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 Anzellotti-Gauss-Green formula and least gradient functions in metric measure spaces, Commun. Contemp. Math. (2023), ahead of print. DOI: 10.1142/S021919972350027X.
  14. 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.
  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.