Solvability conditions for some non-Fredholm operators

Abstract

Abstract. We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger type operators. One of the main results of the work concerns solvability conditions for the equation −∆u + V (x)u −au = f where a ≥ 0. They are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

Solvability Conditions for a Linearized Cahn-Hilliard Equation of Sixth Order

“…As a consequence, the operator does not satisfy the Fredholm property, and solvability conditions of equation (1.4) are not known. We will derive solvability conditions for this equation using the technique developed in our preceding articles [18], [19], [20], [21], [22] and [23]. This method is based on the spectral decomposition of self-adjoint operators.…”

“…[9]), such that neither of these operators has a finite dimensional kernel and a closed image. Solvability conditions for operators of that kind have been studied extensively in recent articles for a single Schrödinger type operator (see [18]), the sums of second order differential operators (see [19]), the Laplacian operator with the drift term (see [20]). Non Fredholm operators arise as well while studying the existence and stability of stationary and travelling wave solutions of certain reaction-diffusion equations (see e.g.…”

“…Note that the first one contains conditions on V (x) analogous to those stated in Assumption 1.1 of [18] (see also [19], [20]). …”

“…By means of Lemma 2.3 of [18], under our Assumption 1 above on the potential function, the operator −∆ − V (x) − a is self-adjoint and unitarily equivalent to −∆ − a on L 2 (R 3 ) via the wave operators (see [10], [14]) Ω ± := s − lim t→∓∞ e it(−∆−V ) e it∆ with the limit understood in the strong L 2 sense (see e.g. [13] p.34, [4] p.90).…”

“…We obtain solvability conditions in H 6 (R 3 ) for a sixth order partial differential equation which is the linearized Cahn-Hilliard problem using the results derived for a Schrödinger type operator without Fredholm property in our preceding article [18]. …”