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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
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.)