The signed mean curvature measure (Q2786813)

The authors study the Dirichlet problem NEWLINE\[NEWLINE\begin{cases} H_1[u]:=div\displaystyle{\Biggl(\frac{Du}{\sqrt{1+|Du|^2}}}\Biggr)=\nu(x) \;\;&\text{in} \;\;\Omega,\\ u=\varphi\;\;&\text{ on} \;\;\partial \Omega,\end{cases}\tag{P}NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with \(C^3\)-boundary, \(\nu\) is a signed Radon measure, and \(\varphi\in C(\partial \Omega)\).NEWLINENEWLINE\( \)NEWLINENEWLINELet \(\phi(\Omega)\) be the set of all \(L^\infty(\Omega)\)-functions which are point-wise limits of sequences \(\{u_j\}\subset W^{1,1}(\Omega)\) such that \(H_1[u_j]\in L^1(\Omega)\) and NEWLINE\[NEWLINE\sup_{j\in \mathbb{N}}\int_\Omega |H_1[u_j]|dx<+\infty.NEWLINE\]NEWLINE Under certain conditions on the measure \(\nu\) and on the mean curvature function of \(\partial \Omega\), the authors prove that problem \((P)\) admits at least a weak solution \(u\in \phi(\Omega)\).NEWLINENEWLINE\( \)NEWLINENEWLINEThe proof of this result is based on an approximation argument and on an existence result of classical solutions to problem \((P)\) for the case of \(\nu\) being a Lipschitz continuous function.NEWLINENEWLINEFor the entire collection see [Zbl 1322.53004].