REVIEW
Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics
Analysis of PDEs · Submitted 10 Sep 2026 · 29 references
Pith's verdict: A technically solid third-order asymptotic expansion for a gradient-constrained free boundary problem, but with a load-bearing assumption whose necessity is unclear.
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Referee report
The paper
The paper claims
For convex energies whose integrand changes at $|\nabla u|=1$, this paper proves a third-order asymptotic expansion at flat two-phase free boundary points, with the remainder converging to a limit that is $C^3$ in each phase.
The reading
What carries the argument
The carrying object is the class of broken polynomials $\Pi(\nu)$: $C^1$ functions that are homogeneous quadratic polynomials $p_+$ above the plane $\{x_n=0\}$ and $p_-$ below it, each solving the corresponding half-space linearized equation $F^m_{p_ip_k}(\nu)D^2_{ik}p=0$, with the sign constraints $\nu\cdot\nabla p_+\geq0$ in $\mathbb{R}^n_+$, $\nu\cdot\nabla p_-\leq0$ in $\mathbb{R}^n_-$, and the nontrivial jump $D^2_{nn}p_+\neq D^2_{nn}p_-$. These functions replace the usual normal-to-the-level-set analysis, because the interface here is a level set of $|\nabla u|$. The decisive identity is the jump relation of Lemma 3.6, $D^2_{nn}p_-=D^2_{nn}p_+ + J\nu_nD^2_{\nu n}p_-/F^1_{p_np_n}(\nu)$, with $J=\nu\cdot(D^2F_1(\nu)-D^2F_2(\nu))\nu^T$. Combined with $D^2_{\nu n}p_0^+\neq D^2_{\nu n}p_0^-$, it forces $\nu_n\neq0$ and $J\neq0$, keeps the free boundaries in a strip converging to $\{x_n=0\}$, and ultimately produces the limiting PDE (4.74) from the weak form of the equation for $q_j$.
Extended reading notes
Core claim
The central claim is that the third-order remainder of a minimizer at a flat two-phase point has a deterministic limit. Writing each $u_j=a_j+\nu_j\cdot x+\delta_jp_j+\delta_j\epsilon_jq_j$, with $p_j\in\Pi(\nu_j)$ obtained from an $L^2$-projection and $\|q_j\|_{L^2(B_1)}=1$, the paper proves that if the second-order defects converge to a broken quadratic $p_0$ with $D^2_{\nu n}p_0^+\neq D^2_{\nu n}p_0^-$ and if $\delta_j/\epsilon_j$ is bounded, then a subsequence satisfies $q_j\rightharpoonup q_0$ in $W^{1,2}_{loc}(B_1)$. The limit obeys the PDE (4.74): a divergence-form equation with coefficients $F^*_{p_ip_k}(\nu,x)$, a bulk force $\tau\sum F^*_{p_ip_kp_l}(\nu,x)D_{ik}p_0D_lp_0$, and an interface term supported on $B'_1\subset\{x_n=0\}$ with amplitude $\frac{\nu\cdot(D^2F_1(\nu)-D^2F_2(\nu))\nu^T}{2\nu_n}(\alpha+\tau|\nabla p_0(x',0)|^2)$, where $\tau=\lim \delta_j/\epsilon_j$ and $\alpha=\lim(|\nu_j|^2-1)/(\delta_j\epsilon_j)$. The same theorem gives $q_0\in C^3(B_r^+)$ and $q_0\in C^3(B_r^-)$ for $0<r<1$. The first version required a bound on $\alpha_j$; the main version removes that assumption, keeping only the boundedness of $\delta_j/\epsilon_j$.
Load-bearing premise
The load-bearing assumption is that the normalized second-order error is at most a fixed multiple of the third-order scale in the expansion of each minimizer; if that ratio can grow without bound, the uniform estimate on the remainders and the passage to the limit no longer follow from this proof.
Editorial extensions
If this is right
- For sequences of two-phase flat minimizers with bounded $\delta_j/\epsilon_j$, the third-order remainders are compact, so a subsequence always converges weakly in $W^{1,2}_{loc}$.
- The limiting remainder $q_0$ is forced to satisfy the explicit PDE (4.74), whose interface term encodes the jump in the Hessian of $F$ across the gradient level set.
- The limit $q_0$ is $C^3$ in $B_r^+$ and $C^3$ in $B_r^-$ for every $r<1$, so any lack of smoothness is confined to the interface plane and is controlled by the trace terms involving $\alpha$ and $\tau$.
- The free boundaries $\Gamma_j$ locally collapse to the flat hyperplane $\{x_n=0\}$ at the analysed scales, with the strip width tending to zero.
- The one-sided $C^3$ estimates are the stated input for a later argument proving $C^{1,\alpha}$ regularity of the free boundary at flat two-phase points.
Reading between the lines
- Because the $\alpha$-term in (4.74) only appears through the $L^2$-projection of the slope $\nu_j$, the paper's simpler expansion with fixed slopes would give $\alpha=0$; this suggests that geometric statements about $\Gamma$ should be invariant under the choice of projection, so the $\alpha$-dependence should cancel once quantities are re-expressed in invariant form.
- If $F_1$ and $F_2$ are both quadratic, all third-derivative terms vanish and $q_0$ solves a linear transmission problem with a jump condition on $\{x_n=0\}$; this is a tractable benchmark in which the convergence statements can be checked by direct computation.
- The arguments are PDE-based and do not use monotonicity formulas or maximum principles; the same route may adapt to systems, parabolic problems, or elastic-plastic models where the gradient threshold varies, which is the paper's stated motivation.
- One-sided $C^3$ control of $q_0$ is stronger than the $C^{1,\alpha}$ regularity sought for the interface, so a natural next check is whether the normal to $\Gamma$ can be recovered directly from the second-order jump of $p_0$ rather than from any derivative of $q_0$.
Original 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 \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} 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 $$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$ where $a \in \mathbb{R}$, $\nu \in \mathbb{R}^n$, $0 < \delta, \epsilon << 1$. 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.
The compiled record
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The abstract, sentence by sentence
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 \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} 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 $$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$ where $a \in \mathbb{R}$, $\nu \in \mathbb{R}^n$, $0 < \delta, \epsilon << 1$. 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.97 | 0.24 | 0.03 | asserted 0.93 |
| s1 | We study the minimizer of the expression \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} where $F(x)$ is a uniformly convex function whose second derivatives might jump at $|x|=1$. | contribution 0.71 | 0.55 | 0.02 | asserted 0.87 |
| s2 | This results in an Euler-Lagrange equation that varies over the free boundary, which we define as $$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. | background 0.50 | 0.76 | 0.03 | asserted 0.99 |
| s3 | $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. | background 0.58 | 0.35 | 0.02 | asserted 0.84 |
| s4 | In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$ where $a \in \mathbb{R}$, $\nu \in \mathbb{R}^n$, $0 < \delta, \epsilon << 1$. | result 0.96 | 0.95 | 0.16 | asserted 0.92 |
| s5 | 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. | result 0.94 | 0.73 | 0.02 | asserted 1.00 |
| s6 | The function $q$ is a rest term with bounded $L^2$ norm. | result 0.80 | 0.70 | 0.02 | asserted 1.00 |
| s7 | 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. | result 0.90 | 0.95 | 0.05 | asserted 1.00 |
| s8 | In addition, we show that $q^0$ is $C^3$ in $B_r^+$ and $B_r^-$, respectively, for $0<r<1$. | result 1.00 | 0.94 | 0.26 | asserted 1.00 |
| s9 | This is the main result of this paper which is intended to be used to show regularity of the free boundary. | future work 0.47 | 0.40 | 0.66 | asserted 0.91 |
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