A uniform gradient bound of minimizers of regularized functionals (Q2706067)

The author of this paper proposes a uniform estimate to the gradient of the minimizer \(u^{(\varepsilon)}\) for an interesting regularized variational problem defined by the functional NEWLINE\[NEWLINEJ_{\varepsilon }(v)=\int_{\Omega }(a^{ij}(v)D_ivD_jv+\chi_{\varepsilon }(v)) d{\mathcal L}^n\quad (i,j=1,2,\dots,n).NEWLINE\]NEWLINE Here, \(\Omega \subset {\mathbf R}^n\) \((n\geq 2)\) is a bounded Lipschitz domain, \(\chi_{\varepsilon }:{\mathbf R} \to{\mathbf R} \) is a smooth function defined for each positive constant \(\varepsilon \) such that \(0\leq \chi_{\varepsilon }\leq 1\) in \(\mathbf R\), \(\chi_{\varepsilon }=0\) in \((-\infty ,0]\), \(\chi_{\varepsilon }=1\) in \([\varepsilon ,+\infty)\), \(\chi_{\varepsilon }'\geq 0\) in \(\mathbf R\) and \(|\chi_{\varepsilon }^\prime|\leq 2/\varepsilon \) in \(\mathbf R\). The coefficient functions \(a^{ij}=a^{ji}\) are differentiable in \(\mathbf R\) and satisfy the uniform elliptic condition. The main problem here is the existence of a minimizer for \(J_{\varepsilon }(v)\) in the function class \(\{v\in W^{1,2}(\Omega)\mid v=\Phi_0\) on \(\partial \Omega \}\) provided a non-negative and bounded \(W^{1,2}(\Omega)\)-function \(\Phi_0\) exists. The supremum of \(|Du^{(\varepsilon)}|\) is estimated locally in \(\Omega \), which is uniform in \(\varepsilon \), that is, \(\sup_{\varepsilon <\varepsilon_0}|Du^{(\varepsilon)}|_{\infty ,U}\leq C\), where \(\varepsilon_0 \) and \(C\) are proper constants.