{"id":"f0d6b9ee-1ef7-4def-92b0-e6d6ad55a9b0","arxiv_id":"2506.23188","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every irregular boundary point for fractional (s,p)-Laplace Dirichlet problems is either semiregular or strongly irregular, never both or neither.","lead":"This paper shows that irregular boundary points for nonlocal nonlinear Dirichlet problems come in exactly two types: semiregular, where solutions have definite limits, and strongly irregular, where they oscillate. It also determines how this classification depends on the fractional order s and the exponent p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The trichotomy is only as secure as the Kellogg property quoted from the authors' preprint [6]; if [6, Theorem 1.5] fails, Case 2 of Theorem 1.1 has no way to produce the required regular boundary points.","rationale":"I reviewed Section 6 with the central claim in mind. In Case 1 the argument is complete: the newly proved removability theorem extends H_g across a zero-capacity hole, giving condition (I), and the finite Wiener integral shows the point is irregular. In Case 2 the exterior-ball construction of regular points is standard, but the first subcase relies on the Kellogg property, i.e. on the quoted capacity-zero conclusion for the set of irregular points. Thus the trichotomy is directly hostage to [6, Theorem 1.5]. I also checked Lemma 6.3; its typeset proof contains an apparent typo, since a small ball around a boundary point cannot satisfy B\\subset\\Omega, and the stated density of regular points fails for the punctured ball. However, Lemma 6.3 is not needed for Theorem 1.1, and its statement can plausibly be repaired by invoking the same Case 2 construction. I found no internal inconsistency in the main proof conditional on the external inputs. The removability theorem has a clear proof and the geometric part of Case 2 is elementary, so the residual risk is concentrated in the unverified Kellogg property. This is exactly the reader's weakest assumption, and I agree with that assessment. Since reliance on a prior theorem is normal in mathematics and I found no evidence of an actual flaw, the ACCEPT verdict should stand.","tokens_in":21862,"tokens_out":20860,"duration_ms":239495,"concrete_test":"Obtain [6] and verify that its Theorem 1.5 is proved for the same kernel class (1.1), the same Sobolev capacity C_{s,p} of Definition 3.2, and the full range 1<p<\\infty, 0<s<1 with sp\\le n, and that the proof does not invoke the trichotomy or any result of the present paper. If the theorem holds, the concern is resolved; if it has a gap, determine whether a weaker density of regular points can be proved directly from the Wiener criterion, and if not, the trichotomy should be treated as conditional on [6].","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 in Section 6 is sound conditional on two inputs: the self-contained removability theorem (Theorem 1.4) and the Kellogg property (Theorem 5.9), quoted from the unpublished preprint [6]. Case 1 works: the removability theorem gives condition (I), and the vanishing condenser capacity makes the Wiener integral finite, so the point is irregular and hence semiregular. Case 2 is the load-bearing step that depends on [6]: whenever C_{s,p}(B_j \\cap \\partial\\Omega)>0, the paper needs a regular boundary point in B_j. That conclusion is exactly the statement that the irregular set has capacity zero. Without [6, Theorem 1.5], the sequence {x_j} of regular points need not exist, and condition (II) cannot be established. The same external theorem is used in Lemma 6.3 and Corollary 5.10. The paper does not reprove or independently verify Kellogg, so the central trichotomy is no stronger than that quoted theorem. This is an external dependency rather than an internal inconsistency; I am not asserting a flaw in [6], but it is the least secure load-bearing assumption in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a boundary regularity classification for Dirichlet problems driven by nonlocal operators of fractional (s,p)-Laplacian type. It introduces two mutually exclusive classes of irregular boundary points: semiregular points, where limits of Sobolev solutions exist for every continuous admissible datum but do not necessarily match the datum, and strongly irregular points, where each datum is attained along some sequence. The main theorem (Theorem 1.1) asserts that every boundary point of a bounded open set is regular, semiregular, or strongly irregular, so that an irregular point cannot fail both existence and attainability. The proof combines a new removability result for solutions and supersolutions in the space V^{s,p} (Theorem 1.4) with the Kellogg property quoted from the authors' companion paper [6]. Sections 6 and 7 provide several characterizations of semiregular and strongly irregular points, including tests using the single function d_{x0}, and Section 8 determines precisely when semiregularity for one pair (s,p) implies semiregularity for another.","tokens_in":22058,"tokens_out":27166,"duration_ms":279433,"significance":"If the results are correct, this is a substantial contribution to the nonlocal nonlinear Dirichlet problem: the trichotomy is new even for the linear case p=2, and the characterizations reduce seemingly all-data properties to a single test function. The removability theorem is proved with a natural sharpness discussion and is a useful standalone tool. The paper is generally careful and well organized, and it gives explicit credit to the companion paper [6] for the Kellogg property. The main caveat is that the central trichotomy is not self-contained: it relies on the Kellogg property and on density of regular boundary points, both quoted from the unpublished preprint [6].","major_comments":[{"comment":"The proof of Theorem 1.1, Case 2, uses the Kellogg property (Theorem 5.9) to conclude that if C_{s,p}(B_j cap partial Omega) > 0, then B_j cap partial Omega contains a regular boundary point. This is exactly the statement that the irregular set has capacity zero, quoted from the authors' unpublished preprint [6, Theorem 1.5]. Lemma 6.3 additionally uses the density of regular boundary points from [6, Remark 10.5]. Since these external results are load-bearing for the trichotomy and are not proved or independently verified here, the manuscript is not self-contained. I am not claiming a flaw in [6], but the refereed paper should either include a proof of the needed consequences of the Kellogg property or clearly state the results as conditional on [6] and make the companion available to the referee.","section":"Section 6, proof of Theorem 1.1 and Theorem 5.9"},{"comment":"The proof of Lemma 6.3 appears to contain a false assertion and a typographical error. The sentence 'R is dense in partial Omega, which does not have any isolated points' is false in general: a punctured ball has an isolated boundary point. The expression 'x0 in R \\ partial Omega' also does not make sense. The lemma is true and can be proved from density of R in partial Omega, but the proof as written should be rewritten.","section":"Section 6, proof of Lemma 6.3"}],"minor_comments":[{"comment":"In the second displayed identity in (1.3), the set notation appears to contain a repeated '\\partial Omega' symbol; please clarify the intended set. The proof suggests the condition is that a small ball around x0 contains no exterior points and C_{s,p}(B(x0,r) cap partial Omega) = 0.","section":"Proposition 1.2, equation (1.3)"},{"comment":"In the typeset version, the overline denoting the closure of the regular set is missing in condition (d) of Theorem 6.1 and in the expression 'R \\ R' in Theorem 7.1(e). Please ensure the closure symbol is visible, since the mathematical meaning depends on it.","section":"Theorem 6.1(d) and Theorem 7.1(e)"}],"recommendation":"major_revision","confidential_remarks":"The editor should obtain the companion preprint [6] (arXiv:2406.05994) as part of the review, since the Kellogg property quoted from it is the key external input to Theorem 1.1. The manuscript appears mathematically sound conditional on [6] and on the removability result for bounded L-harmonic functions quoted from [25], but the central claim cannot be fully verified without access to those sources."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid paper. It proves a trichotomy for irregular boundary points of nonlocal nonlinear Dirichlet problems, and the result is new even for the linear fractional case p=2. The removability theorem for supersolutions with right-hand side is genuinely new, and the sharpness discussion in Remark 4.1 is honest. The proofs are detailed and the architecture is coherent. I read the main proof and found no derivation gap. The definitions of semiregular and strongly irregular are not repackaging the conclusion; Theorem 1.1 is a real dichotomy statement.\n\nThe best part is Section 4. The removability result is proved self-contained and its sharpness is addressed. The characterizations in Sections 6 and 7 are useful, and the universal sequence in Theorem 7.2 is a nice payoff. The s,p-dependence section is a clear extension of known capacity comparison results.\n\nSoft spots: the load-bearing quotation. Theorem 5.9, the Kellogg property, is quoted from the authors' own preprint [6], and it is used in Case 2 of Theorem 1.1 and again in Lemma 6.3. The stress-test note is correct: if [6, Theorem 1.5] fails, the trichotomy has no proof. That said, I do not see this as a fatal flaw. The dependence is explicit, the authors are not hiding it, and the Kellogg property is a known type of result for nonlocal equations. But since the whole trichotomy rests on it, an editor should require the authors either to reprove it in this paper or to point to a peer-reviewed version of [6]. I would not desk-reject over this; I would ask for the dependency to be cleaned up in revision.\n\nThe citation pattern is fine. The local predecessors are cited properly, and the novelty claim for fractional operators, including p=2, stands. The self-citation of [6] is not a problem here; it is the actual source of the quoted result. Minor point: Example 5.11's conclusion uses the trichotomy to identify the point as strongly irregular, but the text also notes the direct route through Theorem 7.1(e), so this is not circular.\n\nThis paper is for researchers in nonlinear potential theory and nonlocal PDEs who care about boundary regularity. It deserves a serious referee. I would accept it for peer review and recommend publication after the [6] dependency is settled.","headline":"A clean structural trichotomy for nonlocal Dirichlet problems, well proved and genuinely new, but the main risk is its load-bearing reliance on a Kellogg property quoted from the authors' own unpublished preprint.","tokens_in":22604,"tokens_out":1786,"would_cite":true,"duration_ms":19099,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","31C15","31C45","35J66"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a trichotomy for boundary points of nonlocal fractional (s,p)-Laplacian Dirichlet problems: every point is regular, semiregular, or strongly irregular.","keywords":["fractional p-Laplacian","nonlocal Dirichlet problem","boundary regularity","semiregular boundary point","strongly irregular boundary point","Sobolev capacity","Perron solution","Kellogg property"],"falsifier":"Find a bounded open set and a boundary point at which some continuous $V^{s,p}$ datum has no limit at the point and, for possibly different datum, no sequence inside the domain carries $H_g$ to the prescribed value; Theorem 1.1 says such a point cannot exist.","tokens_in":21615,"feed_emoji":"📐","tokens_out":17418,"duration_ms":141437,"temperature":0.7,"pith_summary":"Boundary points of Dirichlet problems for nonlocal fractional $(s,p)$-Laplacian type equations need not behave classically: the solution may fail to converge to the prescribed boundary value. This paper proves that every irregular boundary point nevertheless belongs to one of two disjoint classes. A semiregular point is one where every solution has a limit at the point, but for some data that limit is not the prescribed value; a strongly irregular point is one where every prescribed value is attained along some sequence, though limits can oscillate. The main theorem, the trichotomy, says that no boundary point can be irregular and fail both behaviours. The proof rests on the Kellogg property (irregular points form a zero-capacity set) and on a new removability theorem for solutions and supersolutions in the fractional Sobolev space $V^{s,p}$.","feed_headline":"Every boundary point is regular, semiregular, or strongly irregular","feed_subtitle":"A single test function now tells which boundary values are attained for fractional p-Laplacian Dirichlet problems","key_machinery":"Two tools carry the argument. The Kellogg property, quoted from the authors' earlier work, says the set of irregular boundary points has zero Sobolev capacity $C_{s,p}$, the capacity defined by minimizing $\\|u\\|^p_{W^{s,p}(\\mathbb{R}^n)}$ among functions that are $1$ on the set in question. The new removability theorem says that if $E \\subset \\Omega$ is relatively closed with $C_{s,p}(E)=0$, then any solution or supersolution of $Lu=f$ in $\\Omega \\setminus E$ belonging to $V^{s,p}$ extends to a solution or supersolution in all of $\\Omega$. In the proof of the trichotomy, the removability theorem settles the case where some small ball around $x_0$ has complement of zero capacity, forcing all limits to exist; the Kellogg property and the density of regular points settle the opposite case, producing a sequence along which every continuous boundary value is attained. The test function $d_{x_0}(x)=\\min\\{1, |x-x_0|\\}$ then reduces all three classes to one Dirichlet problem.","core_discovery":"On the paper's own terms, the central claim is Theorem 1.1: for a bounded open set $\\Omega$ and a kernel $k$ satisfying the fractional ellipticity bounds, each $x_0 \\in \\partial\\Omega$ is exactly one of regular, semiregular, or strongly irregular. Regularity means that $\\lim_{\\Omega\\ni x \\to x_0} H_g(x) = g(x_0)$ for every $g \\in V^{s,p}(\\Omega) \\cap C(\\mathbb{R}^n)$; the two irregular classes are defined by the two conditions whose conjunction gives regularity, namely that every such limit exists (I) and that for every such $g$ some sequence $y_j \\to x_0$ inside $\\Omega$ carries $H_g(y_j)$ to $g(x_0)$ (II). The theorem's content is that an irregular point cannot fail both (I) and (II). The proof also shows that the set of semiregular points is the largest subset of $\\Omega^c$ of zero Sobolev capacity for which $\\Omega \\cup S$ is open, and that the trichotomy can be read off from the single function $d_{x_0}(x)=\\min\\{1, |x-x_0|\\}$: regular points have $\\lim H_{d_{x_0}}=0$, semiregular points have $\\liminf H_{d_{x_0}}>0$, and strongly irregular points have no limit.","pith_inferences":["Because the proof uses only the Kellogg property and removability, the same trichotomy should be provable for other nonlocal operators for which those two ingredients are available, for example kernels with more general growth or Orlicz-type nonlinearities.","The one-function test suggests a numerical route to classify boundary points in concrete domains: solve the Dirichlet problem for $d_{x_0}$ and inspect the limit at each boundary point, which can be checked against the explicit punctured-ball example.","The removability theorem for supersolutions with a right-hand side may let future constructions first solve in a domain with a small zero-capacity set removed and then extend the solution automatically, effectively simplifying existence proofs.","The $p=2$ proof of the universal sequence for unbounded data uses linearity and a weak Harnack estimate; extending the same conclusion to $p\\ne2$ would require a nonlinear analogue of that estimate, which the paper does not provide."],"forward_implications":["A boundary point can be classified by solving one Dirichlet problem: $\\lim H_{d_{x_0}}=0$ means regular, $\\liminf H_{d_{x_0}}>0$ means semiregular, and nonexistence of the limit means strongly irregular (Theorem 1.3).","The semiregular set has zero $C_{s,p}$ capacity and is removable, and semiregularity is a local property of the domain.","At a strongly irregular point there is a single universal sequence $y_j \\to x_0$ that recovers $g(x_0)$ for all bounded continuous data at once; when $p=2$ the same sequence works for unbounded data.","Semiregularity is monotone in the parameters in an exact way: for $0<s_j\\le 1<p_j$, the implication 'semiregular for $(s_1,p_1)$ implies semiregular for $(s_2,p_2)$' holds for all domains precisely in the three mutually disjoint cases of Proposition 1.5.","For $sp>n$ every boundary point is regular, while for $sp\\le n$ both semiregular points (punctured ball) and strongly irregular points (Example 5.11) occur."],"supporting_citations":[{"why":"Supplies the Kellogg property that the irregular boundary set has zero $C_{s,p}$ capacity, used in Case 2 of the trichotomy proof.","marker":"[6, Theorem 1.5]"},{"why":"Supplies the Wiener criterion characterizing regular boundary points, used to locate regular points in the two cases and in Example 5.11.","marker":"[22, Theorem 1.1 and its proof]"},{"why":"Supplies removability for bounded $L$-harmonic functions, converting the Sobolev trichotomy into the Perron-solution characterizations.","marker":"[25, Theorem 1.1]"},{"why":"Provides existence and uniqueness of the Sobolev solution $H_g$ whose boundary limits define the three classes.","marker":"[26, Theorem 9]"},{"why":"Together with [26, Theorem 9], grounds the Dirichlet problem in $V^{s,p}(\\Omega)$ used throughout.","marker":"[24, Theorem 4.9]"},{"why":"Gives the inclusion relations between zero-capacity sets for different $(s,p)$ that drive Proposition 1.5 on semiregularity.","marker":"[2, Theorem 5.5]"},{"why":"Provides the local $p$-Laplace trichotomy and the $s=1$ characterization of semiregular points used in Section 8.","marker":"[5, Theorem 3.3]"}],"fun_headline_variants":["Every boundary point is regular, semiregular, or strongly irregular","Fractional Laplacian: boundary points split into trichotomy","One test function decides boundary regularity class","Nonlocal Dirichlet: a trichotomy for boundary points","Semiregular vs strongly irregular: a sharp split"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the Kellogg property quoted from the authors' earlier paper, that the set of irregular boundary points has zero Sobolev capacity, together with the related density of regular boundary points; the proof of the trichotomy uses both without reproving them.","fun_headline_variants_meta":{"raw":{"variants":["Every boundary point is regular, semiregular, or strongly irregular","Fractional Laplacian: boundary points split into trichotomy","One test function decides boundary regularity class","Nonlocal Dirichlet: a trichotomy for boundary points","Semiregular vs strongly irregular: a sharp split"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":1264,"prompt_tokens":982,"completion_tokens":282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":598,"completion_tokens_details":{"reasoning_tokens":201}},"tokens_in":598,"tokens_out":282,"duration_ms":26364,"temperature":1.0,"reasoning_tokens":201,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:48:01.898938+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a bounded open set and a boundary point at which some continuous $V^{s,p}$ datum has no limit at the point and, for possibly different datum, no sequence inside the domain carries $H_g$ to the prescribed value; Theorem 1.1 says such a point cannot exist.","supporting_citations":[],"review_version":1}