The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton-Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in L-2, but also measures.

A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators / Giuffre', S.. - In: AXIOMS. - ISSN 2075-1680. - 12:6(2023), pp. 1-13. [10.3390/axioms12060605]

### A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

#### Abstract

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton-Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in L-2, but also measures.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
variational inequalities
obstacle problem
nonlinear monotone operators
Lagrange multipliers
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.12318/141806`