A note on immersions of domains of fractional powers of certain sectorial operators in Sobolev spaces
The authors consider the \(L^p\)-realization \(A_p\) of a uniformly strongly elliptic linear operator of order \(2m\) in a bounded domain \(\Omega\) with \(C^{2m}\)-boundary with boundary conditions given by \(m\) linear boundary operators. For fractional powers they show the embedding \(D(A^{\alpha}_p) \hookrightarrow W_p^{2m(1-\frac{1}{p})}(\Omega)\) for all \(p > 2m\) and \(0 < \alpha < 1\) with \(\frac{p-1}{p} < \alpha < 1\), where \(D(A^{\alpha}_p)\) has the graph norm. The proof uses the analyticity of the semigroup generated by \(-A_p\) together with the \(L^p\)-elliptic regularity of the abstract Cauchy problem and trace results. The article concludes with an application of the derived embedding: the existence of an optimal control for a model for mosquito populations is shown.
domains of powers of sectorial operators
scientific article; zbMATH DE number 6092778