Identification of the singularity of the generalized solution of the Dirichlet problem for an eikonal type equation under the conditions of minimal smoothness of a boundary set

A. A. Uspenskii   ; P. D. Lebedev

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 28 (2018) no. 1, pp. 59-73

Résumé

The subject of the study is pseudo-vertices of a boundary set, which are necessary for the analytical and numerical construction of singular branches of the generalized (minimax) solution of the Dirichlet problem for an eikonal type equation. The case of variable smoothness of the boundary set boundary is considered, under which the order of smoothness at the points of consideration is reduced to the lowest possible value - up to one. Necessary conditions for the existence of pseudo-vertices are obtained, expressed in terms of one-sided partial limits of differential relations, depending on the properties of local diffeomorphisms that determine these points. An example is given that illustrates the application of the results obtained while solving the velocity problem.

Détail

Keywords: first-order partial differential equation, minimax solution, velocity, wave front, diffeomorphism, optimal result function, singular set, symmetry
Mots-clés : eikonal, pseudo-vertex.

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