A shape of eigenfunction of the Laplacian under singular variation of domains. II: The Neumann boundary condition
[For Part I, see ibid. 59, 315-317 (1983; Zbl 0551.35065).] The authors consider the operator \({\mathcal L}\) defined by the quasielliptic partial differential equation with unbounded operator coefficients of the form \[ \sum^{n}_{k=1}(-1)^{\ell_ k} a_ k(x) \partial^{2\ell_ k} u/\partial x_ k^{2\ell_ k\quad}+A(x)u+\sum A\alpha (x) D^{\alpha} u \] on an n-dimensional parallelopiped with homogeneous mixed boundary conditions. Under a number of technical assumptions too numerous to mention, the authors prove that for sufficiently large \(\lambda_ 0>0\) the problem \((L+\lambda_ 0)u=f\), subject to the boundary conditions, is coercively solvable, and the operator \({\mathcal L}+\lambda_ 0\) is positive, i.e., \(\| ({\mathcal L}+\lambda_ 0+s)^{-1}\| \leq M(1+s)^{-1}\) for all \(s\geq 0\). Under some further hypotheses they show that the eigenvectors of \({\mathcal L}+\lambda_ 0\) are complete.
quasielliptic partial differential equation
scientific article; zbMATH DE number 3948861