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On multiquasielliptic equations in \(\mathbb R_n\). (Russian, English) Zbl 1035.35028

Sib. Mat. Zh. 44, No. 5, 1183-1188 (2003); translation in Sib. Math. J. 44, No. 5, 926-930 (2003).
The author investigates the following partial differential equation in \(\mathbb R_n\): \[ P(D_x)u = f(x), \] where \[ P(D_x) = \sum_{\gamma\in M_P}a_{\gamma}D_x^{\gamma} \] is the multiquasielliptic operator with constant coefficients, \(M_P\) denotes the finite set of multi-indexes \(\gamma = (\gamma_1,\dots,\gamma_n)\), \(D_x^{\gamma} = D_1^{\gamma_1}\cdots D_n^{\gamma_n}\), \(D_j = -i\partial/\partial x_j\). Under some reasonable assumptions on the characteristic polynomial associated with the differential operator \(P(D_x)\), the author proves the existence of a generalized solution to the equation under consideration in Sobolev-type spaces. To prove the existence result, the author use the technique of integral operators and a priori estimates.

MSC:

35H30 Quasielliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B45 A priori estimates in context of PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
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