Existence of solutions for $1-$laplacian problems with singular first order terms
Francesco Balducci
TL;DR
This work addresses the existence of finite-energy solutions to the nonlinear Dirichlet problem for the $1$-Laplacian with a gradient term and singular zeroth-order terms: $- abla\,(Du/|Du|) + g(u)|Du| = h(u)f$ in $\Omega$, $u=0$ on $\partial\Omega$, where $g$ and $h$ may blow up at the origin. The authors develop a BV/divergence-measure framework via the Anzellotti–Chen–Frid theory, constructing an approximation scheme with truncated $g_n$, establishing uniform BV bounds, and passing to the limit to identify a vector field $z\in \\mathcal{DM}^\infty_{loc}$ that encodes the singular quotient $Du/|Du|$. Under precise near-zero and infinity growth conditions on $g$ and $h$ (with $0\le\theta<1$, $0\le\gamma\le1$, and $\theta+\gamma\le1$) and $0\le f\in L^N(\Omega)$, existence of a distributional BV-solution is proved; additionally, a small-data regime ensures boundedness of the solution. The paper also handles the case of nonnegative data, adapting the solution concept to accommodate positivity sets via characteristic functions. These results extend prior bounded-$g$ settings and illuminate the interplay between first-order singular terms and source terms in $BV$-type problems, with implications for level-set formulations and regularizing effects.
Abstract
We prove existence of solutions to the following problem \begin{equation*} \begin{cases} -Δ_1 u +g(u)|Du|=h(u)f & \text{in $Ω$,} \\ u=0 & \text{on $\partialΩ$,} \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^N$, with $N\ge2$, is an open and bounded set with Lipschitz boundary, $g$ is a continuous and positive function which possibly blows up at the origin and bounded at infinity and $h$ is a continuous and nonnegative function bounded at infinity (possibly blowing up at the origin) and finally $0 \le f \in L^N(Ω)$. As a by-product, this paper extends the results found where $g$ is a continuous and bounded function. \\We investigate the interplay between $g$ and $h$ in order to have existence of solutions.