Existence of parabolic minimizers to the total variation flow on metric measure spaces

Vito Buffa; Michael Collins; Cintia Pacchiano Camacho

doi:10.1007/s00229-021-01350-2

Existence of parabolic minimizers to the total variation flow on metric measure spaces

Vito Buffa
, Michael Collins
, Cintia Pacchiano Camacho^*

^*Tämän työn vastaava kirjoittaja

Matematiikan ja systeemianalyysin laitos

Friedrich-Alexander University Erlangen-Nürnberg

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

5 Sitaatiot (Scopus)

43 Lataukset (Pure)

Abstrakti

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X, d, μ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T> 0 , we prove existence in the weak parabolic function space Lw1(0,T;BV(Ω)). In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level.

Alkuperäiskieli	Englanti
Sivut	109-145
Sivumäärä	37
Julkaisu	Manuscripta Mathematica
Vuosikerta	170
Numero	1-2
Varhainen verkossa julkaisun päivämäärä	4 tammik. 2022
DOI - pysyväislinkit	https://doi.org/10.1007/s00229-021-01350-2
Tila	Julkaistu - tammik. 2023
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Rahoitus

This work was partially supported by the grant DFG-Project HA 7610/1-1 “Existenz- und Regularitätsaussagen für parabolische Quasiminimierer auf metrischen Maßräumen” and by the Academy of Finland. M. Collins expresses his gratitude to the Friedrich Naumann Foundation for Freedom that supported him through a postgraduate scholarship. He also thanks the Deparment of Mathematics and System Analysis at Aalto University for kindly receiving him in September 2019 and February 2020, while V. Buffa and C. Pacchiano Camacho thank the Department of Mathematics at FAU Erlangen for the kind hospitality during their visit in October 2019.

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10.1007/s00229-021-01350-2Lisenssi: CC BY

s00229-021-01350-2Final published version, 535 KBLisenssi: CC BY

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abstract = "We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X, d, μ) satisfying a doubling condition and supporting a Poincar{\'e} inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T> 0 , we prove existence in the weak parabolic function space Lw1(0,T;BV(Ω)). In this paper, we generalize results from a previous work by B{\"o}gelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level.",

author = "Vito Buffa and Michael Collins and Camacho, \{Cintia Pacchiano\}",

note = "Funding Information: This work was partially supported by the grant DFG-Project HA 7610/1-1 “Existenz- und Regularit{\"a}tsaussagen f{\"u}r parabolische Quasiminimierer auf metrischen Ma{\ss}r{\"a}umen” and by the Academy of Finland. M. Collins expresses his gratitude to the Friedrich Naumann Foundation for Freedom that supported him through a postgraduate scholarship. He also thanks the Deparment of Mathematics and System Analysis at Aalto University for kindly receiving him in September 2019 and February 2020, while V. Buffa and C. Pacchiano Camacho thank the Department of Mathematics at FAU Erlangen for the kind hospitality during their visit in October 2019. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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doi = "10.1007/s00229-021-01350-2",

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AU - Collins, Michael

AU - Camacho, Cintia Pacchiano

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N2 - We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X, d, μ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T> 0 , we prove existence in the weak parabolic function space Lw1(0,T;BV(Ω)). In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level.

AB - We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X, d, μ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T> 0 , we prove existence in the weak parabolic function space Lw1(0,T;BV(Ω)). In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level.

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JF - Manuscripta Mathematica

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Existence of parabolic minimizers to the total variation flow on metric measure spaces

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