On the stability of index of nonlocal elliptic problems (Q1192529)

Let \(Q\) be a bounded domain in \(\mathbb{R}^ n\) with smooth boundary \(\partial Q\). Let \(A^ 0(x,D)\) be a properly elliptic operator on \(\overline {Q}\) of order \(2m\) and suppose boundary operators \(B^ 0_ \mu(x,D)\), \(\mu=1,\dots,m\), of orders \(m_ \mu\leq k+2m-1\) cover \(A^ 0(x,D)\) on \(\partial Q\). A Fredholm operator between Sobolev spaces is induced, namely \[ {\mathcal L}^ 0= (A^ 0,(B^ 0_ \mu)): W^{k+2m}(Q)\to W^ k(Q)\times \prod_ \mu W^{k+2m-m_ \mu-{1\over 2}}(\partial Q). \] Various forms of nonlocal compact perturbations are considered. Indeed \(A=A^ 0+A^ 1\) and \(B_ \mu=B_ \mu^ 0+ B_ \mu^ 1+ B_ \mu^ 2\) where \(A^ 1\) and \(B_ \mu^ 2\) are compact and \(B_ \mu^ 1\) restricted to the null-space of \(A^ 0\) is compact. This includes the case when the support of each \(B_ \mu^ 1\) is contained in the interior of \(Q\). Then \({\mathcal L}=(A,(B_ \mu))\) is also Fredholm and it is proved in section 2 that \({\mathcal L}^ 0\) and \({\mathcal L}\) have the same index. In section 3, mixed boundary value problems are considered, where different boundary operators \(B^ 0_{i\mu}\) act on open \(C^ \infty\)- submanifolds \(\Gamma_ i\) of \(\partial Q\). The boundary \(\partial Q\) is no longer smooth, but has ``corners'' at the boundaries of the \(\Gamma_ i\). Weighted Sobolev spaces are used along with appropriate nonlocal perturbations. It is proved that if \({\mathcal L}^ 0=(A^ 0,(B^ 0_{i\mu}))\) is Fredholm, so is the operator \({\mathcal L}=(A,(B_{i\mu}))\) corresponding to the nonlocal problem. Moreover, \({\mathcal L}\) and \({\mathcal L}^ 0\) have the same index. The index is computed in some special cases in section 4.