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Estimate for the solution to the Cauchy problem for an ultrahyperbolic inequality. (English. Russian original) Zbl 1146.35099
Dokl. Math. 74, No. 2, 751-754 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 410, No. 6, 737-740 (2006).
The author considers the differential inequality \[ |Lu|^2\leq C(F^2+ u^2+ |\nabla_x u|^2), \] where \(x\in\Omega_1\times \Omega_2\equiv\Omega\) with the unknown function \(u(x)\). Here \(x= (x_1,\dots, x_n)\), and \(Lu= \sum^n_{i,j=1} (a_{ij}(x) u_{x_i})_{x_j}\), \(F= F(x)\) and the coefficients of the operator \(L\) satisfy the certain conditions. The author proves Carleman-type inequalities for \(L\) and additionally establishes a Hölder estimate for the corresponding Cauchy problem for \(L\).

MSC:
35R45 Partial differential inequalities and systems of partial differential inequalities
35B45 A priori estimates in context of PDEs
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