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On the uniqueness of \(L^ 2\)-solutions in half-space of certain differential equations. (English) Zbl 0537.35016

The present paper deals with square integrable solutions to the equation \[ \{-\partial^ 2/\partial y^ 2+P(D_ x)+b(y)-\lambda \}u(x,y)=f(x,y) \] in the half-space \(y>0\), \(x\in {\mathbb{R}}^ n\), where \(P(D_ x)\) is a constant coefficient operator. The author proved under certain conditions on \(\lim_{y\to 0} u(x,y),\) b(y), f(x,y) and \(\lambda\) that \(\sup p u=\sup p f\). The main purpose of the paper is to generalize W. Littman’s result [Appl. Math. Optimization 8, 189- 196(1982; Zbl 0484.35062)], primarily by relaxing the assumptions on b(y). In fact the functions b(y) need not vanish as \(y\to \infty\), and indeed they need not go to any limit at all. The essential condition is that \(\int^{\infty}_{0}dy/[1+| b(y)|^{\frac{1}{2}}]=\infty.\)
Reviewer’s remark: It is of worth someone to find weaker conditions especially on b(y) with the hope that these will be better for further use.
Reviewer: T.Rassias

MSC:

35E20 General theory of PDEs and systems of PDEs with constant coefficients
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35G05 Linear higher-order PDEs

Citations:

Zbl 0484.35062
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References:

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