Abstract:
We study the minimization of functionals $$ \int_\Omega f(Du) dx $$ with a convex integrand $f$ of linear growth (such as the area integrand), among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values. Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem.
Assuming radial structure, I explain for the scalar case a necessary and sufficient condition on the integrand such that the Dirichlet problem is solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin. Since for the general case solutions exist only in a suitably generalized sense, I then discuss the extension of the original functional to the space of functions of bounded variation via relaxation and give some results concerning the regularity and uniqueness of such generalized solutions.
The results presented in this talk are based on joined projects with Miroslav Bulíček (Prag), Erika Maringová (Prague), and Thomas Schmidt (Hamburg).
