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On the homotopy classification of elliptic boundary value problems. (English) Zbl 1015.58006
Demuth, Michael (ed.) et al., Partial differential equations and spectral theory. Proceedings of the PDE 2000 conference, Clausthal, Germany, July 24-28, 2000. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 126, 299-305 (2001).
This is a very brief account on homotopy classification of elliptic boundary value problems and corresponding index formulas, which summarize some earlier results of the authors generalizing the well-known classification due to Boutet de Monvel.
First the authors give a homotopy classification of classical elliptic boundary value problems, i.e. problems of the form \[ Du=f,\;Bj^{m-1}_{\partial M}u=g,\;u\in C^\infty(M,E),\;f\in C^\infty (M,F),\;g\in C^\infty (\partial M,G), \] where \(D\) is elliptic of order \(m\), \(j^{m-1}_{\partial M}u\) is the jet of \(u\) of order \(m-1\) in normal direction restricted to the boundary, and \(B\) is a pseudo-differential operator on \(\partial M\), which satisfy the Shapiro-Lopatinskii condition (and thus the boundary problem possesses the Fredholm property). A well-known result of Atiyah-Bott states that the existence for given elliptic \(D\) of such a classical boundary value problem implies vanishing of certain obstruction determined by the principal symbol of \(D\), which lies in \(K^1_c(T^*\partial M)\), and in turn, vanishing of the obstruction implies that near \(\partial M\) the symbol of \(D\) is stably homotopic to a symbol independent of cotangent variables.
The authors obtain an analogous simplification for the whole boundary value problem in a wider class of elliptic operators, which assume the form \(\sum_{0\leq k\leq m}D_k (t) (-i{\partial \over\partial t})^{m-k}\) near the boundary \((t\) is the normal parameter). Their final result is the following isomorphism (stable homotopy classification of elliptic boundary value problems): \[ \chi:Ell (M) \approx K_c \bigl (T^*(M\setminus \partial M)\bigr), \] which leads to the following formula for the index of elliptic boundary value problems: \[ \text{ind}(D,B) =p!\bigl( \chi[D, B]\bigr), \quad p!:K_c \bigl( T^*(M\setminus \partial M)\bigr)\to\mathbb{Z}. \] Here \(Ell (M)\) is the group of stable homotopy classes of elliptic boundary value problems on \(M\), and two elliptic boundary value problems are said to be stably homotopic if they become homotopic after equalizing their orders by composing with an elliptic Fredholm operator on \(M\) with zero index, equal near the boundary to \(D_+ = -{\partial \over\partial t}+ \sqrt{-\Delta_{\partial M}}\) and after adding to them some Dirichlet boundary value problem for the Laplacian on \(M\).
In order to get the classification the authors show that any boundary value problem is stably homotopic to an operator of the form \(\sum_{0\leq k\leq m}D_k(t) (-i{ \partial \over\partial t})^{m-k}\) of order zero. Moreover, the authors show that any first order elliptic boundary value problem for \(D={\partial \over \partial t} +A\) is homotopic to the direct sum of \(D_+\) (without boundary condition) and the Dirichlet problem for \({\partial\over \partial t}+ \sqrt {-\Delta_{\partial M}}\), and higher-order operators are homotopic to first-order ones by an earlier result of Hörmander.
Next the authors extend their results to a more general class of elliptic boundary value problems, which enjoy the Fredholm property but do not satisfy the Atiyah-Bott vanishing condition (thus are not classical). These are the problems of the form \[ Du=f,\;Bj^{m-1}_{ \partial M}u=g,\;g\in\text{Im} P_+\subset C^\infty (\partial M,G), \] where \(P\) is a pseudo-differential projection on \(\partial M\). These boundary problems are shown to be homotopic to a spectral boundary value problem in the style of Atiyah-Patodi-Singer and then, in certain special cases, classified in a manner similar to those cited above.
For the entire collection see [Zbl 0970.00021].
58J32 Boundary value problems on manifolds
58J20 Index theory and related fixed-point theorems on manifolds
19M05 Miscellaneous applications of \(K\)-theory