Variational problems for a class of functionals on convex domains
The author studies minimizers of the integral functional \(J(u)=\int_\Omega [f(|Du|)-u] dx\), where \(\Omega\) is a bounded convex domain in \({\mathbb R}^N\), \(N\geq 2\), \(f~:~[0,\infty[ \mapsto {\mathbb R}\cup {\infty}\), \(f\) is lower semicontinuous but not necessarily convex, \(f(\xi)\geq M\xi -b\) for \(\xi \geq 0\), \(M>0\), \(b\in {\mathbb R}\). Let \(W_\Omega\) be the width of the convex set \(\Omega\) and \({\mathcal K}\) be the subspace in \(W^{1,1}_0(\Omega)\) of functions depending only on the distance from the boundary of \(\Omega\), that is functions of the form \(v(\cdot) = \eta(d(\cdot,\partial \Omega))\), where \(\eta\) is a function from \([0,W_\Omega]\) into \({\mathbb R}\). It is proved that \(J\) admits a unique minimum in \( {\mathcal K}\) provided that the ratio between the Lebesgue measure of \(\Omega\) and the \((N-1)\)-dimensional Hausdorff measure of \(\partial \Omega\) is strictly less than \(M\). This result is a direct consequence of the strict monotonicity of the function \(t \mapsto {\mathcal L}^N(x\in \Omega \mid d(x,\partial \Omega)>t)/{\mathcal H}^{N-1}(x\in \Omega \mid d(x,\partial \Omega)=t)\). The existence and uniqueness result proved in this paper is a generalization of a previous result due to \textit{F. Gazzola} [Arch. Ration. Mech. Anal. 150, 57-76 (1999; Zbl 0949.49010)].
scientific article; zbMATH DE number 1763743