Second-order elliptic equation of divergence form having a compactly-supported solution (Q2773666)

Let \(\varphi(t)\) (\(t\in (0,\varepsilon]\)) be a positive nondecreasing function such that the function \(s\varphi(s)^{-1}\) is also nondecreasing and \(\int_0^\varepsilon \varphi(s)^{-1} ds <\infty\). The main result of the article can be stated as follows:NEWLINENEWLINENEWLINETheorem. Let a function \(\varphi\) meet the above conditions and let \(d\geq 3\). Then there exist a function \(u\in C_0^{\infty}({\mathbb R}^d)\), a number \(\lambda\), and a real positive definite matrix-valued function \(g\) with the property \(|g(x)-g(y)|\leq c\varphi(|x-y|)\) \((x,y\in {\mathbb R}^d,\;|x-y|< \varepsilon)\) such that \(-\text{div} (g\nabla u)=\lambda u.\)NEWLINENEWLINENEWLINEThis theorem implies that the unique continuation property (this property means that every compactly-supported solution of a homogeneous elliptic equation is zero) fails for the matrix-valued functions \(g(x)\) satisfying only the Hölder condition. This holds whenever \(g(x)\) is a Lipschitz matrix-valued function. The author also presents some applications of the above theorem to constructing special solutions to the second-order hyperbolic equations NEWLINE\[NEWLINE \partial^2_{t}v - \text{div} (g\nabla v)=0 NEWLINE\]NEWLINE and special eigenfunctions of the Schrödinger operator with periodic coefficients.