## The 1-Laplacian elliptic equation with inhomogeneous Robin boundary conditions.(English)Zbl 1340.35067

Let $$\Omega$$ be a smooth bounded domain in $$\mathbb R^N$$, and let $$g\in L^2(\partial \Omega)$$. This paper deals with the following Robin problem for the 1-Laplace equation with inhomogeneous condition:
$-\operatorname{div}\left(\frac{D u}{| D u|}\right)=0\text{ in }\Omega, \,\lambda u+\left[\frac{D u}{|D u|},\nu\right]=g\text{ on }\partial \Omega.{(P)}$
Here, $$\lambda$$ is a positive parameter and $$\nu$$ is a unit outward normal vector field defined at $$\mathcal{H}^{N-1}$$-almost every point of $$\partial\Omega$$.
The authors give the following concept of weak solution to problem $$(P)$$: a weak solution to problem $$(P)$$ is a function $$u\in BV(\Omega)\cap L^2(\partial\Omega)$$ such that there exists a vector field $$\mathbf z\in L^\infty(\Omega,\mathbb{R}^N)$$ with the following properties:
$$\bullet$$
$$\|\mathbf{z}\|_\infty\leq 1$$,
$$\bullet$$
$$\operatorname{div}(\mathbf{z})=0$$, in the distributional sense,
$$\bullet$$
$$(\mathbf {z},Du)=| Du|$$, in $$\Omega$$,
$$\bullet$$
$$-[\mathbf z,\nu]=T_1(\lambda u-g)$$, $$\mathcal{H}^{N-1}$$-almost everywhere in $$\partial \Omega$$,

where $$T_1\in C(\mathbb R)$$ is such that $$T_1(r)=r$$ if $$r\in [-1,1]$$ and $$| T_1(r)| =1$$ otherwise.
The main result of this paper states the existence of at least one weak solution to problem $$(P)$$ in the sense specified above. This solution is obtained as the limit as $$p\rightarrow 1^+$$ of the unique weak solution $$u_p$$ of the approximating problem
$-\operatorname{div}(| \nabla u|^{p-2}\nabla u)=0\text{ in }\Omega, \,-| \nabla u|^{p-2}[\nabla u,\nu]=T_1(\lambda u-g)\text{ on }\partial \Omega,$
The authors also establish a connection between the Robin problem $$(P)$$ and the Dirichlet problem
$-\operatorname{div}\left(\frac{D u}{| D u| }\right)=0\text{ in }\Omega, \, u=h\text{ on }\partial \Omega,$
where $$h\in L^2(\partial \Omega)$$. Using this connection jointly to a uniqueness result for the above problem established by the authors, a uniqueness result for problem $$(P)$$ is given under some additional geometric properties on $$\Omega$$ and certain regularity conditions on the datum $$g$$.
Finally, the limiting case $$\lambda=0$$, corresponding to the Neumann problem for the equation
$-\operatorname{div}\left(\frac{D u}{| D u|}\right)=0\text{ in }\Omega,$ is analyzed. In this case, the authors prove that the solution exists if and only if the datum $$g$$ satisfies $\sup\biggl\{\frac{\int_{\partial \Omega} gwd\mathcal{H}^{N-1}}{\int_\Omega| w| dx}, \;\;w\in W^{1,1}(\Omega)\setminus\{0\}, \;\;\int_{\partial \Omega} wd\mathcal{H}^{N-1}=0\biggr\}\leq 1.$ The proofs are based on direct variational methods and approximation arguments.

### MSC:

 35J66 Nonlinear boundary value problems for nonlinear elliptic equations 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35D30 Weak solutions to PDEs