Short-time heat flow and functions of bounded variation in RN
Miranda, Michele Jr1 ; Pallara, Diego1 ; Paronetto, Fabio1 ; Preunkert, Marc2
1 Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy
2 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany.
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 1, pp. 125-145.

On prouve une caractérisation des ensembles avec périmètre fini et des fonctions à variation bornée en termes du comportement du semi-groupe de la chaleur dans RN au voisinage de t=0. On prouve aussi un résultat plus précis pour les ensembles avec frontière assez régulière.

We prove a characterisation of sets with finite perimeter and BV functions in terms of the short time behaviour of the heat semigroup in RN. For sets with smooth boundary a more precise result is shown.

DOI : 10.5802/afst.1142
Miranda, Michele Jr 1 ; Pallara, Diego 1 ; Paronetto, Fabio 1 ; Preunkert, Marc 2

1 Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy
2 Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany. 
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Miranda, Michele Jr; Pallara, Diego; Paronetto, Fabio; Preunkert, Marc. Short-time heat flow and functions of bounded variation in $\mathbf{R}^N$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 1, pp. 125-145. doi : 10.5802/afst.1142. https://www.numdam.org/articles/10.5802/afst.1142/

[1] L. Ambrosio.— Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158, p. 227-260 (2004). | MR | Zbl

[2] L. Ambrosio, N. Fusco, D. Pallara.— Functions of Bounded Variation and Free Discontinuity problems, Oxford U. P., 2000. | MR | Zbl

[3] H. Brézis.— How to recognize constant functions. Connections with Sobolev spaces,Russian Math. Surveys 57, p. 693-708 (2002). | MR | Zbl

[4] J. Dávila.— On an open question about functions of bounded variation,Calc. Var. 15, p. 519-527 (2002). | MR | Zbl

[5] E. De Giorgi.— Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni,Ann. Mat. Pura Appl. (4) 36, p. 191-213 (1954). | MR | Zbl

[6] E. De Giorgi.— Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ric. di Mat.4, p. 95–113 (1955). | MR | Zbl

[7] P. Gilkey, M. van den Berg.— Heat content asymptotics of a Riemannian manifold with boundary, J. Funct. Anal. 120, p. 48-71 (1994). | MR | Zbl

[8] M. Ledoux.— Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math. 118, p. 485-510 (1994). | MR | Zbl

[9] E.H. Lieb, M. Loss.— Analysis, Second edition, Amer. Math. Soc., 2001 | MR | Zbl

[10] M. Miranda (Jr), D. Pallara, F. Paronetto, M. Preunkert.— Heat Semigroup and BV Functions on Riemannian Manifolds, forthcoming.

[11] K. Pietruska-Paluba.— Heat kernels on metric spaces and a characterisation of constant functions, Manuscripta Math. 115 , p. 389–399 (2004). | MR | Zbl

[12] M. Preunkert.— A semigroup version of the isoperimetric inequality, Semigroup Forum 68, p. 233-245 (2004). | MR | Zbl

[13] M. H. Taibleson.— On the theory of Lipschitz spaces of distributions on Euclidean n-spaces I, J. Math. Mech. 13, p. 407-479 (1964). | MR | Zbl

[14] H. Triebel.— Interpolation theory, function spaces, differential operators, North-Holland 1978. | MR | Zbl

[15] J. Wloka.— Partial Differential Equations, Cambridge U. P., 1987. | MR | Zbl

  • Agrachev, Andrei; Rizzi, Luca; Rossi, Tommaso Relative heat content asymptotics for sub-Riemannian manifolds, Analysis PDE, Volume 17 (2024) no. 9, p. 2997 | DOI:10.2140/apde.2024.17.2997
  • Chen, Huangxin; Dong, Piaopiao; Wang, Dong; Wang, Xiao-Ping A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems, Journal of Computational Physics, Volume 511 (2024), p. 113119 | DOI:10.1016/j.jcp.2024.113119
  • Garofalo, Nicola; Tralli, Giulio A New Proof of the Geometric Sobolev Embedding for Generalised Kolmogorov Operators, Kolmogorov Operators and Their Applications, Volume 56 (2024), p. 117 | DOI:10.1007/978-981-97-0225-1_5
  • Yang, Lin; Shao, Dan; Huang, Zhenxing; Geng, Mengxiao; Zhang, Na; Chen, Long; Wang, Xi; Liang, Dong; Pang, Zhi‐Feng; Hu, Zhanli Few‐shot segmentation framework for lung nodules via an optimized active contour model, Medical Physics, Volume 51 (2024) no. 4, p. 2788 | DOI:10.1002/mp.16933
  • Zhao, Nan; Wang, Zhiyong; Li, Pengtao; Liu, Yu Geometric Topics Related to Besov type Spaces on the Grushin Setting, Potential Analysis (2024) | DOI:10.1007/s11118-024-10187-9
  • Meng, Junying; Guo, Weihong; Liu, Jun; Yang, Mingrui Assembling a Learnable Mumford–Shah Type Model with Multigrid Technique for Image Segmentation, SIAM Journal on Imaging Sciences, Volume 17 (2024) no. 2, p. 1007 | DOI:10.1137/23m1577663
  • Alonso Ruiz, Patricia; Baudoin, Fabrice Yet another heat semigroup characterization of BV functions on Riemannian manifolds, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 32 (2023) no. 3, p. 577 | DOI:10.5802/afst.1745
  • Eleuteri, Michela; Lussardi, Luca; Torricelli, Andrea Asymptotic analysis of a family of non-local functionals on sets, ESAIM: Control, Optimisation and Calculus of Variations, Volume 29 (2023), p. 1 | DOI:10.1051/cocv/2022080
  • Kobayashi, Kei; Park, Hyunchul Spectral Heat Content for Time-Changed Killed Brownian Motions, Journal of Theoretical Probability, Volume 36 (2023) no. 2, p. 1148 | DOI:10.1007/s10959-022-01188-8
  • Laux, Tim; Lelmi, Jona Large Data Limit of the MBO Scheme for Data Clustering, PAMM, Volume 22 (2023) no. 1 | DOI:10.1002/pamm.202200308
  • Mazón, José M.; Solera-Diana, Marcos; Toledo-Melero, J. Julián The Heat Flow in Random Walk Spaces, Variational and Diffusion Problems in Random Walk Spaces, Volume 103 (2023), p. 59 | DOI:10.1007/978-3-031-33584-6_2
  • Luo, Shousheng; Tai, Xue-Cheng; Wang, Yang A new binary representation method for shape convexity and application to image segmentation, Analysis and Applications, Volume 20 (2022) no. 03, p. 465 | DOI:10.1142/s0219530521500238
  • Xie, Xiangyun; Liu, Yu; Wang, Haihui Characterizations of sets of finite perimeter using the Ornstein-Uhlenbeck semigroup in the Gauss space, Bulletin des Sciences Mathématiques, Volume 174 (2022), p. 103090 | DOI:10.1016/j.bulsci.2021.103090
  • Amstutz, Samuel; Dapogny, Charles; Ferrer, Alex A consistent approximation of the total perimeter functional for topology optimization algorithms, ESAIM: Control, Optimisation and Calculus of Variations, Volume 28 (2022), p. 18 | DOI:10.1051/cocv/2022005
  • Liu, Jun; Wang, Xiangyue; Tai, Xue-Cheng Deep Convolutional Neural Networks with Spatial Regularization, Volume and Star-Shape Priors for Image Segmentation, Journal of Mathematical Imaging and Vision, Volume 64 (2022) no. 6, p. 625 | DOI:10.1007/s10851-022-01087-x
  • Wang, Dong; Wang, Xiao-Ping The iterative convolution–thresholding method (ICTM) for image segmentation, Pattern Recognition, Volume 130 (2022), p. 108794 | DOI:10.1016/j.patcog.2022.108794
  • Wang, Dong An Efficient Unconditionally Stable Method for Dirichlet Partitions in Arbitrary Domains, SIAM Journal on Scientific Computing, Volume 44 (2022) no. 4, p. A2061 | DOI:10.1137/21m1443406
  • Jacobs, Matt; Kim, Inwon; Mészáros, Alpár R. Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport, Archive for Rational Mechanics and Analysis, Volume 239 (2021) no. 1, p. 389 | DOI:10.1007/s00205-020-01579-3
  • Wang, Dong An Efficient Iterative Method for Reconstructing Surface from Point Clouds, Journal of Scientific Computing, Volume 87 (2021) no. 1 | DOI:10.1007/s10915-021-01457-4
  • Gui, Luying; Ma, Jun; Yang, Xiaoping Shape prior generation and geodesic active contour interactive iterating algorithm (SPACIAL): fully automatic segmentation for 3D lumen in intravascular optical coherence tomography images, Medical Physics, Volume 48 (2021) no. 11, p. 7099 | DOI:10.1002/mp.15201
  • Ma, Jun; Wang, Dong; Wang, Xiao-Ping; Yang, Xiaoping A Characteristic Function-Based Algorithm for Geodesic Active Contours, SIAM Journal on Imaging Sciences, Volume 14 (2021) no. 3, p. 1184 | DOI:10.1137/20m1382817
  • Schilling, Nathanael Short-Time Heat Content Asymptotics via the Wave and Eikonal Equations, The Journal of Geometric Analysis, Volume 31 (2021) no. 2, p. 2172 | DOI:10.1007/s12220-020-00416-z
  • Valverde, Luis Acuña On the heat content for the Poisson kernel over the unit ball in the euclidean space, Bulletin of the London Mathematical Society, Volume 52 (2020) no. 6, p. 1093 | DOI:10.1112/blms.12384
  • Laux, Tim; Otto, Felix Brakke’s inequality for the thresholding scheme, Calculus of Variations and Partial Differential Equations, Volume 59 (2020) no. 1 | DOI:10.1007/s00526-020-1696-8
  • Ruiz, Patricia Alonso; Baudoin, Fabrice; Chen, Li; Rogers, Luke G.; Shanmugalingam, Nageswari; Teplyaev, Alexander Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities, Journal of Functional Analysis, Volume 278 (2020) no. 11, p. 108459 | DOI:10.1016/j.jfa.2020.108459
  • Garofalo, Nicola; Tralli, Giulio Nonlocal isoperimetric inequalities for Kolmogorov-Fokker-Planck operators, Journal of Functional Analysis, Volume 279 (2020) no. 3, p. 108591 | DOI:10.1016/j.jfa.2020.108591
  • Ma, Jun; Nie, Ziwei; Wang, Congcong; Dong, Guoqiang; Zhu, Qiongjie; He, Jian; Gui, Luying; Yang, Xiaoping Active contour regularized semi-supervised learning for COVID-19 CT infection segmentation with limited annotations, Physics in Medicine Biology, Volume 65 (2020) no. 22, p. 225034 | DOI:10.1088/1361-6560/abc04e
  • Song, Yangyang; Peng, Guohua; Sun, Dongwei; Xie, Xiaozhen Active contours driven by Gaussian function and adaptive-scale local correntropy-based K-means clustering for fast image segmentation, Signal Processing, Volume 174 (2020), p. 107625 | DOI:10.1016/j.sigpro.2020.107625
  • Salvador, Tiago; Esedoḡlu, Selim A Simplified Threshold Dynamics Algorithm for Isotropic Surface Energies, Journal of Scientific Computing, Volume 79 (2019) no. 1, p. 648 | DOI:10.1007/s10915-018-0866-8
  • Grzywny, Tomasz; Park, Hyunchul; Song, Renming Spectral heat content for Lévy processes, Mathematische Nachrichten, Volume 292 (2019) no. 4, p. 805 | DOI:10.1002/mana.201800035
  • Wang, Dong; Cherkaev, Andrej; Osting, Braxton; Idema, Timon Dynamics and stationary configurations of heterogeneous foams, PLOS ONE, Volume 14 (2019) no. 4, p. e0215836 | DOI:10.1371/journal.pone.0215836
  • Ferrari, Fausto; Miranda Jr, Michele; Pallara, Diego; Pinamonti, Andrea; Sire, Yannick Fractional Laplacians, perimeters and heat semigroups in Carnot groups, Discrete Continuous Dynamical Systems - S, Volume 11 (2018) no. 3, p. 477 | DOI:10.3934/dcdss.2018026
  • van den Berg, M. Heat Content in Non-compact Riemannian Manifolds, Integral Equations and Operator Theory, Volume 90 (2018) no. 1 | DOI:10.1007/s00020-018-2440-z
  • Kindler, Guy; Kirshner, Naomi; O’Donnell, Ryan Gaussian noise sensitivity and Fourier tails, Israel Journal of Mathematics, Volume 225 (2018) no. 1, p. 71 | DOI:10.1007/s11856-018-1646-8
  • Mazón, Jose Manuel; Rossi, Julio D.; Toledo, Julián The Heat Content for Nonlocal Diffusion with Non-singular Kernels, Advanced Nonlinear Studies, Volume 17 (2017) no. 2, p. 255 | DOI:10.1515/ans-2017-0005
  • Xu, Xianmin; Wang, Dong; Wang, Xiao-Ping An efficient threshold dynamics method for wetting on rough surfaces, Journal of Computational Physics, Volume 330 (2017), p. 510 | DOI:10.1016/j.jcp.2016.11.008
  • Esedoḡlu, Selim; Jacobs, Matt; Zhang, Pengbo Kernels with prescribed surface tension mobility for threshold dynamics schemes, Journal of Computational Physics, Volume 337 (2017), p. 62 | DOI:10.1016/j.jcp.2017.02.023
  • Cygan, Wojciech; Grzywny, Tomasz Heat content for convolution semigroups, Journal of Mathematical Analysis and Applications, Volume 446 (2017) no. 2, p. 1393 | DOI:10.1016/j.jmaa.2016.09.051
  • Esedoḡlu, Selim; Jacobs, Matt Convolution Kernels and Stability of Threshold Dynamics Methods, SIAM Journal on Numerical Analysis, Volume 55 (2017) no. 5, p. 2123 | DOI:10.1137/16m1087552
  • Jacobs, Matt A Fast MBO Scheme for Multiclass Data Classification, Scale Space and Variational Methods in Computer Vision, Volume 10302 (2017), p. 335 | DOI:10.1007/978-3-319-58771-4_27
  • Valverde, Luis Acuña Heat Content for Stable Processes in Domains of $\mathbb {R}^d$, The Journal of Geometric Analysis, Volume 27 (2017) no. 1, p. 492 | DOI:10.1007/s12220-016-9688-9
  • Laux, Tim; Otto, Felix Convergence of the thresholding scheme for multi-phase mean-curvature flow, Calculus of Variations and Partial Differential Equations, Volume 55 (2016) no. 5 | DOI:10.1007/s00526-016-1053-0
  • Acuña Valverde, Luis Heat content estimates over sets of finite perimeter, Journal of Mathematical Analysis and Applications, Volume 441 (2016) no. 1, p. 104 | DOI:10.1016/j.jmaa.2016.03.087
  • Marola, Niko; Miranda, Michele; Shanmugalingam, Nageswari Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces, Potential Analysis, Volume 45 (2016) no. 4, p. 609 | DOI:10.1007/s11118-016-9560-3
  • van den Berg, M.; Gittins, K. On the Heat Content of a Polygon, The Journal of Geometric Analysis, Volume 26 (2016) no. 3, p. 2231 | DOI:10.1007/s12220-015-9626-2
  • Esedoḡ Lu, Selim; Otto, Felix Threshold Dynamics for Networks with Arbitrary Surface Tensions, Communications on Pure and Applied Mathematics, Volume 68 (2015) no. 5, p. 808 | DOI:10.1002/cpa.21527
  • van den Berg, M.; Gittins, K. Uniform bounds for the heat content of open sets in Euclidean space, Differential Geometry and its Applications, Volume 40 (2015), p. 67 | DOI:10.1016/j.difgeo.2015.01.010
  • van den Berg, M.; Gilkey, P. Heat Content with Singular Initial Temperature and Singular Specific Heat, Potential Analysis, Volume 42 (2015) no. 1, p. 1 | DOI:10.1007/s11118-014-9422-9
  • van den Berg, M.; Gilkey, P. Heat Flow Out of a Compact Manifold, The Journal of Geometric Analysis, Volume 25 (2015) no. 3, p. 1576 | DOI:10.1007/s12220-014-9485-2
  • Angiuli, Luciana; Massari, Umberto; Miranda, Michele Geometric properties of the heat content, Manuscripta Mathematica, Volume 140 (2013) no. 3-4, p. 497 | DOI:10.1007/s00229-012-0550-5
  • van den Berg, M. Heat Flow and Perimeter in $\boldsymbol{{\mathbb{R}}^m}$, Potential Analysis, Volume 39 (2013) no. 4, p. 369 | DOI:10.1007/s11118-013-9335-z
  • Bramanti, Marco; Miranda, Michele; Pallara, Diego Two Characterization of BV Functions on Carnot Groups via the Heat Semigroup, International Mathematics Research Notices, Volume 2012 (2012) no. 17, p. 3845 | DOI:10.1093/imrn/rnr170
  • Angiuli, Luciana; Miranda, Michele; Pallara, Diego; Paronetto, Fabio BV functions and parabolic initial boundary value problems on domains, Annali di Matematica Pura ed Applicata, Volume 188 (2009) no. 2 | DOI:10.1007/s10231-008-0076-3
  • Miranda, M; Pallara, D; Paronetto, F; Preunkert, M Heat semigroup and functions of bounded variation on Riemannian manifolds, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2007 (2007) no. 613 | DOI:10.1515/crelle.2007.093

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