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Stability estimates in $H^1_0$ for solutions of elliptic equations in varying domains

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José M. Arrieta, Gerassimos Barbatis

Analysis of PDEs · Submitted 9 May 2012

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

The authors' main claim Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(\Omega\cup\phi(\Omega))}$ in terms of certain measure of vicinity of $\phi$ to the identity map.

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arXiv ID1205.2027 (version 1)
Pith IDAF6G5IMW, Pith's permanent record of this paper and its review
CitationGerassimos Barbatis, Jos\'e M. Arrieta. "Stability estimates in $H^1_0$ for solutions of elliptic equations in varying domains." arXiv:1205.2027 (2012). https://pith.science/paper/AF6G5IMW
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,014 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.56 · human participants 0.01 · controlled comparison 0.16 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz map $\phi$. We consider the solutions $u$ and $\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\in L^2(\Omega\cup\phi(\Omega))$. Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(\Omega\cup\phi(\Omega))}$ in terms of certain measure of vicinity of $\phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $\phi(\Omega)$ and $\Omega$, that is $|\phi(\Omega)\triangle \Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.

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 second-order uniformly elliptic operators subject to Dirichlet boundary conditions. background 0.94 0.29 0.02 asserted 0.82
s1 Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz map $\phi$. background 0.54 0.38 0.02 asserted 0.99
s2 We consider the solutions $u$ and $\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\in L^2(\Omega\cup\phi(\Omega))$. background 0.41 0.78 0.03 asserted 0.95
s3 Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(\Omega\cup\phi(\Omega))}$ in terms of certain measure of vicinity of $\phi$ to the identity map. result 0.92 0.93 0.81 asserted 0.94
s4 For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $\phi(\Omega)$ and $\Omega$, that is $|\phi(\Omega)\triangle \Omega|$. result 0.95 0.91 0.44 asserted 0.97
s5 We provide an example which shows that the estimates obtained are in a certain sense sharp. result 0.78 0.95 0.08 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 consider second-order uniformly elliptic operators subject to Dirichlet boundary conditions. Such operators are considered on a bounded domain $\Omega$ and on the domain $\phi(\Omega)$ resulting from $\Omega$ by means of a bi-Lipschitz map $\phi$. We consider the solutions $u$ and $\tilde u$ of the corresponding elliptic equations with the same right-hand side $f\in L^2(\Omega\cup\phi(\Omega))$. Under certain assumptions we estimate the difference $\|\nabla\tilde u-\nabla u\|_{L^2(\Omega\cup\phi(\Omega))}$ in terms of certain measure of vicinity of $\phi$ to the identity map. For domains within a certain class this provides estimates in terms of the Lebesgue measure of the symmetric difference of $\phi(\Omega)$ and $\Omega$, that is $|\phi(\Omega)\triangle \Omega|$. We provide an example which shows that the estimates obtained are in a certain sense sharp.

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