Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation (Q1943274)

This paper deals with the quasilinear boundary value problem \[ \begin{gathered}\left(\frac{u'}{\sqrt{1+u'{}^2}}\right)'=f(t,u),\quad-r<t<r,\\ u(-r)=a,\;u(r)=b,\end{gathered} \] where \(f:[-r,r]\times{\mathbb R}\to{\mathbb R}\) is continuous and \(a,\,b\in{\mathbb R}\). The main results state the existence of a solution to this problem in the space of functions of bounded variation \(BV(-r,r)\). Such results depend on an adequate set of assumptions about \(f\). For instance, if \[ \liminf_{|s|\to\infty}\frac{F(t,s)}{|s|}\geq-\varphi(t)\quad\text{uniformly in}\;[-r,r],\tag{H} \] where \(F\) is a primitive of \(f\) with respect to the second variable and \(\varphi(t)\leq\frac{1}{r}\), \(\varphi(t)\neq\frac{1}{r}\) is measurable, and \(f\) is increasing in the second variable in a quite general sense, then existence is obtained. It is noted that \((H)\) may be viewed as a control of interference with the first eigenvalue of the minus-one-Laplacian in \([-r,r]\). The solution is a global minimizer of the functional \[ v\mapsto\int_{-r}^r\sqrt{1+|Dv|^2}+\int_{-r}^rF(t,v)\,dt+|v(-r^+)-a|+|v(r^-)-b|, \] where the first integral must be conveniently defined using duality. The proof uses a certain positive definite homogeneous form and convergence via regularized approximations. In addition, information on regularity and on the behaviour of the solution at the endpoints of \([-r,r]\) is provided. The problem and its discussion are motivated by classical results, e.g., of \textit{M. Miranda} [Indiana Univ. Math. J. 24, 227--241 (1974; Zbl 0293.35029)], together with simple examples that show that one cannot expect the existence of \(C^2\)-solutions in general.