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We consider the Dirichlet problem for 2N-order properly elliptic equation with constant coefficients in the unit disc. It is assumed that the characteristic equation has two different roots, both of multiplicity N. We seek the solution of the problem in the class of functions that are 2N-times continuously differentiable in the disc, and continuous up to the boundary along with derivatives of order no more than \(N-1.\) The necessary and sufficient conditions for solvability of this problem are obtained, and if these conditions hold, the solution is determined in an explicit form.

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On a Dirichlet Problem for a Properly Elliptic Equation in the Space of Continuous Functions, in the Case of Multiple Roots

  • 会议论文

  • 发表日期：

    2025年

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  • Armenak Babayan

摘要

We consider the Dirichlet problem for 2N-order properly elliptic equation with constant coefficients in the unit disc. It is assumed that the characteristic equation has two different roots, both of multiplicity N. We seek the solution of the problem in the class of functions that are 2N-times continuously differentiable in the disc, and continuous up to the boundary along with derivatives of order no more than \(N-1.\) The necessary and sufficient conditions for solvability of this problem are obtained, and if these conditions hold, the solution is determined in an explicit form.