Dynamics of Partial Differential Equations

Volume 11 (2014)

Number 2

On the solvability in the sense of sequences for some non-Fredholm operators

Pages: 109 – 124

DOI: http://dx.doi.org/10.4310/DPDE.2014.v11.n2.a1

Authors

Vitali Vougalter (Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa)

Vitaly Volpert (Institut Camille Jordan, Université Claude Bernard Lyon 1, Villeurbanne, France)

Abstract

We investigate solvability of certain linear nonhomogeneous elliptic equations and establish that under reasonable technical conditions the convergence in $L^2(\mathbb{R}^d)$ of their right sides yields the existence and the convergence in $H^2(\mathbb{R}^d)$ of the solutions. The problems involve the sums of second order differential operators without Fredholm property and we apply the methods of spectral and scattering theory for Schrödinger type operators analogously to our previous work.

Keywords

solvability conditions, non Fredholm operators, Sobolev spaces

2010 Mathematics Subject Classification

35J10, 35P10, 47F05

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