Consider the Dirichlet-Laplacian in $Omega:= (0,L)times (0,H)$ and choose another open set $omegasubset Omega$. The estimate $0<C_{omega}leq R_{omega}(u):=frac{Vert uVert^{2}_{L^{2}(omega)}}{Vert uVert^{2}_{L^{2}(Omega)}}leq frac{vol(omega)}{vol(omega)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $forall omega$, the eigenfunctions satisfy $R_{omega}>C_{omega}>0,exists omega, omega ot=emptyset$, such that $inf R_{omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $Rtimes R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added.
تحميل البحث