A uniqueness result for graphs of least gradient

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).