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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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