Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics
Abstract
This is the first part of a project concerning the regularity of the free boundary of a function, where the free boundary depends on the gradient of the function. We study the minimizer of the expression J(u) : = ∫B1 F(∇ u) dx, where F(x) is a uniformly convex function whose second derivatives might jump at |x|=1. This results in an Euler-Lagrange equation that varies over the free boundary, which we define as Γ= ∂ {x ∈ B1: |∇ u| >1 } ∩ ∂ {x ∈ B1: |∇ u| <1 }. We consider two-phase flat points for which the second derivatives of F jump over Γ. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer u can be expressed as u(x)= a + ν· x+ δp(x) + δεq(x), where a ∈ ℝ, ν∈ ℝn, 0 < δ, ε<< 1. Here p is a C1 function, which consists of one polynomial in the upper half ball and another polynomial in the lower half ball. The function q is a rest term with bounded L2 norm. Furthermore, assuming that we have a sequence, uj, of minimizers, we show that qj converges weakly to a function q0 that satisfies a certain PDE. In addition, we show that q0 is C3 in Br+ and Br-, respectively, for 0<r<1. This is the main result of this paper which is intended to be used to show regularity of the free boundary.
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BibTeX
@misc{fejne2026regularity,
title = {{Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics}},
author = {Frida Fejne},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11324},
eprint = {2609.11324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11324},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics
AU - Frida Fejne
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11324
UR - https://arxiv.org/abs/2609.11324
AB - This is the first part of a project concerning the regularity of the free boundary of a function, where the free boundary depends on the gradient of the function. We study the minimizer of the expression J(u) : = ∫_B_1 F(∇ u) dx, where F(x) is a uniformly convex function whose second derivatives might jump at |x|=1. This results in an Euler-Lagrange equation that varies over the free boundary, which we define as Γ= ∂ {x ∈ B_1: |∇ u| >1 } ∩ ∂ {x ∈ B_1: |∇ u| <1 }. We consider two-phase flat points for which the second derivatives of F jump over Γ. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer u can be expressed as u(x)= a + ν· x+ δp(x) + δεq(x), where a ∈ ℝ, ν∈ ℝ^n, 0 < δ, ε<< 1. Here p is a C^1 function, which consists of one polynomial in the upper half ball and another polynomial in the lower half ball. The function q is a rest term with bounded L^2 norm. Furthermore, assuming that we have a sequence, u^j, of minimizers, we show that q^j converges weakly to a function q^0 that satisfies a certain PDE. In addition, we show that q^0 is C^3 in B_r^+ and B_r^-, respectively, for 0<r<1. This is the main result of this paper which is intended to be used to show regularity of the free boundary.
ER -