Affiliation:
1. Department of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, WA6009Crawley, Australia
2. Department of Mathematics, Columbia University, 2990 Broadway, NY 10027New York, USA
Abstract
AbstractThe main goal of this article is to understand the trace properties
of nonlocal minimal graphs in {\mathbb{R}^{3}}, i.e. nonlocal minimal surfaces with a graphical structure.We establish that at any boundary points at which
the trace from inside happens to coincide with
the exterior datum, also the tangent planes
of the traces necessarily coincide with those of the exterior datum.This very rigid geometric constraint is in sharp contrast with the case
of the solutions of the linear equations driven by the fractional Laplacian,
since we also show that, in this case, the fractional normal
derivative can be prescribed arbitrarily, up to a small error.We remark that, at a formal level,
the linearization of the trace of a nonlocal minimal graph
is given by the fractional normal derivative of a fractional Laplace problem,
therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type
present very specific properties which are strikingly different from
those of other problems of fractional type
which are apparently similar, but
diverse in structure, and the nonlinear
case given by the nonlocal minimal graphs
turns out to be significantly more rigid than its linear counterpart.
Subject
Applied Mathematics,General Mathematics
Cited by
10 articles.
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