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Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates

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Pith's one-line read The thesis proves quantitative stability for fractional Serrin problems: a nearly constant fractional normal derivative on the boundary forces a nearly round domain, at a Hölder rate that beats the classical logarithmic rate in the formal…

desk verdict First quantitative nonlocal Serrin stability and a genuine counterexample to Cir+18, but equation (4.8) has a sign error that makes Theorem 4.1 vacuous for f = 1 as printed. read the letter →

arxiv 2507.09219 v1 pith:QPLSFC5G submitted 2025-07-12 math.AP

Jack Thompson This is my paper · ORCID
classification math.AP MSC 35R1135B0635B5035N2547G2035A23
keywords fractionalLaplacianSerrinproblemoverdeterminedproblemsquantitativestabilitymovingplanesmethodantisymmetricfunctionsHarnackinequalitymeancurvature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An overdetermined problem is one with so many boundary conditions that a solution can exist only for special domains; Serrin's classical theorem says the special domain must be a ball. This thesis proves the quantitative, nonlocal version for the fractional Laplacian: if the fractional normal derivative on the boundary deviates from a constant by at most $\delta$, the domain is within distance $C \delta^{1/(s+2)}$ of a ball, for a wide class of nonlinearities. The same rate is established for the related parallel surface problem, and the paper shows this Hölder rate formally improves on the logarithmic stability that the moving-planes method yields in the classical case, so the answer to 'how much error breaks symmetry' is better in the nonlocal setting than in the local one. The tools developed along the way — an antisymmetric barrier, quantitative maximum principles and Harnack inequalities for antisymmetric functions, and a corrected geometric lemma — are reusable for stability analysis of nonlocal overdetermined problems. A closing part proves geometric identities and density estimates for sets of zero fractional mean curvature.

What carries the argument

The engine of the proofs is a new antisymmetric barrier $\varphi(x) = x_1 (\psi_{B_\rho(a)}(x) + \psi_{B_\rho(a^*)}(x))$, assembled from the explicit fractional torsion function of a ball, $\psi_{B_\rho(x_0)} = \gamma_{n,s}(\rho^2 - |x - x_0|^2)^s_+$, for which the thesis proves the pointwise bound $(-\Delta)^s \varphi \le \frac{n+2s}{n} x_1$ in the relevant half-balls. The barrier allows the moving-planes comparison to run uniformly, including when the comparison ball touches the plane of symmetry, where previous constructions failed. Its control rests on Bochner's relation, the identity $(-\Delta)^s v(x) = x_1 (-\Delta)^s_{\mathbb{R}^{n+2}} \tilde w(x_1, 0, 0, x_2, \dots, x_n)$ for antisymmetric $v(x) = x_1 w(x)$, which turns odd functions in $\mathbb{R}^n$ into radial functions in $\mathbb{R}^{n+2}$ and converts boundary-type estimates into interior ones. The barrier feeds a quantitative nonlocal maximum principle for antisymmetric functions and quantitative versions of the nonlocal Hopf lemma and the Serrin corner point lemma, which replace the qualitative steps of the moving-planes method; the improved exponents additionally rest on a corrected geometric lemma bounding the measure of the symmetric difference near the critical hyperplane.

What would settle it

Test the deferred identity directly in one dimension: for $s = 1/2$ and $v(x) = x e^{-x^2}$, evaluate $(-\Delta)^s v$ at several points by high-accuracy quadrature of the singular integral and compare with $x_1 (-\Delta)^s_{\mathbb{R}^3} \tilde w (x_1, 0, 0)$ for the radial lift $\tilde w$ of $w = e^{-x^2}$; agreement to quadrature accuracy would confirm Bochner's relation, while any discrepancy would break Lemma 3.8 and with it the barrier and Theorem 3.1. Independently, compute the counterexample family of Theorem 3.6 and check that the symmetric-difference measure near the critical hyperplane grows like $\varepsilon^{1-1/\alpha}$, contradicting the linear bound of Proposition 3.1(b) in [Cir+18].

Watch

Extended reading notes

Core claim

The paper's headline assertion, Theorem 4.1, is that the fractional Serrin rigidity survives in quantitative form: for a bounded $\mathcal{C}^2$ domain $\Omega$ with a uniform interior sphere condition, a nonnegative weak solution $u$ of $(-\Delta)^s u = f(u)$ in $\Omega$ with $u = 0$ outside $\Omega$, $f$ locally Lipschitz, $f(0) \ge 0$, and $u/\delta^s \in \mathcal{C}^1(\Omega)$, satisfies $\rho(\Omega) \le C\,[\partial^s_\nu u]^{1/(s+2)}_{\partial\Omega}$. Here $\rho(\Omega)$ is the infimum of $R - r$ over all annuli $B_r \subset \Omega \subset B_R$, and the semi-norm measures the oscillation of the fractional normal derivative on the boundary. Theorem 3.1 proves the analogous estimate for the parallel surface problem with an arbitrary locally Lipschitz nonlinearity, extending the linear case $f \equiv 1$ in the literature and relaxing the regularity assumptions on the domain. Under a uniform $\mathcal{C}^\alpha$ boundary condition, both rates improve to $\alpha/(1+\alpha(s+1))$. The thesis also claims that a geometric lemma used in earlier quantitative work on the nonlocal soap-bubble theorem, Proposition 3.1(b) of [Cir+18], is false in the stated generality; it supplies an explicit one-parameter counterexample family (Theorem 3.6) and proves a corrected lemma (Theorem 3.3) whose rate $\gamma(R-r)^{1-1/\alpha}$ it shows to be optimal.

Load-bearing premise

The parallel-surface estimate stands on Bochner's relation, an identity for the fractional Laplacian of antisymmetric functions whose proof is deferred to a companion note that has not yet appeared; the Serrin estimate additionally assumes the solution divided by its boundary-distance power is differentiable up to the boundary, imported from cited regularity theory. If the identity fails, or the note never appears, the proof chain for the parallel-surface stability theorem is incomplete.

Editorial extensions

If this is right

  • Any measured violation of the overdetermined condition, of size $\varepsilon$ in the boundary semi-norm, is paid for by at most $C\varepsilon^{1/(s+2)}$ in how far the domain is from a ball; near-sphericity can therefore be certified from boundary measurements alone.
  • The same modulus of continuity handles arbitrary locally Lipschitz nonlinearities $f$ with $f(0) \ge 0$ in the parallel surface problem, and improves to $\alpha/(1+\alpha(s+1))$ when the boundary is uniform $\mathcal{C}^\alpha$.
  • In the formal limit $s \to 1^-$ (the paper notes the constants in the proof blow up, so the comparison is formal), the Serrin estimate gives Hölder stability with no convexity or domain-class restrictions, a benchmark the local moving-planes method on general domains, with its logarithmic rate, has not reached.
  • The corrected geometric lemma, with rate $\gamma(R-r)^{1-1/\alpha}$, replaces the flawed statement in the nonlocal soap-bubble literature and comes with an explicit family showing the rate is optimal, so any future improvement must pass through a different mechanism.
  • The optimal exponent for the parallel surface problem is bracketed, $1/(s+2) \le \beta(s) \le 1$; the ellipsoid family saturates the linear upper bound, and identifying $\beta(s)$ is left open.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The closed-form ellipsoid computation, where the deviation-to-oscillation ratio tends to a positive constant, suggests the true optimal exponent is $1$ for every $s$, making the proved rate $1/(s+2)$ a conservative bound rather than the final answer.
  • Bochner's relation is operator-agnostic: any nonlocal operator with a known Fourier symbol should admit an antisymmetric Harnack inequality by the same lift to two extra dimensions, which would carry the whole stability machinery beyond the fractional Laplacian.
  • If the formal superiority over the classical logarithmic rate survives scrutiny, the logarithmic bound should be read as an artifact of local perturbation estimates; a local proof matching the Hölder rate on general domains would likely need to import nonlocal-style barriers.
  • The counterexample to Proposition 3.1(b) of [Cir+18] is a warning sign: other quantitative moving-planes proofs built on linear-in-$(R-r)$ measure bounds for symmetric differences should be audited against the corrected rate $\gamma(R-r)^{1-1/\alpha}$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper, a PhD thesis posted on arXiv, develops quantitative stability estimates for two nonlocal overdetermined problems and studies tools around antisymmetric functions for nonlocal operators. The headline result (Theorem 4.1) asserts that for the fractional Serrin problem, under the hypotheses that (-Delta)^s u = f(u) in Omega, u = 0 outside Omega, f is locally Lipschitz with f(0) >= 0, and u/delta^s is C^1, the domain is quantitatively close to a ball with a Holder rate rho(Omega) <= C [partial^s_nu u]^{1/(s+2)}_{partial Omega}. The parallel-surface counterpart (Theorem 3.1) gives the same exponent for general nonlinearities f, and Section 3.6 improves it to alpha/(1+alpha(s+1)) under uniform C^alpha boundary regularity. The paper also claims a counterexample to [Cir+18, Proposition 3.1(b)], computes the stability exponent for an explicit family of ellipsoids, proves antisymmetric Harnack inequalities, and establishes geometric identities and density estimates for fractional mean curvature. The main proofs use the method of moving planes with quantitative maximum principles, antisymmetric barriers, and a new Bochner-type identity.

Significance. If the identified gaps are repaired, the paper would make a substantial contribution: Theorem 4.1 provides the first quantitative stability estimate for the fractional Serrin problem, with an explicit Holder rate that improves, formally as s -> 1, the logarithmic rate available from classical moving-plane arguments. Theorem 3.1 extends the known parallel-surface stability result to general nonlinearities, and the explicit one-parameter ellipsoid computation in Section 3.7 gives a rigorous upper bound on the optimal exponent. The antisymmetric barrier in Theorem 4.3 and the quantitative maximum principles in Proposition 4.4 are likely to be useful beyond the present problems. The paper also contains a careful, self-contained geometric counterexample analysis in Section 3.6. The proofs are long but mostly explicit, with many checkable computations; there is no fitting-as-prediction issue, and the exponents are obtained from optimizing inequalities or from exact ellipsoid computations. The main weaknesses are the sign error in the definition of R in Eq. (4.8), which is load-bearing for Theorem 4.1, and the deferral of the proof of Bochner's relation to an unpublished note.

major comments (3)
  1. [Section 4.1, Eq. (4.8)] The stress-test concern is confirmed. The definition of R contains [f]^{1/(2s)}_{C^{0,1}([0,\|u\|_{L^\infty(\Omega)}])} with a positive exponent. For the constant-torsion case f \equiv 1, which the introduction explicitly presents as a case covered by the new result, [f]=0 and hence R=0. Then the constant C in (4.9) is infinite and Theorem 4.1 is vacuous; moreover Lemma 4.1 cannot be applied because R=0. Independent of the constant case, the dimension is wrong: Remark 4.5 and (4.15) show that condition (4.18) of Proposition 4.4 is satisfied only when \rho \le \kappa_{n,s}^{1/(2s)}|B_1|^{-1/n}\|c^+\|_{L^\infty(U)}^{-1/(2s)}, so the admissible radius must decrease as the Lipschitz constant of f grows. The printed R does the opposite. The parallel-surface definition in (3.8), R_0 = \min\{R, [f]^{-1/(2s)}\}, and the form of Remark 4.5 both indicate that (4.8) should have [f]^{-1/(2s)}. As submitted, Lemmas 4.1, Proposition 4.2, and Theorem 4.1 are not established even for f \equiv 1.
  2. [Section 3.3, Lemma 3.8, Eq. (3.16)] The proof of the Bochner relation (3.16) is deferred to the unpublished note [Dip+24a] by the same authors. This identity is load-bearing: it yields the bound (3.15) on |(-\Delta)^s v|, which is then used in Lemma 3.9 to construct the antisymmetric barrier, in Proposition 3.6 to prove the quantitative maximum principle, and ultimately in Theorem 3.1. The relation is also invoked in Part II as a tool for antisymmetric Harnack inequalities. Since the manuscript cannot be checked on this point without an external unpublished source, please include a complete proof of (3.16), or at least a precise statement with all hypotheses on smoothness and decay, in an appendix or in a published reference.
  3. [Section 4.3, Lemma 4.1, Eq. (4.35)-(4.36)] In the corner-point case (ii), the proof uses the C^1 assumption on u/\delta^s to obtain \delta^s(t\eta)-\delta^s(t\eta^*) = o(t^{1+s}) and \psi(t\eta)-\psi(t\eta^*) = 2t \langle \nabla\psi(0), e_1\rangle + o(t). The argument is plausible because \nabla\delta(0)=e_2 and the Hessian of the distance function vanishes in the normal-tangential block, but this step is stated rather than justified. The quantitative Serrin corner-point lemma must produce a lower bound for v(t\eta)/t^{1+s}, so the limiting inequality \limsup_{t\to 0+}|v(t\eta)|/t^{1+s} \le 2[\partial^s_\nu u]_{\partial\Omega} needs a short proof that the direction e_1 is actually a tangential direction at the corner point. Please add the one-paragraph justification, or state the precise regularity hypotheses on \delta that make the Taylor expansions uniform in t.
minor comments (5)
  1. [Section 2.2, Proposition 2.1] There are several typos in the statement and proof, e.g., 'c : Omega ia a measurable function' and the missing parentheses after 'expanding then using'; the proof is readable but should be copyedited.
  2. [Throughout the thesis] Each chapter has its own numbering, so Proposition 2.1 and Proposition 2.3 appear twice with different content. Since the manuscript is posted as one arXiv document, the duplicate labels create cross-reference ambiguity; consider renumbering the chapters or using chapter-prefixed labels.
  3. [Section 3.5] The discussion leading to (3.31) is explicitly heuristic, but it is easy to read as a proof. Please label the displayed inequality as a formal estimate or a motivating computation rather than a proven lemma, since the passage from the ball Poisson formula to general domains is not justified.
  4. [Section 3.8, claim (3.66)] The proof of (3.66) is a wall of estimates involving the functions g_epsilon and h_epsilon. A short final sentence collecting the uniform-in-epsilon bounds (3.71), (3.72), and the bound on the remaining term would make the chain of inequalities easier to follow.
  5. [Section 4.1, Eq. (4.8)] After correcting the exponent in (4.8), please also check the displayed constant in Theorem 4.1: the expression (diam Omega/R)^{2n+3+6s} R^{n+2-s} becomes singular if R is allowed to be small, and the text should specify that the estimate is understood with R > 0 built from r_Omega and the corrected [f]^{-1/(2s)} term.

Circularity Check

2 steps flagged · score 4.0 of 10

The stability estimates are not fit-to-data circular, but Chapter 3's central proof depends on a load-bearing self-citation to the unpublished note [Dip+24a], and the printed definition of R in (4.8) prevents Theorem 4.1 from going through as stated.

  1. self citation load bearing [Chapter 3, Lemma 3.8, Section 3.3 (equations (3.15)-(3.16)); used in Lemma 3.9, Proposition 3.6, and Theorem 3.1]
    "By Bochner's relation, we have that (-Delta)^s v(x)=x_1 (-Delta)^s_{R^{n+2}} w_tilde(x_1,0,0,x_2,...,x_n) (3.16). ... For more details regarding Bochner's relation and a proof of (3.16), we refer the interested reader to the upcoming note [Dip+24a]."

    The bound (3.15) in Lemma 3.8 is the engine behind the antisymmetric barrier in Lemma 3.9, which in turn drives Proposition 3.6 and the proof of Theorem 3.1. The identity (3.16) that produces (3.15) is not proved in the thesis; its proof is deferred to an unpublished note by the same research group. No machine-checked, code-reproduced, or externally verifiable version is supplied. Thus the central parallel-surface stability theorem inherits its key estimate from a self-citation to a forthcoming paper rather than from an independent proof in the present manuscript. This is a load-bearing provenance gap in the derivation chain.

  2. other [Chapter 4, Theorem 4.1, equation (4.8); used in Lemma 4.1 and Proposition 4.2]
    "Define R := min{r_Omega, kappa^{1/(2s)}_{n,s} |B_1|^{-1/n} [f]^{1/(2s)}_{C^{0,1}([0,||u||_{L^infty(Omega)}])}. ... As noticed in Remark 4.5, condition (4.18) is satisfied in light of (4.8) and (4.15)."

    This is not a circularity by construction, but it is a load-bearing missing-support gap. Condition (4.18) of Proposition 4.4 is guaranteed by the Faber-Krahn bound only when rho is no larger than a constant times ||c^+||_{L^infty}^{-1/(2s)}, as Remark 4.5 states. The printed definition (4.8) instead makes R proportional to [f]^{1/(2s)}. For the constant forcing f identically 1, [f]=0, so R=0, the constant in (4.9) diverges, and Lemma 4.1 cannot apply Proposition 4.4 with rho=R. Consequently the proof of Theorem 4.1 does not go through as printed even in the torsion case; the definition should presumably read [f]^{-1/(2s)}.

full rationale

I found no case where a fitted parameter is renamed as a prediction: the stability exponents 1/(s+2) and alpha/(1+alpha(s+1)) are obtained by optimizing inequalities, and the ellipsoid computation in Section 3.7 is an exact explicit calculation that serves as an external benchmark. The main circularity burden is provenance rather than logic. In Chapter 3, Lemma 3.8's estimate (3.15) relies on identity (3.16), whose proof is deferred to the same group's unpublished note [Dip+24a], and that estimate is load-bearing for the barrier, the quantitative maximum principle, and ultimately Theorem 3.1. Chapter 4 is more self-contained: the new barrier Theorem 4.3 is proved in the text from explicit special-function computations, and the final assembly cites published works for standard moving-plane closure steps. However, equation (4.8) as printed has a sign/exponent error that prevents the proof of Theorem 4.1 from applying Proposition 4.4 for f identically 1; this is a correctness gap rather than a self-referential reduction. Overall the central claims retain independent mathematical content, so the score reflects substantial but not total circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard fractional Sobolev and PDE background, on regularity and geometric assumptions on the domains, and on a small number of results imported from same-group papers, one of them unpublished. No data-fitted constants appear; all exponents are obtained by optimizing inequalities or by exact computation. The Bochner relation (3.16) is the main imported identity that is load-bearing for Chapter 3. The antisymmetric barrier functions are explicit mathematical constructions, not invented physical entities with independent falsifiable handles.

assumptions (7)
  • ad hoc to paper Bochner's relation (3.16): for antisymmetric v(x)=x_1 w(x), (-Delta)^s v(x) = x_1 (-Delta)^s_{R^{n+2}} w_tilde(x_1,0,0,x_2,...,x_n)
    Invoked in Lemma 3.8 (Section 3.3) to bound |(-Delta)^s v| for antisymmetric functions; the proof is deferred to the unpublished note [Dip+24a]. Load-bearing for the antisymmetric barrier (Lemma 3.9), Proposition 3.6, and Theorem 3.1.
  • domain assumption Moving-plane sign lemma: v_mu >= 0 in Omega'_mu for all mu in [lambda, Lambda] (Lemma 3.2, Chapter 3)
    Stated without proof; the text says the proof is part of the proof of Theorem 1.4 in [Dip+24c] by the same research group and is not repeated. Used in Lemma 3.3 to derive the integral estimate (3.21) needed for Theorem 3.1.
  • standard math Explicit fractional torsion solution for balls: (-Delta)^s psi_B = 1 with psi_B = gamma_{n,s} (rho^2 - |x-x_0|^2)^s_+
    Used to build the antisymmetric barrier in Theorem 4.3 and in the ellipsoid computation (Section 3.7); standard results from [Get61; BGR10; Gar19].
  • standard math Hypergeometric function identities used in the proof of Theorem 4.3 (Lemmas 4.1 to 4.3), e.g., [Olv+, Eqs. 10.22.19, 10.22.56, 15.4.20, 15.5.1, 15.8.1]
    Used to compute (-Delta)^s of x_1 psi_{B_rho} outside the ball and to prove the sign inequalities in Lemma 4.3; standard special function identities.
  • standard math Fractional Faber-Krahn inequality lambda_1(A) >= kappa_{n,s} |A|^{-2s/n} (equation (4.15))
    Used to guarantee condition (4.18)/(3.13) for small rho, citing [FJ15; YY13b; BCV20]. Standard in the fractional Laplacian literature.
  • domain assumption Regularity and geometric assumptions on the domains: C^1 or C^2 boundary, uniform interior sphere condition, Minkowski structure Omega = G + B_R, C^alpha boundary regularity for the improved exponent
    Assumptions of Theorems 3.1, 3.3, 3.9, and 4.1, using [Fra00] for the moving-plane geometric facts and [GT01] for distance function regularity. The central results hold only under these hypotheses.
  • domain assumption Structural regularity u/delta^s in C^1(Omega) (condition (4.3)) in Theorem 4.1
    Needed for the Serrin corner point lemma step in Lemma 4.1 case (ii). The thesis notes it is automatic for smooth domains and smooth f via [Gru14; Gru15; RS16b], but it is an extra structural assumption relative to the weak formulation.

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Cite this review

Pith. "Pith review of Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates." pith.science (2026). https://pith.science/paper/QPLSFC5G

@misc{pith2026250709219,
  author       = {Pith},
  title        = {Pith review of: Stability results for nonlocal Serrin-type problems, antisymmetric Harnack inequalities, and geometric estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPLSFC5G}},
  note         = {Machine review of arXiv:2507.09219}
}
read the original abstract

In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE). This thesis is split into three parts. In the first part, we study two overdetermined problems, namely Serrin's problem and the parallel surface problem, driven by the fractional Laplacian. In the second part, we study the Harnack inequality for solutions to nonlocal PDE which are antisymmetric, that is, they have an odd symmetry with respect to reflections across some hyperplane. This topic has a strong motivation coming from proving quantitative stability estimates for nonlocal overdetermined problems. In the third part, we prove several geometric identities and inequalities involving the fractional mean curvature.

Figures

Figures reproduced from arXiv: 2507.09219 by the authors.

Figure 2.1
Figure 2.1. Plot of 𝑓 (𝜀) (𝑥) for 𝜀 = 0.2, 0.4, 0.6, 0.8. Observe that the family of functions in the proof of Corollary 2.2 does not violate the classical strong maximum principle. In our next result, Corollary 2.3, we use Theorem 2.1 to provide a counter-example to the classical strong maximum principle for antisymmetric 𝑠-harmonic functions. This is slightly more delicate to construct than the counter-example in Corollary 2.… view at source ↗
Figure 2.2
Figure 2.2. Plot of 𝑓 as in the proof of Corollary 2.3. 2.3 A Hopf lemma In this section, we prove Proposition 2.5. The main step in the proof is the construction of the following barrier. Lemma 2.1. Suppose that 𝑐 ∈ 𝐿 ∞(𝐵 + 2 ). Then there exists an 𝑥1-antisymmetric function 𝜑 ∈ 𝐶 ∞(ℝ𝑛 ) such that    (−Δ) 𝑠𝜑 + 𝑐𝜑 ⩽ 0 in 𝐵 + 2 \ 𝐵1/2 (𝑒1), 𝜑 = 0 in ℝ𝑛 + \ 𝐵 + 2 , 𝜑 ⩽ 1 in 𝐵1/2 (𝑒1), 𝜕1𝜑(0) > 0. Proof. Let 𝜁 be a sm… view at source ↗
Figure 3
Figure 3. (roughly speaking, the example is obtained by a very small and localized [PITH_FULL_IMAGE:figures/full_fig_p065_3.png] view at source ↗
Figures from the paper (2 more)
Figure 3.1
Figure 3.1. Figure 3.1: Geometry of Ω as described in Section 3.5. Using the antisymmetry of 𝑣, we may rewrite this as 𝑣(𝑥) = ∫ ℝ𝑛 + \𝐵 + 1  1 − |𝑥| 2 |𝑦| 2 − 1 𝑠  1 |𝑥 − 𝑦| 𝑛 − 1 |𝑥 ′ − 𝑦| 𝑛  𝑣(𝑦) 𝑑𝑦 = ∫ Ω+\𝐵 + 1  1 − |𝑥| 2 |𝑦| 2 − 1 𝑠  1 |𝑥 − 𝑦| 𝑛 − 1 |𝑥 ′ − 𝑦| 𝑛  𝑢(𝑦) 𝑑𝑦, where 𝑥…
Figure 3.2
Figure 3.2. Figure 3.2: Diagram of Ω𝜀 in Theorem 3.6. in (0, 1/2) × (−3/2, −1/2), let 𝜂 ∈ 𝐶 ∞ 0 (ℝ) be such that    𝜂(𝜏) = 2𝜏 in (−1/4, 1/4), 𝜂 = 0 in ℝ \ (−1, 1), 𝜂(−𝜏) = −𝜂(𝜏) in ℝ, |𝜂| ⩽ 1 in ℝ. Let also 𝜂𝜀(𝜏) := 𝜀𝜂 𝜏 − 𝜀 1− 1 𝛼 𝜀 1/𝛼 ! and 𝜓𝜀(𝜏) := − √ 1 − 𝜏 2 − 𝜂𝜀(𝜏) a…

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