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Non-smooth variational problems and applications.

Victor A Kovtunenko1, Hiromichi Itou2, Alexander M Khludnev3

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Summary
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This study explores advanced mathematical methods for complex nonlinear variational problems. It develops novel techniques for non-smooth and ill-posed equations in mechanics and physics.

Keywords:
continuum mechanicsnon-smooth variational methods

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Area of Science:

  • Applied Mathematics
  • Mathematical Physics
  • Engineering Sciences

Background:

  • Variational methods are crucial for solving partial differential equations across various scientific fields.
  • Classical mathematical tools often fail for complex nonlinear, non-smooth, and ill-posed problems.
  • Singular and unilaterally constrained problems in mechanics and physics present significant challenges.

Purpose of the Study:

  • To address a wide class of nonlinear variational problems, including static/evolution equations, inverse problems, and optimization.
  • To focus on singular and unilaterally constrained problems governed by complex variational equations and inequalities.
  • To advance the mathematical theory of non-smooth variational problems and their applications in engineering.

Main Methods:

  • Development of non-standard well-posedness analysis for challenging mathematical problems.
  • Application of numerical methods, asymptotic techniques, and approximation methods, including homogenization.
  • Utilizing both primal and dual variational formalisms for theoretical and computational approaches.

Main Results:

  • Successful application of variational methods to a broad range of nonlinear and non-smooth problems.
  • Demonstration of effective numerical and approximation techniques for complex systems.
  • Advancement in understanding the physical consistency and simulation of these problems.

Conclusions:

  • The study provides significant advances in the mathematical theory and application of non-smooth variational problems.
  • The developed methods offer robust solutions for problems previously intractable with classical approaches.
  • This work bridges mathematical theory with practical applications in engineering and physical sciences.