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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.

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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