Minimal Nodal Domains for Strictly Elliptic Partial Differential Equations with Homogeneous Boundary Conditions

This work presents a proof of the dependence of the first eigenvalue for uniformly elliptic partial differential equations on the domain in a less abstract setting than that of Ivo Babushka and Rudolf Vyborny in 1965. The proof contained here, under rather mild conditions on the boundary of the domain, �Ω, demonstrates that the first eigenvalue of elliptic partial differential equation [ �� + �� = 0 �� Ω [ � = 0 �� �Ω depends continuously on the domain in the following sense. If a sequence of domains is such that, then the corresponding first eigenvalues satisfy is the first eigenvalue for [ �� + �� = 0 �� Ω [ � = 0 �� �Ω The work also reviews and utilizes the Sturmian comparison results of John G. Heywood, E. S. Noussair, and Charles A. Swanson. For a continuously parameterized family of domains, say with μ ∈ = [a, b], the continuous dependence of the eigenvalue on the domain combined with the Sturmian comparison results provide a theorem that insures, under certain conditions, that the elliptic partial differential equation [ �� = 0 �� Ω [ � = 0 �� �Ω has a solution which is positive on a nodal domain That is there is a least value of μ [a, b] so that a positive solution u exists for [ �� = 0 �� Ωμ [ � = 0 �� �Ωμ Beyond these results the work contains a theorem that shows for certain types of domains, rectangles in , among them, that there is a critical dimension smaller than which, no solution to the problem [ �� + �� = 0 �� Ω [ � = 0 �� �Ω exists when the eigenvalue is fixed. During the investigations taken up in this work, certain observations were made regarding linear approximations to eigenvalue problems in R2 using a standard numerical approximation scheme. One such observation is that if a linear approximation to an eigenvalue problem contains an incorrect estimate for an eigenvalue, the resulting graphical approximation seems to betray whether or not the estimate was low or high. The observations made do not appear to exist in the literature