TY - JOUR

T1 - Non-smooth variational problems and applications

AU - Kovtunenko, Victor A.

AU - Itou, Hiromichi

AU - Khludnev, Alexander M.

AU - Rudoy, Evgeny M.

N1 - Funding.The authors gratefully acknowledge the support of the Japan Society for the Promotion of Science(JSPS) and the Russian Foundation for Basic Research (RFBR) under the Japan–Russia Research CooperativeProgram (project no. JPJSBP120194824).

PY - 2022/11/14

Y1 - 2022/11/14

N2 - Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences. This article is part of the theme issue 'Non-smooth variational problems and applications'.

AB - Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences. This article is part of the theme issue 'Non-smooth variational problems and applications'.

KW - continuum mechanics

KW - non-smooth variational methods

KW - non-smooth variational methods

UR - http://www.scopus.com/inward/record.url?scp=85138524003&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/b168e9ed-538e-3476-99c4-86d326eb9201/

U2 - 10.1098/rsta.2021.0364

DO - 10.1098/rsta.2021.0364

M3 - Review article

C2 - 36154476

AN - SCOPUS:85138524003

VL - 380

JO - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0962-8428

IS - 2236

M1 - 20210364

ER -