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

Planned 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. W. Górny and J.M. Mazón, A duality-based approach to gradient flows of linear growth functionals, Publ. Mat., to appear (preprint available at arXiv:2212.08725).
  4. W. Górny, Strongly anisotropic Anzellotti pairings and their applications to the anisotropic p-Laplacian preprint (2023), available at arXiv:2305.18876.
  5. W. Górny, Weak solutions to gradient flows of functionals with inhomogeneous growth in metric spaces, preprint (2023), available at arXiv:2307.13456.
  6. M. Friedrich, W. Górny and U. Stefanelli, A characterization of l1 double bubbles with general interface interaction, preprint (2023), available at arXiv:2311.07782.
  7. S. Buccheri and W. Górny, A metric counterpart of the Gu-Yung formula, preprint (2024), available at arXiv:2403.13475.
  8. M. Friedrich, W. Górny and U. Stefanelli, The l1 double-bubble problem in three dimensions, preprint (2024), available at arXiv:2403.19295.
  9. W. Górny, J.M. Mazón, J. Toledo, Evolution problems with perturbed 1-Laplacian type operators on random walk spaces, preprint (2024), available at arXiv:2410.15203.