Interpolation of weighted Sobolev spaces (Q2760729)

Let \(\Omega\) be a domain of the space \(\mathbb R^n\), \(\Psi=\{\omega_\alpha(x)\}\), and \(\omega(x)\) be finite collection of positive continuous functions on \(\Omega\). We further let \(H^m_{p,\psi}(\Omega)\) and \(L_{p,\omega}(\Omega)\) stand for the Sobolev weighted spaces with the respective norms given by NEWLINE\[NEWLINE \begin{gathered} \| u\|_{H^m_{p,\psi}(\Omega)}= \biggl(\int\limits_\Omega\sum\limits_{| \alpha| \leq m} \omega_\alpha(x)| D^\alpha u(x)| ^p\,dx\biggr)^{1/p}, \\ \| u\|_{L_{p,\omega}(\Omega)}= \biggl(\int\limits_\Omega \omega(x)| u(x)| ^p\,dx\biggr)^{1/p}, \end{gathered}NEWLINE\]NEWLINE and let \(\overset\circ{H}^m_{p,\psi}(\Omega)\) be the completion of \(C^\infty_0(\Omega)\) with respect to the norm of \(H^m_{p,\psi}(\Omega)\).NEWLINENEWLINE The main results of the article consist in describing the properties of interpolation spaces NEWLINE\[NEWLINE H^s_{p,\psi}(\Omega)=\bigl(H^m_{p,\psi}(\Omega), L_{p,\omega}(\Omega)\bigr)_{1-s/m,p} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \overset\circ{H}^s_{p,\psi}(\Omega)= \bigl(\overset\circ{H}^m_{p,\psi}(\Omega), L_{p,\omega}(\Omega)\bigr)_{1-s/m,p} \quad \bigl(s\in(0,m),\;1<p<+\infty\bigr). NEWLINE\]NEWLINE The suggested mathematical apparatus seems to be applicable to studying the elliptic spectral problems with a weight of definite sign, i.e., problems of the form NEWLINE\[NEWLINE \begin{gathered} Lu=\lambda Bu,\quad x\in\Omega\subset \mathbb R^n, \\ B_ju|_\Gamma,\quad j=1,\dots,m,\;\Gamma=\partial\Omega. \end{gathered} NEWLINE\]NEWLINE Here \(L\) is an elliptic operator of degree \(2m\), \(B_j\) is a normal system of differential operators on \(\Gamma\), \(Bu=g(x)u\), and \(g(x)\) is a measurable function changing sign on \(\Omega\).