On solvability conditions for overdetermined systems of differential equations in Sobolev spaces (Q881211)

By using iterations of Green's integrals [\textit{M. Nacinovich} and \textit{A. Shlapunov}, Math. Nachr. 180, 243--284 (1996; Zbl 0871.35066)], local solvability conditions for the equation \(A_iu= f\) are studied. Here \(\{A_i, E_i\}\) is an elliptic complex such that \(A_i: C^\infty(E_1)\to C^\infty(E_{i+1})\), \(0\leq i\leq N\), is a differential operator of degree \(m\), \(E_i\) is a vector bundle over a smooth compact \(n\)-dimensional manifold \(X\) with smooth boundary \(\partial X\). The interior of \(X\) is denoted by \(\mathring X\). Let \(H^m(X, E_i)\) be the \(m\)th Sobolev space of the sections of \(E_i\), \(\mathring H^m(X, E_i)\) the closure of compact support sections of \(E_i\) in \(H^m(X, E_i)\), and let \({\mathcal H}_i(X)=\{u\in\mathring H^m(X, E_i)|A^*_{i-1} u= A_i u= 0\}\). Then there exist orthogonal projectors \[ \Pi^{(i)}: H^{-m}(X, E_i)\to{\mathcal H}_i(X),\;\Phi_i: H^{-m}(X, E_i)\to \mathring H^m(X, E_i)\cap{\mathcal H}^\perp_i(X), \] such that \({\mathfrak A}_i\Pi^{(i)}= \Phi_i\Pi^{(i)}= \Pi^{(i)}\Pi_i= 0\) and \(\Phi_i\Delta_i u= u- \Pi^{(i)}u\), \(\Delta_i\Phi_i w= w- \Pi^{(i)}w\) [\textit{B.-W. Schulze}, \textit{A. Shlapunov} and \textit{N. Tarkhanov}, Ann. Global Anal. Geom. 24, 131--160 (2003; Zbl 1067.58019)]. Here \({\mathfrak A}_i= A_i+ A^*_{i-1}\) and \(\Delta_i= A^*_i A_i+ A_{i-1} A^*_{i-1}\). By using these operators and the Green operator \(G_{A_i}\), etc., for \(A_i\), etc., [\textit{N. N. Tarkhanov}, The parametrix method in the theory of differential complexes, Novosibirsk: Nauka. (1990; Zbl 0758.58002)] operators on \(C^\infty(\overline D, E_i)\), etc., \(D\Subset\mathring X\) \[ ({\mathcal G}^{(i,1)}u)(x)= -\int_{\partial D} G_{A_i}(^tA^*_i \Phi_i(x,\cdot), u), \] \[ (T^{(i,1)}f)(x)= \int_D \langle^t A^*_i \Phi_i(x,\cdot), f\rangle_{x,i+1}\,dx, \] and \({\mathcal G}^{(i,2)}\), \(T^{(i,2)}\), which are defined similar to \({\mathcal G}^{(i,1)}\), \(T^{(i,1)}\), but replaced \(^tA^*_i\) by \(^tA_{i-1}\), are introduced. Then to set \({\mathcal G}^{(i)}={\mathcal G}^{(i,1)}+{\mathcal G}^{i,2)}\), the Green formula \[ (\chi_D u)(x)= ({\mathcal G}^{(i)}u)(x)+ (T^{(i,1)} A_iu)(x)+ (T^{(i,2)} A^*_{i-1} u)(x)+ (\Pi^{(i)}_D u)(x), \] is shown (Lemma 1). Here \(\chi_D\) is the characteristic function of \(D\). Then for the strong operator topologies of \({\mathcal L}(H^m(D, E_i))\) and \({\mathcal L}(L^2(D, E_{i+1})\), it is shown \[ \begin{aligned} I&= \pi_{S^m(A_i, D)}+ \sum^\infty_{\nu= 0} (\Pi^{(i)}_D+{\mathcal G}^{(i)}+ T^{(i,2)} A^*_{i-1})^\nu T^{(i,1)} A_i,\\ I&= \pi_{\ker T}^{(i,1)}+ \sum^\infty_{\nu= 0} A_i(\Pi^{(i)}_D+{\mathcal G}^{(i)}+ T^{(i,2)} A^*_{i-1})^\nu T^{(i,1)}, \end{aligned} \] (Theorem 2). Here \({\mathcal L}(H)\) is the Banach space of continuous linear operators on a Hilbert space \(H\) and \(S^m(A_i, D)\) is the space of weak solutions of the equation \(A_iu= 0\) in \(D\). Then it is shown that the equation \(A_u= f\), \(f\in L^2(D, E_{i+1})\) has a solution \(u\in H^m(D, E_i)\) if and only if \(f\perp\ker T^{(i,1)}\) and the series \[ R^{(i)} f= \sum^\infty_{\nu= 0} (\Pi^{(i)}_D+{\mathcal G}^{(i)}+ T^{(i,2)} A^*_{i-1})^\nu T^{(i,1)} f, \] converges in \(H^m(D, E_i)\) (\S5. Theorem 3). Introducing the harmonic space \({\mathcal H}^{i+1}(D)= \{g\in S^0(A^*_i, D)\cap S^0(A_{i+1}, D)|\nu_i(g)= 0\}\) of the complex \(\{A_i, E_i\}\), the first condition of Th.3 is refined to \(f\perp{\mathcal H}^{i+1}(D)\) and \(A_{i+1} f= 0\) (Corollary 4). The author remarks that \({\mathcal H}^{i+1}(D)\) is not a finite-dimensional space in general. To show these results, the Hermitian form \[ h^{(i)}_D(u,v)= \int_X ({\mathfrak A}_i e_i(u),{\mathfrak A}_i e_i(v))_x \,dx+ \int_X (\Pi^{(i)} e_i(u), \Pi^{(i)} e_i(v))_x \,dx, \] where \(e_i(u)\) is obtained from \(u\) by adopting suitable Dirichlet problem of \(\Delta_i\), which is shown to give an equivalent norm as the standard Sobolev \(m\)-norm (\S2. Theorem 1) is heavily used. The author remarks for the Dolbeault complex, \({\mathcal G}^{(i)}\), etc. are similar-to the Martinelli-Bochner integral [\textit{A. V. Romanov}, Sov. Math., Dokl. 19, 1211--1215 (1978); translation from Dokl. Akad. Nauk SSSR 242, 780--783 (1978; Zbl 0434.35066)]. In \S3, after computing \({\mathcal G}^{(i,k)}\), it is shown \({\mathcal G}^{(i,1)}\) is not self-adjoint with respect to \(h^{(i)}_D\) if \(D\) is the unit ball \({\mathcal B}_1\) (Lemma 3). It is also remarked that the pointwise convergence of the iteration of \({\mathcal G}^{(i)}+\Pi^{(i)}_D\) in spaces other than \({\mathcal L}(H^m(D, E_i))\) is impossible, in general (\S4. Proposition).