The penalized obstacle problem. II: \(C^{1,\alpha}\) regularity of level sets
[Common review for Part I (cited above) and Part II.] Consider the problem of minimizing the functional \(E\) defined by \[ E(v)= \int_{\Omega}(| \nabla v|^2+ 2v) dx \] over all nonnegative functions \(v\in H^1\) with given nonnegative boundary values \(g\). It was proved by \textit{H. Lewy} and \textit{G. Stampacchia} [Commun. Pure Appl. Math. 22, 153-188 (1969; Zbl 0167.11501)] that all minimizers have bounded weak second derivatives by introducing the penalized problem \(\Delta u^{\varepsilon}= \beta_{\varepsilon}(u^{\varepsilon})\) in \(\Omega\), \(u_{\varepsilon}= g\) on \(\partial\Omega\); the major conditions on the family \((\beta_{\varepsilon})\) are that \(\beta_{\varepsilon}(t)\to -\infty\) for \(t<0\) and \(\beta_{\varepsilon}(t)\to 1\) for \(t>0\). In these two papers, the author shows that the level sets of the minimizers are \(C_{1,\alpha}\) by considering the penalized problem of Lewy and Stampacchia.
scientific article; zbMATH DE number 179229
scientific article