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Approximation in \(L_p\) by solutions of quasi-elliptic equations. (Russian) Zbl 1052.35058
The following theorem is proven: Let \(P(x,D)\) be a quasi-elliptic differential operator of order \(\vec{l}\) with \(C^{\infty}\) coefficients defined on an open subset \(\Omega\subset \mathbb R^n\). If \(P(x,D)\) has a fundamental solution and \(K\subset\Omega\) is compact, then the following statements are equivalent: (i) the space \(\eta(K)\) of distributions \(u\) such that \(P(x,D) u=0\) on a neighborhood of \(K\) is dense in \(\eta^p(K)=L^p(K)\cap \eta(\overset\circ K)\); (ii) \(C_0^{\infty}(K) \) is dense in \((L^{\vec{l}}_p)_K\); (iii) \(C_0^{\infty}(\mathbb R^n\setminus K) \) is dense in \((L^{-\vec{l}}_p\,)_{\mathbb R^n\setminus K}\); (iv) \((u,f)=0\) for all \(u\in (L^{-\vec{l}}_p\,)_{\mathbb R^n\setminus K}\) and \(f\in (L^{\vec{l}}_p)_K\).
35H30 Quasielliptic equations
35A35 Theoretical approximation in context of PDEs
46E15 Banach spaces of continuous, differentiable or analytic functions
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