Basic information

Principal investigator: Wojciech Górny

Title: "Inhomogeneous-growth problems including a linear-growth term"

Funding institution: Austrian Science Fund

Grant number: ESP 88

Location: University of Vienna

Awarded funds: 294 015,98 €

Duration: 36 months

Starting date: 01.10.2022

End date: 30.09.2025

Information on the FWF website: link.

Summary

The project is concerned with the study of elliptic and parabolic problems related to functionals which have inhomogeneous growth, in the case when some of the terms have linear growth. The main underlying difficulty is the possible lack of reflexivity and separability of natural energy spaces in problems with linear or inhomogeneous growth. Problems covered by the project include the elliptic and parabolic problems related to the (1,q)-Laplacian operator, the p(x)-Laplacian operator without a lower bound on p(x), and the p-Laplacian with p equal to one on some of the coordinates. In this project, we study existence, regularity and qualitative properties of solutions to problems of this type, including asymptotics for evolution equations. We also plan to study the regularity of solutions in the fully anisotropic superlinear case, without assuming any lower bound on the rate of growth.

Associated publications

  1. 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.
  2. W. Górny and J.M. Mazón, Weak solutions to the total variation flow in metric measure spaces, in: Ferreira et al. (eds.), Proceedings book. XXVII Congreso de Ecuaciones Diferenciales y XVII Congreso de Matemática Aplicada, Zaragoza, July 18th–22nd, 2022. Prensas de la Universidad de Zaragoza, pp. 55--62, 2023. DOI: 10.26754/uz.978-84-18321-66-5.
  3. M. Friedrich, W. Górny and U. Stefanelli, A characterization of l1 double bubbles with general interface interaction, Adv. Calc. Var. 18 (2025), 609-637. DOI: 10.1515/acv-2023-0131.
  4. W. Górny, Weak solutions to gradient flows of functionals with inhomogeneous growth in metric spaces, J. Evol. Equ. 25 (2025), Art. 44. DOI: 10.1007/s00028-025-01071-z.
  5. W. Górny, J.M. Mazón, J. Toledo, Evolution problems with perturbed 1-Laplacian type operators on random walk spaces, Math. Ann. 392 (2025), 3895-3957. DOI: 10.1007/s00208-025-03180-z.
  6. W. Górny, Strongly anisotropic Anzellotti pairings and their applications to the anisotropic p-Laplacian, J. Math. Anal. Appl. 552 (2025), 129734. DOI: 10.1016/j.jmaa.2025.129734.
  7. W. Górny and J.M. Mazón, A duality-based approach to gradient flows of linear growth functionals, Publ. Mat. 69 (2025), 341-365. DOI: 10.5565/PUBLMAT6922504.
  8. M. Friedrich, W. Górny and U. Stefanelli, The l1 double-bubble problem in three dimensions, J. Geom. Anal. 35 (2025), Art. 323. DOI: 10.1007/s12220-025-02151-9.
  9. W. Górny, M. Łasica, A. Matsoukas, Euler–Lagrange equations for variable-growth total variation, Nonlinear Anal. 263 (2026), 113984. DOI: 10.1016/j.na.2025.113984.
  10. S. Buccheri and W. Górny, A metric counterpart of the Gu-Yung formula, Rev. Mat. Complut., to appear, preprint available at arXiv:2403.13475.
  11. W. Górny, M. Łasica, A. Matsoukas, Adaptive double-phase Rudin–Osher–Fatemi denoising model, preprint (2025), available at arXiv:2510.04382.

Presentation of the results of the project

Invited talks at conferences:
  1. The double-bubble problem for the l1 norm at the “Annual 2025 ÖMG-DMV Meeting”, section “Partial Differential Equations”, Linz, 1-5 September 2025.
  2. Euler–Lagrange equations for variable-growth total variation with applications to image processing at the “Annual 2025 ÖMG-DMV Meeting”, minisymposium “Applied Operator Theory”, Linz, 1-5 September 2025.
  3. Characterisation of weak solutions to gradient flows of general linear growth functionals at “Degenerate and Singular PDEs”, Vienna, 24-28 February 2025.
  4. Evolution equations on two overlapping random walk structures at “Recent Progress in PDEs”, Rome, 20-21 February 2025.
  5. Optimal transport techniques in geometric problems at the “VIII Symposium on Nonlinear Analysis”, Toruń, 17-21 June 2024.
  6. Weak solutions to gradient flows in metric measure spaces at “Nonlinear Partial Differential Equations 2023”, conference on the occasion of J.M.Mazón’s 70th birthday, València, 24-26 October 2023.
  7. Duality methods for gradient flows of linear growth functionals at the “10th International Congress on Industrial and Applied Mathematics” (ICIAM 2023), minisymposium “Frontiers of gradient flows: well-posedness, asymptotics, singular limits”, Tokyo, 20-25 August 2023.
  8. Geometric aspects of the planar least gradient problem at “International Banach Prize Mini-Conference”, Banach Center, Warsaw, 18-19 May 2023.
Presentations at seminars:
  1. Partial differential equations with linear growth at the Colloquium of MIM, University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, on 4 December 2025.
  2. Variable-growth total variation denoising at the seminar of the Mathematical Physics Equations Group, University of Warsaw, Faculty of Mathematics, Informatics and Mechanics, on 6 November 2025.
  3. Adaptive double-phase total variation denoising at the Calculus of Variations seminar, Universität Wien, Faculty of Mathematics, on 27 October 2025.
  4. Random walk spaces and their applications to nonlocal PDEs at TU Wien, Department of Mathematics, on 14 January 2025.
  5. A formula relating the Lp- and weak Lp-norms at the Calculus of Variations seminar, Universität Wien, Faculty of Mathematics, on 13 May 2024.
  6. Gradient flows of functionals with linear growth at the PDE Afternoon, Universität Wien, Faculty of Mathematics, on 1 March 2023.
Miscellaneous:
  1. A double-bubble problem for the l1 anisotropy, talk at the “3rd Austrian Calculus of Variations Day”, Vienna, 23-24 November 2023.