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On the regularity of solutions in the Pucci-Serrin identity. (English) Zbl 1046.35039
In this well-written paper the authors consider only first order variational problems and in this setting extend the Pucci-Serrin identity (Proposition 1 of [P. Pucci and J. Serrin, Indiana Univ. Math. J. 35, 681–703 (1986; Zbl 0625.35027)]) by removing the \(C^2(\Omega)\) regularity of solutions \(u\) of \((\mathcal P)\) and by replacing the \(C^1\) assumption on \(\nabla_\xi\mathcal L\) with the strict convexity of \(\mathcal L(x,s,\cdot)\). For applications of the Pucci-Serrin variational identity to higher order problems see Section 5 of the original paper and also, e.g., [P. Pucci and J. Serrin, J. Math. Pures Appl. (9), 69, 55–83 (1990; Zbl 0717.35032)].
In particular, in this paper the authors consider the second order Euler equation \[ \begin{cases} -\text{div}(\nabla_\xi\mathcal L(x,u,\nabla u))+D_s\mathcal L(x,u,\nabla u)=f\quad\text{in }\Omega,\\u=0\quad\text{on }\Omega,\end{cases}\tag{\(\mathcal P\)} \] where \(\Omega\) is a bounded domain of \(\mathbb R^n\), with \(C^1\) boundary, the functional \(\mathcal L=\mathcal L(x,s,\xi):\overline\Omega\times \mathbb R\times \mathbb R^n\to \mathbb R\) is of class \(C^1\) and strict convex in \(\xi\) for all \((x,s)\in\overline\Omega\times \mathbb R\), while \(f\in C(\overline\Omega)\). The fact that the \(C^1(\overline\Omega)\) regularity of weak solutions \(u\) of \((\mathcal P)\) was enough in the radial case was already known and used by Pucci and Serrin, see for instance [P. Pucci and J. Serrin, Asymptotic Anal. 4, 97–160 (1991; Zbl 0733.34042)]. The paper contains a number of useful discussions and also particular situations of independent interest in which the strict convexity of \(\mathcal L(x,s,\cdot)\), so natural in applications, can be relaxed to solely convexity.

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
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