FIXED POINTS IN THE CONSTRUCTION OF A MINIMAX SOLUTION FOR A CLASS OF BOUNDARY VALUE PROBLEMS FOR HAMILTON–JACOBI EQUATIONS

Pavel D. Lebedev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Alexander A. Uspenskii     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)

Abstract


This paper deals with analytical and numerical methods for constructing a minimax (generalized) solution to the Dirichlet problem for the Hamilton–Jacobi equation. The case of a closed planar nonconvex boundary set is considered, where the boundary points have a smoothness defect in the coordinate functions with respect to third-order derivatives. These points belong to the pseudo-vertices of the boundary set. Pseudo-vertices generate branches of a singular set, which are one-dimensional manifolds where the smoothness of the minimax solution breaks down. To construct a branch of a singular set, it is necessary to find markers, i.e., numerical characteristics of the corresponding pseudo-vertex. The markers (left and right ones) establish a link between the characteristics of the Hamilton–Jacobi equation and the geometry of the boundary set. For the markers, a relation with the structure of the equation at a fixed point is obtained. An iterative procedure for calculating a solution based on Newton's method is proposed. The convergence of the procedure to the pseudo-vertex marker is proved. An example of constructing a minimax solution is given, demonstrating the effectiveness of the developed approaches for solving nonsmooth boundary value problems.

Keywords


Hamilton–Jacobi equation, Minimax solution, Speed-in-action, Singular set, Wavefront, Diffeomorphism, Eikonal, Pseudo-vertex

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References


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DOI: http://dx.doi.org/10.15826/umj.2025.2.011

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