Solvability and optimization for a class of mixed variational problems

ABSTRACT

We consider an abstract mixed variational problem governed by a nonlinear operator A and a bifunctional J, in a real reflexive Banach space X. The operator A is assumed to be continuous, Lipschitz continuous on each bounded subset of X, and generalized monotone. First, we pay attention to the unique solvability of the problem. Next, we prove a continuous dependence result of the solution with respect to the data. Based on this result, we prove the existence of at least one solution for an associated optimization problem. Finally, we apply our abstract results to the well-posedness and the optimization of an antiplane frictional contact model for nonlinearly elastic materials of Hencky type.

Keywords:

• Mixed variational problem
• Lagrange multiplier
• optimization problem
• Hencky material
• antiplane frictional contact problem
• weak solution

AMS subject classifications:

• 35J65
• 49J20
• 65K10
• 49K27
• 49J40
• 74M10
• 74M15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.