Non-smooth variational problems and applications

Abstract

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’.

On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition

“…Under the assumption of small deformations, the only substantial injectivity requirement that is usually taken into account is the non-penetration of the opposite crack faces, which is ensured by the local Signorini-type boundary conditions (e.g. [1][2][3][4]). Applying such conditions, papers [5][6][7][8][9] study volumetric rigid inclusions with cracks on their boundaries.…”

Non-smooth variational problems and applications in mechanics

“…After successful publishing in 2022 in Philosophical Transactions of the Royal Society A of the theme issue 2236 on Non-smooth variational problems and applications [1], we were facing the continuation since many relevant topics remained not included there. During the International Congress on Industrial and Applied Mathematics 1 held from 20 to 25 August 2023 at Waseda University in Tokyo, Japan, we organized a mini-symposium on Singular problems in mechanics , 2 where recent achievements in the field were reported.…”