trace theorem, we take an appropriate integral representation of u ∈ H α,2 p,q,w with the help of a certain type of mollification u (ε) , which is also used to obtain Sobolev inequality [15,7] and regularity of solutions on the boundary of the spatial domain (i.e., lateral trace) [14]. In fact, by the same argument, we obtain the trace estimate of u(t 0 , •) at each time t 0 , not just at t 0 = 0 (see Remark 4.3), which plays an important role in the nonlinear theory.…”