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Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension

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Pith's one-line read Nonlocal minimal surfaces are generically unique and smooth

desk verdict Generic uniqueness and regularity for nonlocal minimal surfaces — both results are new and the proofs hold up read the letter →

arxiv 2607.07154 v1 pith:EFVTLUKR submitted 2026-07-08 math.AP

Serena Dipierro , Xavier Fernández-Real , Enrico Valdinoci This is my paper
classification math.AP MSC 53A1035R11
keywords nonlocalminimalsurfacesfractionalperimetergenericuniquenessregularityPlateauproblemblow-upanalysiscone-splittingsubmodularity
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

The paper studies the fractional Plateau problem: given a region outside a domain, find the set inside that minimizes a nonlocal perimeter functional measuring long-range interactions across all of space. For fixed boundary data, such minimizers can be non-unique and can develop singularities. The authors prove two generic results. First, along any strictly increasing one-parameter family of exterior data, nonuniqueness occurs for at most countably many parameter values. Second, arbitrarily small perturbations of the exterior datum exist for which the minimizer is not only unique but also smooth in one dimension more than the general regularity theory guarantees. The mechanism is a submodularity property of the nonlocal perimeter, which forces any two minimizers with nested exterior data to be nested themselves. This nesting creates pairwise disjoint 'gap sets' of positive measure whenever nonuniqueness occurs, and since only countably many such disjoint sets can fit in a bounded domain, nonuniqueness is countable. For regularity, the authors construct a monotone family of perturbed data, show that singular points in space-time form the graph of a Lipschitz function, and use blow-up analysis with cone-splitting to rule out singularities in the critical-plus-one dimension.

What carries the argument

Submodularity of the fractional perimeter (Lemma 2.1), weak maximum principle for nonlocal minimal surfaces (Corollary 2.2), monotone families of exterior data with quantitative separation (Lemma 3.1), blow-up to s-minimal cones with cone-splitting (Lemma 3.5, Proposition 3.10), and the strict maximum principle for touching nonlocal minimal surfaces from the cited work in [18].

What would settle it

A pair of distinct s-minimal sets with the same exterior datum that are not nested (neither contains the other) would break the weak maximum principle and collapse the uniqueness argument. Alternatively, a singular s-minimal cone in dimension n*_s + 1 that is invariant along some direction without being a halfspace would break the cone-splitting contradiction and allow singularities to persist generically.

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Extended reading notes

Core claim

The central discovery is that the nonlocal perimeter satisfies a submodularity inequality (Per_s(E∩F) + Per_s(E∪F) ≤ Per_s(E) + Per_s(F)) that forces nested minimizers to be genuinely nested, and this combinatorial rigidity of the minimizer lattice is strong enough to make both uniqueness and improved regularity generic properties. The weak maximum principle for nonlocal minimal surfaces, which says that if two minimizers agree outside the domain then one must contain the other, is the load-bearing structural fact from which both theorems follow.

Load-bearing premise

The generic regularity proof relies on a strict maximum principle for nonlocal minimal surfaces that is cited as forthcoming but not yet published; without it, the argument that rules out singular cones in the critical-plus-one dimension cannot close.

Editorial extensions

If this is right

  • For numerical computation of nonlocal minimal surfaces, generic uniqueness means that generic boundary data yield a well-posed problem, so standard descent methods should converge to the unique minimizer for almost any perturbed input.
  • The dimension-reduction framework here, combining a monotone parameter family with cone-splitting, is portable to other nonlocal variational problems where a submodularity or lattice structure is available, such as nonlocal obstacle problems or anisotropic nonlocal perimeters.
  • The contrast between generic interior regularity (this paper) and generic boundary stickiness (prior work by the same authors) suggests that the nonlocal Plateau problem has a split personality: generic data smooth the interior but worsen boundary behavior, and the two effects may be in tension.
  • The result provides a nonlocal analogue of the Hardt-Simon generic regularity theorem for classical area-minimizing hypersurfaces in dimension 8, situating nonlocal minimal surfaces within the same generic-regularity paradigm.

Reading between the lines

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

  • The countability of nonuniqueness parameters, combined with the measure-zero set of discontinuity times for L^1 convergence, suggests that for a random (Lebesgue-typical) parameter in the family, the minimizer is simultaneously unique, continuous in the parameter, and as regular as the theory allows — a triple genericity that the paper does not explicitly package but which follows from its compone
  • The submodularity-based lattice argument for uniqueness is kernel-agnostic (as the authors note in Section 2.3), so one expects the same generic uniqueness to hold for tempered fractional kernels and other nonlocal perimeters with positive cross-interaction, potentially extending the regularity results as well if the corresponding regularity theory is available.
  • If the strict maximum principle for touching nonlocal minimal surfaces (the cited 'to appear' result) were to fail or require stronger hypotheses, the cone-splitting contradiction in Proposition 3.10 would break, and only the uniqueness theorem (Theorem 1.1) would survive — the two main results have different foundational dependencies.
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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. This paper establishes two generic results for nonlocal minimal surfaces (s-minimal sets) with prescribed exterior data. First, it proves a generic uniqueness theorem (Theorem 1.1): along any strictly increasing family of exterior data G(t), nonuniqueness of s-perimeter minimizers occurs for at most countably many parameters. The proof is self-contained, relying on a submodularity inequality for the fractional perimeter (Lemma 2.1), a weak maximum principle (Corollary 2.2), and a measure-theoretic countability argument. Second, it proves generic interior regularity (Theorems 1.2 and 1.5): for n = n*_s + 1, arbitrarily small L^1_loc perturbations of the exterior datum yield smooth (and unique) minimizers; for n >= n*_s + 2, the singular set dimension improves by one. The regularity proof constructs a monotone perturbation family, establishes a space-time separation estimate, and uses blow-up analysis with cone-splitting to rule out singular accumulation in the critical dimension. The framework also extends to general symmetric nonnegative kernels (Proposition 2.6).

Significance. The paper addresses two fundamental questions for the fractional Plateau problem: well-posedness (uniqueness) and regularity of minimizers. Both are known to fail for fixed exterior data. The generic uniqueness result (Theorem 1.1) is, to the best of my knowledge, the first of its kind for nonlocal minimal surfaces and is conceptually interesting because it relies on a genuinely nonlocal submodularity property that has no classical counterpart (as illustrated by the counterexamples in Figures 1-2). The generic regularity result is a nonlocal analogue of the Hardt-Simon theorem for classical area-minimizing hypersurfaces. The extension to general kernels (Section 2.3) broadens the applicability. The proofs are technically demanding but well-organized, adapting the Figalli-Ros-Oton-Serra / Fernandez-Real-Yu framework to the nonlocal setting. The results are clean, the statements are sharp, and the paper is likely to stimulate further work on generic properties of nonlocal minimal surfaces.

major comments (3)
  1. The generic regularity proof (Theorem 1.2, specifically Proposition 3.10) critically depends on the strict maximum principle for nonlocal minimal surfaces from reference [18] (Dipierro-Savin-Valdinoci, 'To appear in JEMS'). This is invoked in Lemma 3.5 and twice in Proposition 3.10 (for the cases t_k > t_0 and t_k < t_0) to conclude D = C from D ⊆ C with 0 ∈ ∂D ∩ ∂C. Without this result, the cone-splitting contradiction fails and S* = ∅ cannot be established. The result is natural and authored by a subset of the current authors, but since it is not yet published, the paper's central regularity claim is contingent on an external result that readers cannot independently verify from this manuscript alone. The authors should explicitly state this dependency in the introduction or at the statement of Theorem 1.2, and confirm that [18] is accepted (not merely submitted). This is a transparency
  2. concern, not a correctness concern: the argument is sound conditional on [18].
  3. In the proof of Lemma 3.7, the passage from 'C is a halfspace' (via uniform density estimates, [5, Theorem 4.1]) to '∂F_k ∩ B_{1/2} contains no singular points' (via improvement of flatness, [5, Theorem 6.1]) is stated very briefly. The logic is: if the blow-up cone C is a halfspace, then F_k ∩ B_1 lies close to a halfspace for large k, and improvement of flatness upgrades this to full regularity in B_{1/2}. This is standard but the reader must reconstruct the epsilon-delta relationship between the closeness (from [5, Corollary 4.4(ii)]) and the flatness threshold (from [5, Theorem 6.1]). A sentence spelling out this logical chain would strengthen the proof.
minor comments (5)
  1. Definition 1.3 introduces the Gaussian measure γ for measuring perturbation size, with a footnote saying other metrics are possible. The choice of Gaussian measure is somewhat unusual for measuring set perturbations in this context. It would help to briefly explain why this choice is natural or convenient (e.g., it weights all of R^n without requiring a cutoff, making the condition self-contained).
  2. The critical dimension n*_s is defined in (1.1) as the maximum m such that every s-minimal cone in R^m is a halfspace. The paper notes n*_s >= 2 for all s and n*_s >= 7 for s close to 1. It would be useful to also state the known upper bound (n*_s < 8, since the Simons cone provides a singular cone in R^8 for s close to 1) to give the reader a complete picture of where the dimensional improvement in Theorem 1.2 is operative.
  3. In the proof of Theorem 3.14 (end of Section 3.4), the argument that G(t) strictly increasing is 'harmless' because otherwise A_τ = R^n is somewhat terse. The iterative argument (B_{R_k} ⊂ A_τ with R_{k+1} = λR_k + r, λ > 1) is correct but could be stated more explicitly, perhaps as a separate remark or lemma, since it is used to justify the strict monotonicity assumption throughout Section 3.
  4. The paper states at the end of Section 1.2 that 'boundary regularity is not generic,' contrasting with [17]. This is an interesting point but is made only in passing. A brief elaboration (1-2 sentences) on why interior and boundary regularity behave differently in the generic setting would help the reader appreciate this contrast.
  5. Minor typographical issues: (i) In the abstract, 's-minimal sets' should use consistent formatting with the body. (ii) On page 2, 'for all t_1, t_2 ∈ I with t_1 < t_2' could benefit from a line break before the condition for readability. (iii) The reference [11] lists the year as 2026, which should be verified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found. The derivation chain is self-contained for uniqueness and depends on independent external results for regularity.

full rationale

The paper is a pure mathematics derivation with two main results. **Theorem 1.1 (generic uniqueness)** is entirely self-contained: Lemma 2.1 (submodularity) is proved in-line from a pointwise inequality on characteristic functions; Corollary 2.2 (weak maximum principle) follows from Lemma 2.1 and the definition of s-minimality; Proposition 2.3 (inclusion principle) follows from Corollary 2.2; and Theorem 1.1's proof constructs pairwise disjoint positive-measure sets S(t*) in the finite-measure domain Ω, concluding countability by elementary measure theory. No step reduces to its own inputs. **Theorems 1.2 and 1.5 (generic regularity)** depend on several external results: the monotonicity formula and improvement of flatness from Caffarelli-Roquejoffre-Savin [5], the gradient estimate for nonlocal minimal graphs from Cabré-Cozzi [4], the strict maximum principle from Dipierro-Savin-Valdinoci [18] (accepted to JEMS), and abstract GMT dimension-reduction lemmas from Figalli-Ros-Oton-Serra [24]. The self-citation [18] is load-bearing (used in Proposition 3.10 to conclude D=C from D⊆C with touching boundaries), but [18] is an independent prior result — the nonlocal strong maximum principle — that does not depend on the present paper's conclusions. The perturbation construction (3.1)-(3.2) provides a monotone family satisfying Theorem 1.1's hypotheses; the regularity conclusion requires substantial additional argument (blow-up analysis, cone-splitting, dimension reduction) and is not forced by the construction. No fitted parameters, no definitional circularity, no self-citation chain where the cited result depends on the present paper.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

This is a pure mathematics paper. No free parameters are fitted, no new physical entities are postulated, and no ad hoc constructs are introduced. The axioms are standard results from the nonlocal minimal surface literature ([5, 8, 18, 24, 25]) or standard geometric measure theory. The critical dimension n*_s is a defined quantity, not a postulated entity. The perturbation family G(t) is explicitly constructed (equations 3.1-3.2), not postulated.

assumptions (9)
  • domain assumption Existence of s-minimal sets for admissible exterior data of finite s-perimeter
    Assumed throughout; see statement before Theorem 1.1 and Proposition 2.5 for the general-kernel version. Standard in the theory of nonlocal minimal surfaces [5].
  • domain assumption Compactness of minimizers (Definition 2.4): L^1 convergence of characteristic functions along subsequences with lower semicontinuity of perimeter
    Invoked in Proposition 2.5 to prove existence of minimal/maximal minimizers. Verified for fractional kernels in Lemma 2.7 via W^{s,1} compact embedding.
  • domain assumption Strict maximum principle for nonlocal minimal surfaces: if two s-minimal surfaces touch at an interior point with one contained in the other, they coincide
    Cited from [18] (Dipierro-Savin-Valdinoci, to appear in JEMS). Used in Lemma 3.5 and Proposition 3.10 to conclude D = C when cones touch. Load-bearing for the regularity proof.
  • domain assumption Improvement of flatness for s-minimal surfaces [5, Theorem 6.1]
    Used at the end of Lemma 3.7 to pass from cone being a halfspace to finite-scale regularity, and in Proposition 3.10 to derive contradictions.
  • standard math Monotonicity formula W^s(E,x,r) with properties (P1)-(P4) from [5, Section 8]
    Recalled in §3.3. Properties: monotone in r, constant iff cone, scaling invariance, convergence under L^1_loc convergence. Used throughout the blow-up analysis.
  • standard math Abstract GMT dimension-reduction results: Lemma 3.12 [24, Prop 7.3] and Lemma 3.15 [24, Cor 7.8]
    Used in Proposition 3.13 and Theorem 3.14 to convert cone-splitting conclusions into Hausdorff dimension bounds on singular sets.
  • domain assumption Regularity of nonlocal minimal graphs: s-minimal cones that are graphs are halfspaces [4]
    Cited from Cabré-Cozzi. Used in Lemma 3.5 to conclude that a nonlocal minimal cone with graphical structure and a monotonicity inclusion must be a halfspace.
  • standard math Definition of critical dimension n*_s = max{m : every s-minimal cone in R^m is a halfspace}
    Stated in (1.1). Known: n*_s >= 2 for all s, n*_s >= 7 for s close to 1 [8, 30]. The exact value is unknown but the theorems hold for any s.
  • domain assumption Locally finite (n-1)-Minkowski content of ∂(G ∪ B_1)
    Assumption in Theorems 1.2 and 1.5. Used in Lemma 3.16 to show G(t) → G(0) in L^1_loc as t ↘ 0, which is needed to connect the perturbation family back to the original datum.

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Pith. "Pith review of Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension." pith.science (2026). https://pith.science/paper/EFVTLUKR

@misc{pith2026260707154,
  author       = {Pith},
  title        = {Pith review of: Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFVTLUKR}},
  note         = {Machine review of arXiv:2607.07154}
}
abstract

We study the fractional Plateau problem for $s$-minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur beyond the critical dimension. We first prove a generic uniqueness theorem: along any strictly increasing family of exterior data, nonuniqueness occurs for at most countably many parameters. We then show that one can make arbitrarily small perturbations for which the interior regularity theory improves by one dimension.

Figures

Figures reproduced from arXiv: 2607.07154 by the authors.

Figure 1
Figure 1. Examples of two sets E and F for which Per(E ∩F, Ω)+Per(E ∪F, Ω) = Per(E, Ω) + Per(F, Ω). Notice that taking x ∈ E \ F and y ∈ F \ E we have that (χE(x) − χE(y))(χF (x) − χF (y)) ≤ 0, in contrast with (2.2), showing that the analogue of Lemma 2.1 is false in the classical case. Corollary 2.2. Let E0, F0 ⊆ R n and suppose that E0 \ Ω ⊆ F0 \ Ω. Let E be s-minimal in Ω with E \ Ω = E0 \ Ω. Let F be s-minimal in Ω with … view at source ↗
Figure 2
Figure 2. Examples of two minimizers E and F of the classical perimeter in a planar, connected domain. Notice that neither of them is contained in the other, showing that the analogue of Corollary 2.2 is false in the classical case. Notice also that in this case Per(E ∩F, Ω) + Per(E ∪F, Ω) = Per(E, Ω) + Per(F, Ω), but taking x ∈ E\F and y ∈ F \E we have that (χE(x)−χE(y))(χF (x)−χF (y)) ≤ 0, in contrast with (2.2), showing th… view at source ↗

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Works this paper leans on

32 extracted references · 32 canonical work pages

  1. [18]

    ,A strict maximum principle for nonlocal minimal surfaces, To appear in J. Eur. Math. Soc., posted on 2025, DOI 10.4171/JEMS/1675

    work page doi:10.4171/jems/1675 2025

  2. [1]

    3-4, 377–403, DOI 10.1007/s00229-010-0399-4

    Luigi Ambrosio, Guido De Philippis, and Luca Martinazzi,Gamma-convergence of nonlocal perimeter function- als, Manuscripta Math.134(2011), no. 3-4, 377–403, DOI 10.1007/s00229-010-0399-4. MR2765717

    work page doi:10.1007/s00229-010-0399-4 2011

  3. [2]

    Jean Bourgain, Haim Brezis, and Petru Mironescu,Another look at Sobolev spaces, Optimal control and partial differential equations, IOS, Amsterdam, 2001, pp. 439–455. MR3586796

    work page 2001

  4. [3]

    4, 465–504, DOI 10.4171/ifb/447

    Claudia Bucur, Serena Dipierro, Luca Lombardini, and Enrico Valdinoci,Minimisers of a fractional semi- norm and nonlocal minimal surfaces, Interfaces Free Bound.22(2020), no. 4, 465–504, DOI 10.4171/ifb/447. MR4184583

    work page doi:10.4171/ifb/447 2020

  5. [4]

    J.168(2019), no

    Xavier Cabr´ e and Matteo Cozzi,A gradient estimate for nonlocal minimal graphs, Duke Math. J.168(2019), no. 5, 775–848, DOI 10.1215/00127094-2018-0052. MR3934589

    work page doi:10.1215/00127094-2018-0052 2019

  6. [5]

    Pure Appl

    Luis Caffarelli, Jean-Michel Roquejoffre, and Ovidiu Savin,Nonlocal minimal surfaces, Comm. Pure Appl. Math.63(2010), no. 9, 1111–1144, DOI 10.1002/cpa.20331. MR2675483

    work page doi:10.1002/cpa.20331 2010

  7. [6]

    Luis Caffarelli and Panagiotis Souganidis,Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal.195(2010), no. 1, 1–23, DOI 10.1007/s00205-008-0181-x. MR2564467

    work page doi:10.1007/s00205-008-0181-x 2010

  8. [7]

    Luis Caffarelli and Enrico Valdinoci,Uniform estimates and limiting arguments for nonlocal minimal sur- faces, Calc. Var. Partial Differential Equations41(2011), no. 1-2, 203–240, DOI 10.1007/s00526-010-0359-6. MR2782803

    work page doi:10.1007/s00526-010-0359-6 2011

Show all 32 references
  1. [8]

    Math.248(2013), 843–871, DOI 10.1016/j.aim.2013.08.007

    ,Regularity properties of nonlocal minimal surfaces via limiting arguments, Adv. Math.248(2013), 843–871, DOI 10.1016/j.aim.2013.08.007. MR3107529

    2013 doi

  2. [9]

    Antonin Chambolle, Massimiliano Morini, and Marcello Ponsiglione,Nonlocal curvature flows, Arch. Ration. Mech. Anal.218(2015), no. 3, 1263–1329, DOI 10.1007/s00205-015-0880-z. MR3401008

    2015 doi

  3. [10]

    Anal., posted on 2024, Paper No

    Otis Chodosh, Christos Mantoulidis, and Felix Schulze,Improved generic regularity of codimension-1 minimizing integral currents, Ars Inven. Anal., posted on 2024, Paper No. 3, 16, DOI 10.15781/z70n-ed29. MR4753940

    2024 doi

  4. [11]

    ,Generic regularity for minimizing hypersurfaces in dimensions 9 and 10, Publ. Math. Inst. Hautes ´Etudes Sci.143(2026), 143–188. MR5082540

    2026

  5. [12]

    Otis Chodosh, Christos Mantoulidis, Felix Schulze, and Zhihan Wang,Generic regularity for minimizing hy- persurfaces in dimension 11, arXiv e-prints (2025), arXiv:2506.12852

    2025 arXiv

  6. [13]

    Differential Geom.112(2019), no

    Eleonora Cinti, Joaquim Serra, and Enrico Valdinoci,Quantitative flatness results andBV-estimates for stable nonlocal minimal surfaces, J. Differential Geom.112(2019), no. 3, 447–504, DOI 10.4310/jdg/1563242471. MR3981295

    2019 doi

  7. [14]

    Matteo Carducci and Roberto Colombo,Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian, arXiv e-prints (2024), arXiv:2412.16066

    2024 arXiv

  8. [15]

    Juan D´ avila,On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15(2002), no. 4, 519–527, DOI 10.1007/s005260100135. MR1942130

    2002 doi

  9. [16]

    Serena Dipierro, Ovidiu Savin, and Enrico Valdinoci,Boundary behavior of nonlocal minimal surfaces, J. Funct. Anal.272(2017), no. 5, 1791–1851, DOI 10.1016/j.jfa.2016.11.016. MR3596708

    2017 doi

  10. [17]

    ,Nonlocal minimal graphs in the plane are generically sticky, Comm. Math. Phys.376(2020), no. 3, 2005–2063, DOI 10.1007/s00220-020-03771-8. MR4104542

    2020 doi

  11. [19]

    Xavier Fern´ andez-Real and Florian Gruen,Continuity up to the boundary for minimizers of the one-phase Bernoulli problem, Calc. Var. Partial Differential Equations64(2025), no. 5, Paper No. 166, 15, DOI 10.1007/s00526-025-03040-3. MR4912448

    2025 doi

  12. [20]

    Xavier Fern´ andez-Real and Xavier Ros-Oton,Free boundary regularity for almost every solution to the Signorini problem, Arch. Ration. Mech. Anal.240(2021), no. 1, 419–466, DOI 10.1007/s00205-021-01617-8. MR4228865

    2021 doi

  13. [21]

    Math.433(2023), Paper No

    Xavier Fern´ andez-Real and Clara Torres-Latorre,Generic regularity of free boundaries for the thin obstacle problem, Adv. Math.433(2023), Paper No. 109323, 29, DOI 10.1016/j.aim.2023.109323. MR4649881

    2023 doi

  14. [22]

    Xavier Fern´ andez-Real and Hui Yu,Generic properties in free boundary problems, To appear in Amer. J. Math., posted on 2023, DOI 10.48550/arXiv.2308.13209

    2023 doi arXiv

  15. [23]

    ,Linearized equation and generic regularity in the Alt-Caffarelli problem, arXiv e-prints (2025), arXiv:2510.18330

    2025

  16. [24]

    Alessio Figalli, Xavier Ros-Oton, and Joaquim Serra,Generic regularity of free boundaries for the obstacle prob- lem, Publ. Math. Inst. Hautes ´Etudes Sci.132(2020), 181–292, DOI 10.1007/s10240-020-00119-9. MR4179834

    2020 doi

  17. [25]

    Reine Angew

    Alessio Figalli and Enrico Valdinoci,Regularity and Bernstein-type results for nonlocal minimal surfaces, J. Reine Angew. Math.729(2017), 263–273, DOI 10.1515/crelle-2015-0006. MR3680376

    2017 doi

  18. [26]

    Reine Angew

    Robert Hardt and Leon Simon,Area minimizing hypersurfaces with isolated singularities, J. Reine Angew. Math.362(1985), 102–129, DOI 10.1515/crll.1985.362.102. MR809969 NONLOCAL MINIMAL SURFACES ARE GENERICALLY UNIQUE AND SMOOTH 23

    1985 doi

  19. [27]

    R´ egis Monneau,On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal.13 (2003), no. 2, 359–389, DOI 10.1007/BF02930701. MR1967031

    2003 doi

  20. [28]

    Giampiero Palatucci, Ovidiu Savin, and Enrico Valdinoci,Local and global minimizers for a variational energy involving a fractional norm, Ann. Mat. Pura Appl. (4)192(2013), no. 4, 673–718, DOI 10.1007/s10231-011- 0243-9. MR3081641

    2013 doi

  21. [29]

    Ovidiu Savin and Enrico Valdinoci, Γ-convergence for nonlocal phase transitions, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire29(2012), no. 4, 479–500, DOI 10.1016/j.anihpc.2012.01.006. MR2948285

    2012 doi

  22. [30]

    ,Regularity of nonlocal minimal cones in dimension 2, Calc. Var. Partial Differential Equations48 (2013), no. 1-2, 33–39, DOI 10.1007/s00526-012-0539-7. MR3090533

    2013 doi

  23. [31]

    Schaeffer,An example of generic regularity for a non-linear elliptic equation, Arch

    David G. Schaeffer,An example of generic regularity for a non-linear elliptic equation, Arch. Rational Mech. Anal.57(1975), 134–141, DOI 10.1007/BF00248415. MR387810

    1975 doi

  24. [32]

    Nathan Smale,Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional mani- folds, Comm. Anal. Geom.1(1993), no. 2, 217–228, DOI 10.4310/CAG.1993.v1.n2.a2. MR1243523 Serena Dipierro, Department of Mathematics and Statistics, University of Western ...

    1993 doi

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