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On membership of solutions of differential equations in Nikol’skij spaces. (English. Russian original) Zbl 0624.35026
Sov. Math., Dokl. 34, 210-214 (1987); translation from Dokl. Akad. Nauk SSSR 289, 1289-1293 (1986).
Let $$a_{ij}$$, f denote trigonometric polynomials, $$a_{ij}=a_{ji}$$, and consider the equation $(1)\quad Au:=- \sum^{m}_{i,j=1}\partial_ i(a_{ij}\partial_ ju)+a_{00}u=f$ on the torus $$T^ m=[0,2\pi]^ m$$ assuming (for some number $$b=\rho^ 2>0)$$ (2) (Av,v)$$\leq b\| v\|^ 2$$ on $$L^ 2(T^ m)$$. Let $$R_ 0$$ denote the radius of convergence on $$T^ m$$ of the expansion in a series of powers of t of the solution v(t,x) of the Cauchy problem $\partial^ 2_ tv=Av,v|_{t=0}=f,\quad \partial_ tv|_{t=0}=0$ with A and f as in (1), (2).
Then the author’s main result is: If $\| \cosh (t\sqrt{A})f\|^ 2\leq C\cdot M_ p\cdot (R_ 0-t)\quad (as\quad t\to R_ 0)$ for some $$p<1$$, where $$M_ p(\tau):=\exp (e^{\tau^{-p}})$$, then a generalized solution $$u\in L^ 2(T_ m)$$ belongs to the space $$H_ 2^{2R_ 0\rho /\pi}(T_ m)$$.
Reviewer: M.Fuchs
##### MSC:
 35J25 Boundary value problems for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35B10 Periodic solutions to PDEs 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)