Well-Posed Problems for the Laplace–Beltrami Operator
Dosmagulova K. Kanguzhin B.
September 2025Multidisciplinary Digital Publishing Institute (MDPI)
Symmetry
2025#17Issue 9
Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from (Formula presented.) embedded in (Formula presented.). The presence of the cut introduces singular perturbations of the domain, and we develop an analytical framework to characterize well-posed problems in this setting. Our approach combines Green’s functions, spectral analysis, and Sobolev space methods to establish solvability criteria and uniqueness results. In particular, we identify explicit conditions for the existence of solutions with data supported near the cut, and extend the formulation to include delta-type perturbations supported on the removed circle. These results generalize earlier work on punctured two-dimensional spheres and provide a foundation for the study of PDEs on manifolds with localized singularities.
Green’s functions for elliptic equations , Laplace–Beltrami operator , three-dimensional punctured sphere , well-posed problems
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Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Department of Mathematics, Faculty of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Department of Mathematics and Computer Modeling, International IT University, Almaty, 050040, Kazakhstan
Institute of Mathematics and Mathematical Modeling
Department of Mathematics
Department of Mathematics and Computer Modeling
10 лет помогаем публиковать статьи Международный издатель
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