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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
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