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Journal of Convex Analysis, Vol. 4, No. 2, pp. 257-280 (1997)
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Existence of Regular Solutions for a One-Dimensional Simplified Perfect-Plastic Problem with a Unilateral Gradient Constraint

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Thierry Astruc

Laboratoire d'Analyse Non-Linéaire Appliquée, UFR Sciences et Techniques, Université de Toulon et du Var, BP 132, 89130 La Garde cedex, France, astruc@univ-tlv.fr

**Abstract:** This work is devoted to the study of the existence of "regular" solutions for a one-dimensional problem with unilateral constrained gradient in Perfect-Plasticity.

The particularity of this problem consists in the fact, that the tools usually employed to prove the Inf-Sup equality between the displacement problem and the stress problem do not work.

In a first part, we establish this equality by the mean of a penalty method which employs the theory of the convex functions of measures.

In a second part, we find the regular limit loads, between which the displacement problem possesses at least a solution which is in $W^{1,1}$, verifiyng the boundary conditions and the constraint on the gradient. We give an example where these loads are infinite.

**Full text of the article:**

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