The
project is located in the mainstream of Partial Differential Equations and Applied Mathematics, and concentrates on mathematically challenging and practically important classes of nonlinear
differential variational-hemivariational inequalities, evolution inclusions and various constrained multivalued problems in infinite dimensional Banach spaces.
The
project addresses detailed theoretical questions on well-posedness, classification of inequalities, the properties of the solution set, generic regularity and smoothness of solutions, sensitivity
analysis, associated optimal control and inverse problems, etc.
More
appropriate formulations are based on two relatively new concepts: the differential variational inequality introduced in 2008 infinite dimensions
and the hemivariational inequality initiated in the 1980's. The concept of hemivariational inequality has been originated from variational descriptions of physical processes, and has a precise physical
meaning expressing the principle of virtual work or power introduced by Fourier in 1823, and is based
on the notion of generalized subgradient of Rockafellar-Clarke for locally lipschitz functions.
The
goal of the project is to achieve a progress in the development of a general theory
of differential variational-hemivariational inequalities. We study a plethora of impressive applications of new
mathematical models with nonsmooth potentials described by differential variational-hemivariational
inequalities to contact mechanics of solids (in elasticity, viscoelasticity, viscoplasticity, piezoelectricity,
thermoviscoelasticity, electro-elasticity) and of fluids (non-Newtonian fluids, rheological models),
and other areas (electrical circuits, economical dynamics, complementarity problems, trafic network
problems, biomechanics, medicine, biology, geophysics, etc.).