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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.

MSC:
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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