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Solution of the Dirichlet problem for the equation of curvature of order m. (English. Russian original) Zbl 0699.35096
Sov. Math., Dokl. 37, No. 2, 322-325 (1988); translation from Dokl. Akad. Nauk SSSR 299, No. 1, 35-38 (1988).
The paper announces the existence of classical solutions of the Dirichlet problem for the equation of curvature of order m in a strictly convex domain in \(R^ n\). In geometrical terms this means that there exists at least one surface with given boundary and curvature of order m. Under some additional conditions a uniqueness result is also true. The proofs (not contained in the paper) are based on the theory of quasilinear elliptic equations. The main difficulty is to obtain suitable a priori estimates.
Reviewer: J.Madjarova

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
53A05 Surfaces in Euclidean and related spaces
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