On the regularity of solutions in the Pucci-Serrin identity
type: Item
instance of: scholarly article
title: On the regularity of solutions in the Pucci-Serrin identity
zbMATH Open document ID: 1046.35039
author: Marco Degiovanni
author: A. Musesti
author: Marco Squassina
published in: Calculus of Variations and Partial Differential Equations
publication date: 2004-07-07T00:00:00Z
review text: 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 [\textit{P. Pucci} and \textit{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., [\textit{P. Pucci} and \textit{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 [\textit{P. Pucci} and \textit{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.
reviewed by: Patrizia Pucci
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 2078243
zbMATH Keywords: Pucci-Serrin identity
zbMATH Keywords: Euler equations
zbMATH Keywords: nonlinear elliptic differential equations
MaRDI profile type: Publication
full work available at URL: s00526-003-0208-y
OpenAlex ID: W2010387725
DOI: 10.1007/S00526-003-0208-Y
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description: scientific article; zbMATH DE number 2078243