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On the bounded slope condition and the validity of the Euler Lagrange equation. (English) Zbl 1013.49015
If $$\Omega\subset \mathbb{R}^n$$ is a bounded convex set and if $$u^0$$ denotes a boundary datum such that the bounded slope condition $$(\text{BSC})_K$$ holds with positive constant, then G. Stampacchia [Commun. Pure Appl. Math. 16, 383-421 (1963; Zbl 0138.36903)] proved that any minimizer $$u$$ of $\int_\Omega f(\nabla u) dx\to \min\quad\text{on }u- u^0\in W^{1,1}(\Omega),\tag{$${\mathcal P}$$}$ among all Lipschitz functions satisfy $$|\nabla u(x)|\leq K$$ provided $$u$$ is of class $$C^1(\Omega)\cap H^2(\Omega)$$ and $$f$$ is a regular integrand, i.e., of class $$C^2$$ such that $$D^2f$$ is everywhere positive. In the present paper the hypotheses concerning the minimizer are weakened as follows: suppose that $$u$$ is a $$W^{1,1}(\Omega)$$ solution of $$({\mathcal P})$$ being continuous in $$\Omega$$. Then $$u$$ is Lipschitz and $$|\nabla u(x)|\leq K$$ a.e. The main tool used in the proof is a generalized version of the weak maximum principle.

##### MSC:
 49K20 Optimality conditions for problems involving partial differential equations 49K10 Optimality conditions for free problems in two or more independent variables 49J45 Methods involving semicontinuity and convergence; relaxation
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