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We consider the Allen--Cahn equation with a constraint. Our constraint is provided by the subdifferential of the indicator function on a closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier for our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem.

Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier