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Asymptotic flatness of Morrey extremals

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Ryan Hynd, Francis Seuffert

Analysis of PDEs · Submitted 16 May 2019

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

The authors' main claim In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$.

how it is stated · asserted 1.00 read the record →

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arXiv ID1905.07060 (version 5)
Pith IDWWODRLIQ, Pith's permanent record of this paper and its review
CitationFrancis Seuffert, Ryan Hynd. "Asymptotic flatness of Morrey extremals." arXiv:1905.07060 (2019). https://pith.science/paper/WWODRLIQ
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,835 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.31 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v5. Results and the main claim are underlined; .

We study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-\Delta_pu=c(\delta_{x_0}-\delta_{y_0})$ for some $c\in \mathbb{R}$ and distinct $x_0,y_0\in \mathbb{R}^n$. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded $p$-harmonic functions on exterior domains of $\mathbb{R}^n$ for $p>n$.

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 study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. contribution 0.95 0.91 0.07 asserted 0.69
s1 In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$. result 0.99 0.95 0.75 asserted 1.00
s2 To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-\Delta_pu=c(\delta_{x_0}-\delta_{y_0})$ for some $c\in \mathbb{R}$ and distinct $x_0,y_0\in \mathbb{R}^n$. method 1.00 0.60 0.06 asserted 1.00
s3 More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded $p$-harmonic functions on exterior domains of $\mathbb{R}^n$ for $p>n$. contribution 0.82 0.88 0.23 asserted 0.99

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

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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 study the limiting behavior as $|x|\rightarrow \infty$ of extremal functions $u$ for Morrey's inequality on $\mathbb{R}^n$. In particular, we compute the limit of $u(x)$ as $|x|\rightarrow \infty$ and show $|x||Du(x)|$ tends to $0$. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form $-\Delta_pu=c(\delta_{x_0}-\delta_{y_0})$ for some $c\in \mathbb{R}$ and distinct $x_0,y_0\in \mathbb{R}^n$. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded $p$-harmonic functions on exterior domains of $\mathbb{R}^n$ for $p>n$.

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