Lagrange multipliers and nonlinear variational inequalities with gradient constraints
Sofia Giuffrè1, Attilio Marcianò1
1D.I.I.E.S., Mediterranea University of Reggio Calabria, Località Feo di Vito, 89122 Reggio Calabria, Italy.
Summary
This study establishes an equivalence between gradient-constrained nonlinear monotone variational inequalities and a double obstacle problem. It proves the existence of Lagrange multipliers, advancing non-smooth variational problems.
Area of Science:
- Optimization Theory
- Mathematical Analysis
- Applied Mathematics
Background:
- Variational inequalities are fundamental in modeling various mathematical and physical phenomena.
- Gradient constraints introduce complexity, requiring advanced analytical techniques.
- Non-smooth problems present unique challenges in theoretical and computational approaches.
Purpose of the Study:
- To establish a novel equivalence between nonlinear monotone variational inequalities with gradient constraints and a double obstacle problem.
- To demonstrate the existence of Lagrange multipliers for the studied class of inequalities.
- To contribute to the understanding of non-smooth variational problems and their applications.
Main Methods:
- Development and application of a new strong duality principle.
- Reformulation of the variational inequality problem into a double obstacle problem.
- Analysis of existence conditions for Lagrange multipliers.
Main Results:
- A proven equivalence between the gradient-constrained variational inequality and a double obstacle problem.
- Demonstration of the existence of Lagrange multipliers, denoted as [Formula: see text].
- The findings are part of a broader theme on non-smooth variational problems.
Conclusions:
- The established equivalence offers new perspectives and potential solution methods for gradient-constrained variational inequalities.
- The existence of Lagrange multipliers is a significant theoretical advancement.
- This research enhances the toolkit for addressing complex non-smooth variational problems.
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