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Structure of shape derivatives around irregular domains and applications

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Jimmy Lamboley (IRMAR), Michel Pierre (IRMAR)

Optimization and Control · Submitted 19 Sep 2006

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

The authors' main claim This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal).

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arXiv IDmath/0609526 (version 1)
Pith IDJBHGLA7H, Pith's permanent record of this paper and its review
CitationJimmy Lamboley (IRMAR), Michel Pierre (IRMAR). "Structure of shape derivatives around irregular domains and applications." arXiv:math/0609526 (2006). https://pith.science/paper/JBHGLA7H
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,964 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

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 In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. contribution 0.99 0.86 0.15 asserted 0.98
s1 This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). result 0.85 0.94 0.43 asserted 1.00
s2 We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set. contribution 0.91 0.92 0.10 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · 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

In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivity of the shape derivative and we show it implies its continuity with respect to the uniform norm when the boundary is Lipschitz (this restriction is essentially optimal). We apply this idea to various cases including the perimeter-type functionals for convex and pseudo-convex shapes or the Dirichlet energy of an open set.

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