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PreprintCC BY 4.0arXiv

Sobolev--Poincaré inequalities for piecewise W2,p functions over general polytopic meshes

  • Michele Botti
  • Lorenzo Mascotto
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Abstract

We establish Sobolev--Poincaré inequalities for piecewise W2,p functions in two and three dimensions over families of general polytopic (including shape-regular simplicial and Cartesian) meshes, allowing for degenerate facets. The proposed inequalities feature control by either the broken Hessian or, more naturally for general fourth-order problems, the broken Laplacian. Important tools in their derivation are a Helmholtz decomposition in Banach--Sobolev spaces and Sobolev--Poincaré inequalities for piecewise W1,p functions; we avoid the use of any inverse inequality. We provide estimates with fairly general Lebesgue exponents. Applications to the stability analysis of nonconforming discretizations for second- and fourth-order, linear and nonlinear problems are exhibited.

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Cite

BibTeXRIS
BibTeX
@misc{botti2026sobolevpoincaré,
  title = {{Sobolev--Poincaré inequalities for piecewise $W^2,p$ functions over general polytopic meshes}},
  author = {Michele Botti and Lorenzo Mascotto},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06646},
  eprint = {2611.06646},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06646},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sobolev--Poincaré inequalities for piecewise W^2,p functions over general polytopic meshes
AU  - Michele Botti
AU  - Lorenzo Mascotto
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06646
UR  - https://arxiv.org/abs/2611.06646
AB  - We establish Sobolev--Poincaré inequalities for piecewise W^2,p functions in two and three dimensions over families of general polytopic (including shape-regular simplicial and Cartesian) meshes, allowing for degenerate facets. The proposed inequalities feature control by either the broken Hessian or, more naturally for general fourth-order problems, the broken Laplacian. Important tools in their derivation are a Helmholtz decomposition in Banach--Sobolev spaces and Sobolev--Poincaré inequalities for piecewise W^1,p functions; we avoid the use of any inverse inequality. We provide estimates with fairly general Lebesgue exponents. Applications to the stability analysis of nonconforming discretizations for second- and fourth-order, linear and nonlinear problems are exhibited.
ER  -