Basic information

Principal investigator: Wojciech Górny

Title: "Anisotropic least gradient problem"

Funding institution: National Science Centre, Poland

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

Location: University of Warsaw

Awarded funds: 106 400 PLN

Duration: 51 months

Starting date: 17.07.2018

End date: 16.10.2022

Information on the NCN website: link.

Summary

The anisotropic least gradient problem is on the borderline between the calculus of variations, partial differential equations and geometric measure theory. It is the following minimization problem: given some boundary data, one wants to minimize the integral of the gradient of a function inside the domain. It is calculated with respect to an anisotropic norm, which may additionally depend on location in a continuous way. This project comprises of a few issues. Firstly, existing results concerning existence of solutions require continuity of the boundary data. The goal is to extend known existence results to some classes of discontinuous boundary data, under suitable geometric assumptions on the domain which generalize strict convexity to the anisotropic case). The second issue is the structure of solutions for discontinuous boundary data. In the isotropic case, there may be multiple solutions, but in low dimensions they share a similar structure of level sets: the difference of two solutions is a locally constant function. The second objective is to examine if for sufficiently smooth anisotropic norms it is also the case and on what sets of lower dimension do the jumps of this function concentrate, and if some related structure results hold in greater generality. The third aim is to study uniqueness of solutions on the plane for continuous boundary data without any additional assumptions on the regularity of the anisotropic norm.

Associated publications

  1. W. Górny, Lp regularity of least gradient functions, Proc. Amer. Math. Soc. 148 (7) (2020), pp. 3009-3019. DOI: 10.1090/proc/15031.
  2. W. Górny, Least gradient problem with respect to a non-strictly convex norm, Nonlinear Anal. 200 (2020), 112049. DOI: 10.1016/j.na.2020.112049.
  3. W. Górny and J.M. Mazón, Least gradient functions in metric random walk spaces, ESAIM:COCV 27 (2021), S28. DOI: 10.1051/cocv/2020087.
  4. W. Górny, Existence of minimisers in the least gradient problem for general boundary data, Indiana Univ. Math. J. 70, no. 3 (2021), pp. 1003-1037. DOI: 10.1512/iumj.2021.70.8420.
  5. W. Górny, Bourgain-Brezis-Mironescu approach in metric spaces with Euclidean tangents, J. Geom. Anal. 32 (4) (2022), Art. 128. DOI: 10.1007/s12220-021-00861-4.
  6. W. Górny, Local and nonlocal 1-Laplacian in Carnot groups, Ann. Fenn. Math. 47 (1) (2022), pp. 427-456. DOI: 10.54330/afm.114742.
  7. S. Dweik and W. Górny, Least gradient problem on annuli, Analysis & PDE 15 (3) (2022), pp. 699-725. DOI: 10.2140/apde.2022.15.699.
  8. W. Górny and J.M. Mazón, On the p-Laplacian evolution equation in metric measure spaces, J. Funct. Anal. 283 (2022), 109621. DOI: 10.1016/j.jfa.2022.109621.
  9. W. Górny and J.M. Mazón, Weak solutions to gradient flows in metric measure spaces, Proc. Appl. Math. Mech. 22:1 (2022), e202200099. DOI: 10.1002/pamm.202200099.
  10. W. Górny, The trace space of anisotropic least gradient functions depends on the anisotropy, Math. Ann. 387 (2023), 1343–1365. DOI: 10.1007/s00208-022-02488-4.
  11. S. Dweik and W. Górny, Optimal transport approach to Sobolev regularity of solutions to the weighted least gradient problem, SIAM. J. Math. Anal. 55 (2023), no. 3, 1916-1948. DOI: 10.1137/21M1468358.
  12. W. Górny, Applications of optimal transport methods in the least gradient problem, Ann. Scu. Norm. Sup. Pisa Cl. Sci. (5) 24 (2023), pp. 1817-1851. DOI: 10.2422/2036-2145.202105_049.
  13. W. Górny and J.M. Mazón, The Neumann and Dirichlet problems for the total variation flow in metric measure spaces, Adv. Calc. Var. 17 (2024), 131-164. DOI: 10.1515/acv-2021-0107.
  14. W. Górny and J.M. Mazón, The Anzellotti-Gauss-Green formula and least gradient functions in metric measure spaces, Commun. Contemp. Math. 26 (2024), no. 6, 2350027. DOI: 10.1142/S021919972350027X.
  15. M. Friedrich, W. Górny and U. Stefanelli, The double-bubble problem on the square lattice, Interfaces Free Bound. 26 (2024), no. 1, pp. 79-134. DOI: 10.4171/ifb/510.
  16. W. Górny, Least gradient problem with Dirichlet condition imposed on a part of the boundary, Calc. Var. Partial Differential Equations 63 (2024), Art. 58. DOI: 10.1007/s00526-023-02646-9.

Book related to the topic of the project

  1. W. Górny and J.M. Mazón, Functions of Least Gradient, Monographs in Mathematics, vol. 110, 2024, ISBN 978-3-031-51880-5. DOI: 10.1007/978-3-031-51881-2.

Presentation of the results of the project

Invited talks at conferences:
  1. A new notion of solutions to gradient flows in metric measure spaces, “Nonuniformly elliptic problems”, Banach Center, Warsaw, 5-9 September 2022.
  2. Weak solutions to gradient flows in metric measure spaces, “92nd Annual Meeting of the International Association of Applied Mathematics and Mechanics” (GAMM Annual Meeting 2022), minisymposium “Evolution equations with gradient flow structure”, Aachen, 15-19 August 2022.
  3. Weak solutions to the total variation flow in metric measure spaces, “XXVII Congress of differential equations and applications / XVI Congress of applied mathematics” (XXVII CEDYA/XVII CMA), minisymposium “New trends on the 1-Laplacian”, Zaragoza, 18-22 July 2022.
  4. Zagadnienie najmniejszego gradientu, "Polish Mathematical Society Mini-Conference", Banach Center, Warsaw, 2-3 June 2022.
  5. Geometric aspects of the 1-Laplacian, "XII Forum of Partial Differential Equations", Banach Center, Będlewo, 19-25 September 2021.
  6. Least gradient problem and minimal surfaces, "Juliusz Schauder Medal Awarding Ceremony", Toruń (online), 18 June 2021.
  7. Optimal transport methods in the least gradient problem, "Nonlocal diffusion problems, nonlocal interface evolution", Banach Center, Warsaw (online), 1-3 October 2020.
  8. The least gradient problem with respect to a non-smooth or non-strictly convex norm, "9th International Congress on Industrial and Applied Mathematics" (ICIAM 2019), minisymposium "A broad view of the least gradient problems", Valencia, 15-19 July 2019.
Conference posters:
  1. 1-Laplacian on metric random walk spaces, poster at "Winterschool on Analysis and Applied Mathematics", Münster (online), 22-26 February 2021.
  2. Least gradient problem on unbounded domains, poster at "VII School of Analysis in memory of A. Pełczyński", Banach Center, Będlewo, 28-31 March 2019.
  3. Existence and regularity of minimizers in the anisotropic least gradient problem, poster at "Joint Meeting of UMI-SIMAI-PTM", Wrocław, 17-20 September 2018.
Presentations at seminars:
  1. A discrete version of the double-bubble problem at the Universitat de València, Department of Mathematical Analysis, on 25 May 2022.
  2. A discrete double-bubble problem on Z^2 at the Calculus of Variations seminar, Universität Wien, Faculty of Mathematics, on 24 March 2022.
  3. Least gradient problem in 2D and optimal transport at the Łódź University of Technology, Institute of Mathematics, on 16 November 2021 (online).
  4. A discrete double bubble problem on the square lattice at the WEMM Seminar, Wien-Erlangen-München-Münster, on 9 November 2021 (online).
  5. Planar least gradient problem at the American University of Beirut, Department of Mathematics, on 26 October 2021 (online).
  6. Linear structures and gradient flows in metric spaces at the Calculus of Variations seminar, Universität Wien, Faculty of Mathematics, on 31 May 2021.
  7. Geometric aspects of the least gradient problem at the PDE Afternoon, Universität Wien, Faculty of Mathematics, on 3 March 2021.
  8. Structure of solutions to the least gradient problem at the seminar of the Mathematical Physics Equations Group, University of Warsaw, Faculty of Mathematics, Informatics and Mechanics on 23 January 2020.
  9. The least gradient problem at the Calculus of Variations seminar, Universität Wien, Faculty of Mathematics, on 15 January 2020.
  10. Least gradient problem for discontinuous boundary data at the Universitat de València, Department of Mathematical Analysis, on 10 April 2019.
  11. Least gradient problem on unbounded domains at the seminar of the Mathematical Physics Equations Group, University of Warsaw, Faculty of Mathematics, Informatics and Mechanics on 21 March 2019.