## 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 $$f\in (L^{\vec{l}}_p)_K$$.

### MSC:

 35H30 Quasielliptic equations 35A35 Theoretical approximation in context of PDEs 46E15 Banach spaces of continuous, differentiable or analytic functions

### Keywords:

homogeneous metric
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