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The Neumann and Dirichlet problems for the total variation flow in metric measure spaces

  • Wojciech Górny ORCID logo und José M. Mazón ORCID logo EMAIL logo
Veröffentlicht/Copyright: 29. Juni 2022

Abstract

We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L 1 initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss–Green formula.


Communicated by Juha Kinnunen


Funding source: Austrian Science Fund

Award Identifier / Grant number: FR 4083/3-1/I4354

Award Identifier / Grant number: CZ 01/2021

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2017/27/N/ST1/02418

Award Identifier / Grant number: PGC2018-094775-B-100

Funding statement: The first author has been partially supported by the DFG-FWF project FR 4083/3-1/I4354, by the OeAD-WTZ project CZ 01/2021, and by the project 2017/27/N/ST1/02418 funded by the National Science Centre, Poland. The second author has been partially supported by the Spanish MCIU and FEDER, project PGC2018-094775-B-100.

References

[1] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set Valued Anal. 266 (2002), 111–128. 10.1023/A:1016548402502Suche in Google Scholar

[2] L. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, J. Funct. Anal. 266 (2014), no. 7, 4150–4188. 10.1016/j.jfa.2014.02.002Suche in Google Scholar

[3] L. Ambrosio, N. Gigli and G. Savaré, Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam. 29 (2013), no. 3, 969–996. 10.4171/RMI/746Suche in Google Scholar

[4] L. Ambrosio, M. Miranda, Jr. and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi, Quad. Mat. 14, Seconda Università degli Studi di Napoli, Caserta (2004), 1–45. Suche in Google Scholar

[5] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations 14 (2001), no. 3, 321–360. 10.57262/die/1356123331Suche in Google Scholar

[6] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The Dirichlet problem for the total variation flow, J. Funct. Anal. 180 (2001), no. 2, 347–403. 10.1006/jfan.2000.3698Suche in Google Scholar

[7] F. Andreu, V. Caselles, J. I. Díaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal. 188 (2002), no. 2, 516–547. 10.1006/jfan.2001.3829Suche in Google Scholar

[8] F. Andreu-Vaillo, V. Caselles and J. M. Mazón, Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progr. Math. 223, Birkhäuser, Basel, 2004. 10.1007/978-3-0348-7928-6Suche in Google Scholar

[9] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318. 10.1007/BF01781073Suche in Google Scholar

[10] P. Bénilan and M. G. Crandall, Completely accretive operators, Semigroup Theory and Evolution Equations (Delft 1989), Lecture Notes Pure Appl. Math. 135, Dekker, New York (1991), 41–75. 10.1201/9781003419914-4Suche in Google Scholar

[11] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Math. 17, European Mathematical Society, Zürich, 2011. 10.4171/099Suche in Google Scholar

[12] V. Bögelein, F. Duzaar and P. Marcellini, A time dependent variational approach to image restoration, SIAM J. Imaging Sci. 8 (2015), no. 2, 968–1006. 10.1137/140992771Suche in Google Scholar

[13] H. Brezis, Operateurs Maximaux Monotones, North Holland, Amsterdam, 1973. Suche in Google Scholar

[14] V. Buffa, M. Collins and C. P. Camacho, Existence of parabolic minimizers to the total variation flow on metric spaces, Manuscripta Math. (2022), 10.1007/s00229-021-01350-2. 10.1007/s00229-021-01350-2Suche in Google Scholar

[15] V. Buffa, G. E. Comi and M. Miranda Jr., On BV functions and essentially bounded divergence-measure fields in metric spaces, Rev. Mat. Iberoam. (2021), 10.4171/RMI/1291. 10.4171/RMI/1291Suche in Google Scholar

[16] L. Bungert and M. Burger, Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type, J. Evol. Equ. 20 (2020), no. 3, 1061–1092. 10.1007/s00028-019-00545-1Suche in Google Scholar

[17] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. 10.2307/2373376Suche in Google Scholar

[18] S. Di Marino, Recent advances on BV and Sobolev Spaces in metric measure spaces, Ph.D. Thesis, Scoula Normale Superiore, Pisa, 2014. Suche in Google Scholar

[19] I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Stud. Appl. Math., North-Holland, Amsterdam, 1976. Suche in Google Scholar

[20] N. Gigli, On the differential structure of metric measure spaces and applications, Mem. Amer. Math. Soc. 236 (2015), no. 1113, 1–91. 10.1090/memo/1113Suche in Google Scholar

[21] N. Gigli, Lecture notes on differential calculus on 𝖱𝖢𝖣 spaces, Publ. Res. Inst. Math. Sci. 54 (2018), no. 4, 855–918. 10.4171/PRIMS/54-4-4Suche in Google Scholar

[22] N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 (2018), no. 1196, 1–161. 10.1090/memo/1196Suche in Google Scholar

[23] W. Górny and J. M. Mazón, On the p-Laplacian evolution equation in metric measure spaces, preprint (2021), https://arxiv.org/abs/2103.13373. Suche in Google Scholar

[24] W. Górny and J. M. Mazón, The Anzellotti–Gauss–Green formula and least gradient functions in metric measure spaces, preprint (2021), https://arxiv.org/abs/2105.00432. 10.1051/cocv/2020087Suche in Google Scholar

[25] H. Hakkarainen, J. Kinnunen, P. Lahti and P. Lehtelä, Relaxation and integral representation for functionals of linear growth on metric measure spaces, Anal. Geom. Metr. Spaces 4 (2016), no. 1, 288–313. 10.1515/agms-2016-0013Suche in Google Scholar

[26] D. Hauer and J. M. Mazón, Regularizing effects of homogeneous evolution equations: The case of homogeneity order zero, J. Evol. Equ. 19 (2019), no. 4, 965–996. 10.1007/s00028-019-00502-ySuche in Google Scholar

[27] J. Kinnunen and C. Scheven, On the definition of solution to the total variation flow, Calc. Var. Partial Differential Equations 61 (2022), no. 1, Paper No. 40. 10.1007/s00526-021-02142-ySuche in Google Scholar

[28] P. Lahti, L. Malý and N. Shanmugalingam, An analog of the Neumann problem for the 1-Laplace equation in the metric setting: Existence, boundary regularity, and stability, Anal. Geom. Metr. Spaces 6 (2018), no. 1, 1–31. 10.1515/agms-2018-0001Suche in Google Scholar

[29] P. Lahti and N. Shanmugalingam, Trace theorems for functions of bounded variation in metric spaces, J. Funct. Anal. 274 (2018), no. 10, 2754–2791. 10.1016/j.jfa.2018.02.013Suche in Google Scholar

[30] A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations 30 (1978), no. 3, 340–364. 10.1016/0022-0396(78)90005-0Suche in Google Scholar

[31] D. Lučić and E. Pasqualetto, Infinitesimal Hilbertianity of weighted Riemannian manifolds, Canad. Math. Bull. 63 (2020), no. 1, 118–140. 10.4153/S0008439519000328Suche in Google Scholar

[32] L. Malý, N. Shanmugalingam and M. Snipes, Trace and extension theorems for functions of bounded variation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), no. 1, 313–341. 10.2422/2036-2145.201511_007Suche in Google Scholar

[33] J. M. Mazón, M. Solera and J. Toledo, The total variation flow in metric random walk spaces, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 29. 10.1007/s00526-019-1684-zSuche in Google Scholar

[34] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. 10.1016/S0021-7824(03)00036-9Suche in Google Scholar

[35] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D. 60 (1992), 259–268. 10.1016/0167-2789(92)90242-FSuche in Google Scholar

Received: 2021-12-17
Revised: 2022-03-12
Accepted: 2022-03-16
Published Online: 2022-06-29
Published in Print: 2024-01-01

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