Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics

Mohamed El Morsalani; Mohammed Barkatou

REVIEW 4 major objections 5 minor 16 references

This paper shows that the nested level sets of an elliptic PDE can be globally encoded by a thickness function over a convex core, and that their boundary normals decompose into a static radial part, the thickness gradient, and small correc

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 19:33 UTC pith:KHLNBWLZ

load-bearing objection The decomposition idea is nice, but the main displacement expansion has a factor-of-two error because the proof switches reflection laws, and the numerics don't stand up. the 4 major comments →

arxiv 2607.16912 v1 pith:KHLNBWLZ submitted 2026-07-18 math.AP

Geometric Gradient Flows from Elliptic Level Sets: Normal Decomposition and Reflection Dynamics

Mohammed Barkatou , Mohamed El Morsalani This is my paper

classification math.AP MSC 35J2535B6553A0549Q10

keywords elliptic level setsnormal graphradial monotonicitythickness functionspecular reflection mapgradient flowgeometric energyshape calculus

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the nested level sets of a solution to -Δu=f, with f supported in a strictly convex inner core, can be coerced into a global graph over the core boundary. The thickness of each level shell obeys a clean evolution law, and the paper derives an explicit decomposition of the level-set unit normal into the static radial normal, the surface gradient of the thickness, and small curvature/PDE corrections. It then proves that the specular reflection of the normal back to the core moves a base point by -2 d ∇d plus a quadratically small remainder, so the discrete reflection orbits approximate a continuous gradient flow to first order. The upshot is that the family of superlevel sets is not merely nested but is dynamically driven by the thickness gradient, with a geometric energy that decreases strictly along both continuous and discrete evolution.

Core claim

Under quantitative radial monotonicity, every small-t level boundary ∂Ω_t is a smooth normal graph over the convex core ∂C, and the inward unit normal obeys n = ν - ∇_{∂C} d_t + G + P, with -∇_{∂C} d_t the leading kinematic term and G, P small curvature/PDE corrections. Along t_n=2^{-n}, the specular reflection map F_n satisfies F_n(p)-p = -2 d_n(p)∇_{∂C} d_n(p) + R_n(p) with quadratic remainder control. Hence the discrete reflection orbits approximate a continuous non-autonomous gradient flow driven by the thickness gradient, and the geometric energy E(t)=∫ d_t^2 is strictly dissipated.

What carries the argument

The load-bearing object is the thickness function d_t on the core boundary, defined by u(c+d_t(c)ν(c))=t. The normal graph representation turns each level surface into a function on a fixed manifold, and the exact identity ∇_{∂C} d_t = -∇_{∂C} u/∂_r u couples the PDE to the geometry. The reflection map F_n is obtained by firing a ray along the specular reflection of the level-set normal back to the core; the displacement expansion F_n(p)-p=-2d_n∇d_n+R_n is the central quantitative result. The factor of 2 comes from the metric structure of tubular coordinates, and the remainder is controlled by the Weingarten map and Schauder bounds.

Load-bearing premise

The whole framework rests on Lemma 4.3, which asserts — citing an earlier paper rather than proving — that u decreases strictly along every normal ray from the core to the outer boundary with no critical points; if this global radial monotonicity fails, the normal-graph representation, transversality, and return map all collapse.

What would settle it

Compute the solution for a smooth, strictly convex core and a non-radial source supported inside the core, and check along an explicit normal ray whether ∂_r u ever changes sign between the core and the boundary. A single sign change falsifies the global monotonicity premise and breaks the normal-graph theorem; alternatively, independently implementing the specular return map for a harmonic level shell and comparing F_n(p)-p to -2d_n∇d_n should show agreement to O(d^2), and any deviation at order d would falsify the expansion.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Each level boundary gains a global, smooth coordinate system over the core, so boundary variations can be computed on a fixed manifold.
The kinematic term -∇d_t acts independently of core curvature and source profile, separating the driving mechanism from geometry and PDE details.
Discrete reflection orbits track the continuous non-autonomous gradient flow at rate O(2^{-n}) after a time rescaling.
Geometric energy ∫ d_t^2 decreases strictly along both continuous evolution and the discrete map, providing a Lyapunov function near the boundary.
Fully radial configurations are stationary states with vanishing remainders, confirming the calibrations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same gradient-flow correspondence should hold for any elliptic operator with a comparable quantitative radial monotonicity, so the paper supplies a template for level-set dynamics beyond the Laplacian; a test would be the p-Laplacian in a thin shell.
The factor-2 displacement relation suggests a discrete-to-continuous limit in which the reflection map is a forward-Euler step with time step 2d_n; comparing this with known curvature-flow discretizations could reveal a deeper connection to mean curvature motion.
If global monotonicity fails in a region, the local thin-shell expansions might still hold away from critical points, offering a path to study level-set flow past singularities by patching charts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

The decomposition idea is nice, but the main displacement expansion has a factor-of-two error because the proof switches reflection laws, and the numerics don't stand up.

read the letter

I read the paper carefully, and I'm fairly confident the reader is right: the main theorem is wrong, and not by a small margin. The defined reflection map F_n sends p along a ray with specular direction v = n - 2(n·ν)ν. Substituting their own normal expansion n = ν - ∇d + ... gives v = -ν - ∇d + ..., so the ray hits the core at parameter λ ≈ d and at the point p - d∇d + .... The metric projection to ∂C does not double this, to first order. You get -d∇d, not -2d∇d. The factor 2 only appears in Appendix C by using a different operation, point reflection across the tangent plane of the level set, followed by projection. That is not the map defined in Section 7. Check the flat model: if ∂C is y=0 and the level set is y=d(x), the specular ray gives exactly -d d'(x), not -2d d'(x). So Proposition 8.4 is internally inconsistent with its own definitions, and the gradient flow identification and energy dissipation rate, both of which rest on the coefficient 2, collapse.

There is real substance here, though. The normal decomposition n = ν - ∇d + G + P is a reasonable way to organize the geometry of a level-set graph over a convex core, and the first-order variation formulas in Section 6 are correct and clearly derived. The idea of linking the level-set thickness to a discrete dynamical system is worth exploring. The writing is clean, and the literature is discussed seriously, though perhaps too heavily self-referential.

Other soft spots: Lemma 4.3, the global radial monotonicity, is simply asserted with a citation to the author's earlier work. That property is the ground for the normal graph representation and the whole return map, so an unproved lemma at this location is a real gap, not a technicality. And the numerics worry me. The reported errors match the theoretical formulas to the last digit, which would be remarkable even with well-controlled experiments; the C^1 norm claimed for the ellipse geometry is inconsistent with the stated parameters; and no code or data are provided. I would not rely on this as independent validation.

Who is this for? Someone studying level-set geometry of elliptic problems might mine the decomposition idea, but they should treat the dynamical claims with suspicion. I would not cite it in its current form. It deserves a serious referee—the error is clear enough to be fixable in principle (change the reflection law or the expansion), and the authors have a useful idea underneath. My recommendation: reject for now, but invite a rewrite if the factor-of-two issue and the numerics can be resolved.

Referee Report

4 major / 5 minor

Summary. The paper studies superlevel sets Ω_t={u>t} of the solution to the elliptic Dirichlet problem -Δu=f in Ω, with f≥0 supported in a strictly convex core C⊂Ω. Under a quantitative radial monotonicity condition it claims that each ∂Ω_t is a C^{1,α} normal graph over ∂C with thickness function d_t, that the inward unit normal admits the decomposition n=ν-∇_{∂C}d_t+G+P, that the discrete specular reflection map F_n on ∂C satisfies F_n(p)-p=-2d_n(p)∇_{∂C}d_n(p)+R_n(p) with quadratic remainder control, and that the resulting orbits approximate a non-autonomous gradient flow with strict dissipation of the energy E(t)=∫_{∂C}d_t^2. Finite element experiments are presented in support of these claims.

Significance. The paper aims at an attractive bridge between elliptic level-set geometry, discrete reflection dynamics, and gradient-flow approximation, and the proposed normal decomposition is a useful organizing idea. The explicit energy dissipation formulas and the attempt at machine-free asymptotic control are also valuable. However, the central displacement expansion does not follow from the reflection map actually defined in Section 7, and the foundational monotonicity lemmas contain a circular or unsupported step. Since the advertised gradient-flow interpretation and the numerical validation test exactly this displacement expansion, the core contribution is not established as written.

major comments (4)

[§7, Eq. (38); Prop. 8.4; Appendix C.3] For the map actually defined in (38)-(39), the reflection direction is v=n-2(n·ν)ν. Inserting the paper's own normal expansion n=ν-∇_{∂C}d+O(‖d‖²_{C¹}) gives v=-ν-∇_{∂C}d+...; solving ρ(p+dν+λv)=0 yields λ=d+O(d²) and the unprojected contact point y=p-d∇_{∂C}d+O(d²). Metric projection onto ∂C gives F_n(p)-p=-d_n(p)∇_{∂C}d_n(p)+O(d²), not -2d_n(p)∇_{∂C}d_n(p). Appendix C obtains the factor 2 from the different operation (90), point reflection of x across the level tangent plane followed by projection along ν; this is not the ray in (39). In the flat model ∂C={y=0}, level set y=d(x), law (38) gives exactly -dd', not -2dd'. Thus Proposition 8.4 and the subsequent gradient-flow identification in Theorem 8.5 are not supported by the defined dynamics. The numerical error operator (76) subtracts 2d∇d and would not vanish if applied to the map defined in Section 7.
[Lemma 4.1 and Lemma 4.3] The proof of Lemma 4.1 applies the Hopf boundary lemma at c0∈∂C by asserting that c0 is a strict boundary minimum of u on the tangent ball B_R(y). The only justification is 'u(x)>u(c0) for all x∈B_R(y) due to the strict outward decline of the elliptic profile toward the zero boundary ∂Ω'; this is exactly the monotonicity that the lemma is supposed to establish. With arbitrary non-negative f supported in C, the assertion is not automatic. Moreover, Lemma 4.3's proof is self-referential: it invokes 'the localized non-degeneracy from Lemma 4.3' in its own proof and then cites [1] for the global monotonicity. Since Theorem 5.1 and all subsequent normal-graph and reflection results rely on these lemmas, the foundational rigidity premise is not proven.
[Theorem 8.5 proof] The proof uses the unproved structural relation ‖∇_{∂C}d_n‖_{L∞}≤C_2‖d_n‖_{L∞}. Assumption 2.3 only postulates ‖d_n‖_{C¹(∂C)}≪1, which does not imply such a bound: functions like d_n=ε sin(x/ε) have ‖d_n‖∞=ε but ‖∇d_n‖∞=O(1). Without this relation, the normalized remainder ‖∇d_n‖²_{L∞}/‖d_n‖_{L∞} in the proof may fail to vanish, so the convergence to the gradient flow limit in Theorem 8.5 is not established from the stated hypotheses.
[Section 7, Theorem 8.2] There is a sign/orientation inconsistency that affects the interpretation of the reflection dynamics. The vector n is introduced as the inward unit normal, but the expansion in Theorem 8.2 and Appendix C.2 gives n·ν=1-½‖∇d‖²+...>0, which is the outward normal of the level surface relative to the core. This positive sign is exactly what makes v in (38) point toward the core. With a genuinely inward normal, n·ν would be negative and v would point outward, so Lemma 7.1 would fail. The convention needs to be fixed and reconciled with the 'inward' terminology.

minor comments (5)

[Remark 2.5] 'Theorems 8.2, 8.2' is a duplicated reference; the intended target is likely Proposition 8.4 or Theorem 8.2 alone.
[Corollary 4.2] The proof cites Lemma 4.3 for quantitative radial monotonicity, but the needed bound is Lemma 4.1; Lemma 4.3 itself is redundant if Lemma 4.1 holds, and its proof is circular.
[Section 11.3.2] The numerical section does not provide enough detail to determine whether the implemented 'exact specular ray reflection' is the map of Eq. (38)-(39) or the point-reflection map of Appendix C.3. No solver code or pseudocode is supplied, so Table 1 cannot be used to validate Proposition 8.4 independently.
[Remark 6.4] The claimed direct proof of Shahgholian's rigidity theorem assumes ∂_r u=-1 on ∂Ω from ∂_ν u=-1; this is only true when the boundary normal is radial. For a non-parallel graph, ν_∂Ω has a tangential component, so the argument as stated is incomplete.
[Section 12, Open Problem 4] 'While Theorem guarantees...' is missing a theorem number; this is a typographical gap in an otherwise readable list.

Circularity Check

2 steps flagged

Central displacement expansion F_n−p=−2d_n∇d_n is not derived from the specular map (38) the paper defines: the proof's own intermediate y_n=p−d∇d gives coefficient 1, while the factor 2 is imported from a different point-reflection formula (90) in Appendix C. Lemma 4.3's global monotonicity also rests on the authors' own [1] and on a self-reference.

specific steps

other [Section 7 eqs. (38),(45); Proposition 8.4; Appendix C.3 eqs. (90)-(92)]
"F_n(p)=P(y_n(p)) ... v_n(p)=n(x_n(p))−2(n(x_n(p))·ν(p))ν(p) ... y_n(p)=p−d(p)∇_{∂C}d(p)+O(d^2+d|∇_{∂C}d|) [yet] the geometric pullback through the metric projection tensor removes the nominal normal variations and doubles the driving tangential displacement ... F_n(p)−p=−2d(p)∇_{∂C}d(p)+R_n(p). ... x* = x−2⟨n, x−p⟩n ... This completes the derivation of the explicit factor of 2."

The return map is defined by shooting along the specular law (38). Inserting the paper's own normal expansion n=ν−∇d gives v=−ν−∇d+..., and the proof itself computes the contact point as y_n=p−d∇d. Metric projection onto ∂C is the identity on tangent vectors to first order, so F_n−p=−d∇d, not −2d∇d. The factor 2 is obtained only in Appendix C from a different transformation — point reflection x*=x−2⟨n,x−p⟩n across the level-set tangent plane (eq. 90) — followed by an asserted 'sign orientation flip.' The gradient-flow identification (Theorem 8.5) and the energy-dissipation pairing (D.5) rest on this coefficient, so the central prediction is constructed by the auxiliary operation rather than derived from the defined map.
self citation load bearing [Lemma 4.3 (Global radial monotonicity framework)]
"This uniform decay is an established structural feature for solutions of elliptic ring problems under the geometric normal property. Following Barkatou [1], the combination of the localized non-degeneracy from Lemma 4.3, the vanishing boundary condition u=0 on ∂Ω, and the absence of external source terms (f=0 in Ω\C) prevents the formation of local internal minima or critical points along the normal rays."

The global monotonicity r↦u(c+rν(c)) strictly decreasing on [0,d_0(c)] is the premise for the normal-graph representation (Theorem 5.1) and for applying the implicit function theorem globally over ∂C. As written, its proof cites the authors' own paper [1] and invokes 'the localized non-degeneracy from Lemma 4.3' — i.e., the very lemma being proved. Mitigation: since G4 gives d_0<δ_0, the locally proven Lemma 4.1 bound ∂_r u≤−η on [0,δ_0] already implies the needed monotonicity, so this self-citation is repairable and only mildly load-bearing; nonetheless the paper's written derivation chain relies on it.

full rationale

The analytic core is largely non-circular in the statistical sense: no parameter is fitted to data and then renamed a prediction; the normal decomposition (Theorem 8.2) is a genuine Taylor expansion of n=∇u/|∇u| in tubular coordinates, with the kinematic term −∇d identified from the exact implicit-function identity ∇d=−∇u/∂_r u; the energy computation is an integration-by-parts exercise. The numerical FEM work is internal validation, not benchmarking with fitted parameters. However, the headline displacement proposition (8.4) does not follow from the map actually defined in Section 7. Substituting the paper's own expansion into the specular law (38) yields F_n−p=−d∇d; the proof even computes y_n=p−d∇d and then asserts the projection 'doubles' it. The factor 2, on which the gradient-flow identification and the energy-dissipation balance are built, comes from a different operation in Appendix C (point reflection across the level-set tangent plane, eq. (90)) and a sign flip. The numerical 'validation' cannot discriminate: the error field E_k=F_k−p+2d_k∇d_k in (76) would absorb a true −d∇d term as d∇d, which is also O(‖d‖²_C1), so Tables 1–3 are compatible with either coefficient. Separately, the global radial monotonicity (Lemma 4.3), the foundational premise for transversality and the normal-graph representation, is justified only by the authors' own [1] and a self-referential phrase; this is repairable from Lemma 4.1 plus d_0<δ_0, so it is mild. There are also unsupported assumptions (e.g., ‖∇d_n‖≤C₂‖d_n‖ asserted in Theorem 8.5) that are gaps, not circularity. Overall: partial circularity — the central displacement/gradient-flow prediction is constructed via an auxiliary reflection law, not derived from the stated dynamics.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The analytic paper has no fitted constants, but it depends on unproved global monotonicity, an explicit thin-shell smallness condition, and an ad hoc gradient-to-thickness bound. The numerical section provides no independent data and therefore adds no evidential weight.

free parameters (1)

structural smallness tolerance ε* and implied threshold t0(ε*) = not quantified
Assumption 2.3 / Remark 2.5 define the expansion regime by requiring ∥d0∥_{C^1} < ε*/2 for an unspecified ε*. No quantitative bound is derived; the entire asymptotic theory is conditional on this hidden smallness.

axioms (5)

domain assumption Quantitative local radial monotonicity: ∂_r u ≤ −η < 0 on ∂C × [0, δ0] (Lemma 4.1)
The proof uses Hopf's lemma, but the key assertion that u(x) > u(c0) throughout the interior ball is not demonstrated. This non-degeneracy is needed for the global normal graph.
domain assumption Global radial monotonicity along every ray to d0(c) (Lemma 4.3)
The proof cites the author's paper [1] and asserts that no internal critical points form. It is load-bearing for transversality, normal-graph representation, and the return map.
domain assumption Small-thickness hypothesis ∥d0∥_{C^1(∂C)} << 1 (Assumption 2.3)
Needed for all remainder expansions. It is not a consequence of the PDE and fails for the numerical ellipse, whose boundary thickness has nonzero gradient near the core.
ad hoc to paper Uniform gradient bound ∥∇_{∂C} d_n∥ ≤ C_2 ∥d_n∥ used in Theorem 8.5
Invoked 'under the active thin-shell framework' without proof. For non-radial d_0 with nonzero gradient this is not automatic, so the convergence claim depends on an unstated quantitative condition.
standard math Classical background: Schauder estimates, strong maximum principle / Hopf lemma, Federer reach/injectivity, Poincaré inequality, implicit function theorem
Used throughout Sections 4-5 and Appendices A-D as standard machinery. These are reasonable background assumptions.

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read the original abstract

We investigate the asymptotic geometry of shifting superlevel sets $\Omega_t = \{x \in \Omega : u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-\Delta u = f$ in $\Omega$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset \Omega$. Under a quantitative radial monotonicity condition, each boundary $\partial\Omega_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,\alpha}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{\Omega_t} = \nu - \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $\nu$ is the static radial normal, $-\nabla_{\partial C} d_t$ is the kinematic driving vector, and $\mathcal{G}, \mathcal{P}$ are curvature and PDE Hessian remainder operators.In a thin-shell configuration $(\Vert{}d_0\Vert{}_{C^1} \ll 1)$, we formalize a discrete specular point-reflection mapping $F_n$ on $\partial C$. We prove the tangential displacement satisfies $F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$, with quadratic control $\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$. Using the energy $\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N-1}$, we show these orbits approximate a continuous gradient flow driven by $+\nabla_{\partial C} d_{\tilde{t}}(p)$ to first order. Finite element computations (FEniCS) validate these convergence rates.

Figures

Figures reproduced from arXiv: 2607.16912 by Mohamed El Morsalani, Mohammed Barkatou.

**Figure 2.** Figure 2: Logarithmic decay trajectories tracking the continuous geometric energy functional [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗

**Figure 3.** Figure 3: Logarithmic error trajectory tracking the convergence metrics [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 linked inside Pith

[1]

Some geometric properties for a class of non-Lipschitz domains,

M. Barkatou, "Some geometric properties for a class of non-Lipschitz domains,”New York J. Math.8(2002), 189–213

2002
[2]

Alexandrov reflection for expanding curvature flows in hyperbolic space,

T. Bourni, J. M. Espinar, and A. Mishra, "Alexandrov reflection for expanding curvature flows in hyperbolic space,” preprint (2026), arXiv:2602.12186 [math.DG]

Pith/arXiv arXiv 2026
[3]

Shapes and Geometries: Analysis, Differential Calculus and Optimization

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus and Optimization",SIAM, 2001

2001
[4]

Global Convergence of the Return Dynamics in the ClassO C,

M. El Morsalani and M. Barkatou, "Global Convergence of the Return Dynamics in the ClassO C,” preprint (2026), arXiv:2603.28453 [math.DS]

Pith/arXiv arXiv 2026
[5]

Curvature measures,

H. Federer, "Curvature measures,”Trans. Amer. Math. Soc.93(1959), 418–491

1959
[6]

Elliptic Partial Differential Equations

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations",Springer, 2001

2001
[7]

Über lineare Funktionaloperationen,

E. Helly, "Über lineare Funktionaloperationen,”Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse121(1912), 265– 297

1912
[8]

Convexity of level sets for solutions to elliptic ring problems,

N. J. Korevaar, "Convexity of level sets for solutions to elliptic ring problems,”Comm. Partial Differential Equations15(1990), no. 4, 541–556

1990
[9]

Capacitary functions in convex rings,

J. L. Lewis, "Capacitary functions in convex rings,”Arch. Rational Mech. Anal.66(1977), 201–224

1977
[10]

Dynamics of Schwarz reflections: the mating phenomena,

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, "Dynamics of Schwarz reflections: the mating phenomena,”Annales Scientifiques De l’ENS, Fasc.6, T. 56, (2023), 1826–1881

2023
[11]

The convexity of level sets for solutions to elliptic partial differential equations,

X. Ma and B. Ou, "The convexity of level sets for solutions to elliptic partial differential equations,”Trends in Partial Differential Equations, ALM10(2010), 295–322

2010
[12]

Quadrature surfaces and the moving plane method,

H. Shahgholian, "Quadrature surfaces and the moving plane method,”Ark. Mat.32(1994), 495–508

1994
[13]

Counterexample to the convexity of level sets of solutions to the mean curva- ture equation,

X.-J. Wang, "Counterexample to the convexity of level sets of solutions to the mean curva- ture equation,”J. Eur. Math. Soc.15(2013), 1173–1181

2013
[14]

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,

O. C. Zienkiewicz and J. Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,”Int. J. Numer. Methods Eng.33(1992), 1331–1364

1992
[15]

The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity,

O. C. Zienkiewicz and J. Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity,”Int. J. Numer. Methods Eng.33(1992), 1365–1382

1992
[16]

A Posteriori Error Estimation in Finite Element Analysis

M. Ainsworth and J. T. Oden, "A Posteriori Error Estimation in Finite Element Analysis", Wiley, 2000. 46

2000

[1] [1]

Some geometric properties for a class of non-Lipschitz domains,

M. Barkatou, "Some geometric properties for a class of non-Lipschitz domains,”New York J. Math.8(2002), 189–213

2002

[2] [2]

Alexandrov reflection for expanding curvature flows in hyperbolic space,

T. Bourni, J. M. Espinar, and A. Mishra, "Alexandrov reflection for expanding curvature flows in hyperbolic space,” preprint (2026), arXiv:2602.12186 [math.DG]

Pith/arXiv arXiv 2026

[3] [3]

Shapes and Geometries: Analysis, Differential Calculus and Optimization

M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries: Analysis, Differential Calculus and Optimization",SIAM, 2001

2001

[4] [4]

Global Convergence of the Return Dynamics in the ClassO C,

M. El Morsalani and M. Barkatou, "Global Convergence of the Return Dynamics in the ClassO C,” preprint (2026), arXiv:2603.28453 [math.DS]

Pith/arXiv arXiv 2026

[5] [5]

Curvature measures,

H. Federer, "Curvature measures,”Trans. Amer. Math. Soc.93(1959), 418–491

1959

[6] [6]

Elliptic Partial Differential Equations

D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations",Springer, 2001

2001

[7] [7]

Über lineare Funktionaloperationen,

E. Helly, "Über lineare Funktionaloperationen,”Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse121(1912), 265– 297

1912

[8] [8]

Convexity of level sets for solutions to elliptic ring problems,

N. J. Korevaar, "Convexity of level sets for solutions to elliptic ring problems,”Comm. Partial Differential Equations15(1990), no. 4, 541–556

1990

[9] [9]

Capacitary functions in convex rings,

J. L. Lewis, "Capacitary functions in convex rings,”Arch. Rational Mech. Anal.66(1977), 201–224

1977

[10] [10]

Dynamics of Schwarz reflections: the mating phenomena,

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, "Dynamics of Schwarz reflections: the mating phenomena,”Annales Scientifiques De l’ENS, Fasc.6, T. 56, (2023), 1826–1881

2023

[11] [11]

The convexity of level sets for solutions to elliptic partial differential equations,

X. Ma and B. Ou, "The convexity of level sets for solutions to elliptic partial differential equations,”Trends in Partial Differential Equations, ALM10(2010), 295–322

2010

[12] [12]

Quadrature surfaces and the moving plane method,

H. Shahgholian, "Quadrature surfaces and the moving plane method,”Ark. Mat.32(1994), 495–508

1994

[13] [13]

Counterexample to the convexity of level sets of solutions to the mean curva- ture equation,

X.-J. Wang, "Counterexample to the convexity of level sets of solutions to the mean curva- ture equation,”J. Eur. Math. Soc.15(2013), 1173–1181

2013

[14] [14]

The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,

O. C. Zienkiewicz and J. Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique,”Int. J. Numer. Methods Eng.33(1992), 1331–1364

1992

[15] [15]

The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity,

O. C. Zienkiewicz and J. Z. Zhu, "The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity,”Int. J. Numer. Methods Eng.33(1992), 1365–1382

1992

[16] [16]

A Posteriori Error Estimation in Finite Element Analysis

M. Ainsworth and J. T. Oden, "A Posteriori Error Estimation in Finite Element Analysis", Wiley, 2000. 46

2000