Abstract
Abstract We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ ( t ) {\varphi(t)} is a non-negative convex function vanishing only at t = 0 {t=0} . We show that this property is always satisfied in dimension n = 2 {n=2} , while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ ( t ) = c t 2 {\varphi(t)=ct^{2}} ) in dimension n ≥ 4 {n\geq 4} . The validity of the quadratic rigidity, which we prove in dimension n = 2 {n=2} , implies the existence of the trace of a divergence-measure vector field ξ on an ℋ 1 {\mathcal{H}^{1}} -rectifiable set S, as soon as its weak normal trace [ ξ ⋅ ν S ] {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.
Highlights
The structure and the properties of vector fields, whose distributional divergence either vanishes or is represented by a locally finite measure, are of great interest in Mathematics and in Physics
The validity of the quadratic rigidity, which we prove in dimension n = 2, implies the existence of the trace of a divergencemeasure vector field ξ on an H1-rectifiable set S, as soon as its weak normal trace [ξ ⋅ νS] is maximal on S
We deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense
Summary
The structure and the properties of vector fields, whose distributional divergence either vanishes or is represented by a locally finite measure, are of great interest in Mathematics and in Physics. Assuming that Ω is weakly-regular (that is, the perimeter of Ω is finite and coincides with the (n − 1)-dimensional Hausdorff measure of ∂Ω, see Definition 2.5) one can show [34, Section 3] that there exists a function [ξ ⋅ νΩ] ∈ L∞(∂Ω; Hn−1) such that the following Gauss–Green formula holds for all ψ ∈ C1c (Rn):. Despite there exist counterexamples to the quadratic rigidity in Rn when n ≥ 4, we cannot exclude that Theorem 1.4 might be true in any dimension Were it false, one should be able to construct a vector field with maximal normal trace at some (n − 1)-submanifold S, whose blow-ups at most points of S are not unique.
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