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A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

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Francois Hamel, Xavier Ros-Oton, Yannick Sire, Enrico Valdinoci

Analysis of PDEs · Submitted 7 Jul 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional.

how it is stated · asserted 1.00 read the record →

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arXiv ID1507.01830 (version 2)
Pith ID2VUDPEU7, Pith's permanent record of this paper and its review
CitationEnrico Valdinoci, Francois Hamel, Xavier Ros-Oton, Yannick Sire. "A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane." arXiv:1507.01830 (2015). https://pith.science/paper/2VUDPEU7
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The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,553 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.24 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. background 0.78 0.53 0.03 asserted 0.86
s1 Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. result 0.99 0.95 0.91 asserted 1.00
s2 The proof is based on a Liouville type approach. method 1.00 0.86 0.03 asserted 1.00
s3 A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions. contribution 0.87 0.94 0.05 asserted 0.89

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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not examined
  • full text
  • figures and tables
  • references and citing sentences
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  • P abstract
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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