A note on ``On a critical point theory for multivalued functionals and application to partial differential inclusions''
The article by \textit{M. Frigon} [ibid. 31, 735-753 (1998; Zbl 0901.47033)] alluded to in the title, contained an approach via critical points to Dirichlet problems for nonlinear elliptic partial differential equations with discontinuous right-hand side. Such equations can be viewed as partial differential inclusions. Frigon (loc. cit.) used a definition of critical point based on the notion of ``weak slope'' [cf., \textit{M. Degiovanni} and \textit{M. Marzocchi}, Ann. Mat. Pura Appl., IV. Ser. 167, 73-100 (1994; Zbl 0828.58006)]. The present authors propose a simpler approach: they consider a partial differential inclusion \[ -\Delta u\in F(x,u(x)) \quad \text{in }\Omega, \quad u\mid \partial\Omega=0 \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^{n}\) with piecewise smooth boundary and \(F(x,\cdot):\mathbb{R}\to\mathbb{R}\) is an upper semicontinuous multivalued mapping with compact convex values, and a solution is an element \(u\in H_0^1(\Omega)\) such that \(\Delta u\in L^{q}(\Omega)\) for some \(q\) and such that \(-\Delta u(x)\in F(x,u(x))\) for almost all \(x\in\Omega\). The authors assume that \(\min F\) and \(\max F:\Omega\times\mathbb{R}\to\mathbb{R}\) are \(\mathcal{L}\times\mathcal{B}\)-measurable, where \(\mathcal{L}\) is the Lebesgue \(\sigma\)-algebra in \(\Omega\) and \(\mathcal{B}\) are the Borel sets in \(\mathbb{R}\). Moreover, they assume that there is a \(\sigma\in[0,\frac{n+2}{n-2}]\) and constants \(a,b\) such that \(|F(x,t)|:=\max\{|z|\mid z\in F(x,t)\}\leq a+b|t|^\sigma\). The authors define a multivalued functional \(J:L^{\sigma+1}(\Omega)\to\mathbb{R}\) by assigning to \(u\) the set \(\int_\Omega(\int_0^{u(x)}f(x,s) ds) dx\) where \(f\) ranges over all measurable selections of \(F\). The authors then prove that the critical points of \(I(u):=\frac 12\int_\Omega\|\text{grad }u\|^2 -J(u)\) are solutions to \(-\Delta u\in F(x,u(x))\), \(u\mid \partial\Omega=0\).
scientific article; zbMATH DE number 1545347
scientific article