Approximation of Lipschitz functions preserving boundary values. (English) Zbl 1429.26004
Authors’ abstract: We studied the problem of approximating Lipschitz functions defined on an open subset of a Banach space by differentiable Lipschitz functions preserving both the Lipschitz constant and the boundary value. In order to do that, we first obtained the following purely metric result, that can be of independent interest: Given a real-valued Lipschitz function on a metric space, such that its restriction to a given closed subset has a better Lipschitz constant, we can approximate it uniformly by a function with better Lipschitz constant on bounded sets, and which coincides with the initial function on the given closed subset. This intermediate result allowed us to give a positive answer to our problem, when the Lipschitz constant on the boundary of the function to be approximated is smaller than its global Lipschitz constant. The order of differentiability of the approximating functions depends on the regularity of the partitions of unity of the pertinent space, and then the approximations can be taken infinitely many times differentiable in finite-dimensional and Hilbert spaces. These results yield approximation of 1-Lipschitz functions by everywhere-differentiable functions, which satisfy the Eikonal equation almost everywhere and coincides in the boundary with the initial function, provided that the restriction of the function to the boundary has Lipschitz constant less than 1. We proved the optimality of these results by exhibiting an example of a 1-Lipschitz function defined on the boundary of an open subset of a two-dimensional normed space, which does not admit any 1-Lipschitz differentiable extension. The question whether the main problem of this paper, without restrictions on the boundary value of the function to be approximated, has a positive solution in a finite-dimensional Euclidean space remains open. A related question would be to find conditions on the norm of a finite-dimensional space for which our problem has a positive solution, without restrictions on the boundary value of the initial function.
Reviewer: Bogdan Szal (Zielona Gora)
MSC:
| 26A16 | Lipschitz (Hölder) classes |
| 26B05 | Continuity and differentiation questions |
| 41A29 | Approximation with constraints |
| 41A30 | Approximation by other special function classes |
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 54C30 | Real-valued functions in general topology |
| 58C25 | Differentiable maps on manifolds |
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