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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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