On eigencurves of elliptic boundary value problems

Abstract

SynopsisLet 7 be a selfadjoint uniformly elliptic partial differential operator on a bounded domain in R", and let S be a (possibly indefinite) L°° multiplication operator. Estimates of the form ak + o(k) and ak + P + o(l) are sought for the eigenvalues ft(k) of kS -7 as k-* ±°°. A necessary and sufficient condition is also obtained for existence of linear eigencurves, i.e. p(k) = ok + ft.

Eigenvalue problems for the wave equation with strong damping

“…(2.5) Jn In order to be able to find the intersections of the graph of the function on the right-hand side of (2.4) with the eigencurves of the family L p , it is now convenient to know the asymptotic behaviour of the latter. This is given by the following result which can be found in [2]. In order to describe the structure of the spectrum of T in different cases, the following definition will be useful.…”

“…Proof. If Xe C\IR, then from (4.4) one has that (aG + sA) 2 -4(G -B) < 0. Since A and B are bounded, this can only happen if G is also bounded.…”

“…In particular, the analysis can be extended to the case where the functions a and b differ only by a constant. From the results in [2], it is known that the only case where the eigencurves of the operator L p may contain line segments is when the function c(x) is essentially constant. In any other case, one has c~ <c + and it follows that there must always exist intersections between the eigencurves (for large enough n in equation (2.4)) and either the right branch of the function g(p, a, e) or the left part of the branch which is asymptotic to y = p/( ae ) -I/a 2 .…”

“…The results presented throughout the paper can easily be adapted to more general operators than the Laplacian, and in particular to elliptic operators of order 2r, with the appropriate boundary conditions (see [2] for the conditions needed so that the asymptotic results in Lemma 2.4 apply).…”

On the second eigencurve for the p-laplacian opertor with weight

“…The purpose of this article is to study the following problem: For α ∈ R Find the existence of real numbers β(α) such that (α, β(α)) ∈ C 2 and the asymptotic behavior of β(α) as |α| → +∞, Many results have been obtained on this kind of problems (see; [4], [5], [8], in [5] the authors proved some properties related to the first eigencurve C 1 such as concavity, differentiability and the asymptotic behavior, this last property can not be adapted to the other eigencurves, in [8] the authors have studied this class of problems under the following assumptions…”