RESEARCH, PREPRINTS & PUBLICATIONS

RESEARCH INTERESTS

Nonlinear PDE & the Calculus of Variations Here, a particular focus is on the understanding of relaxations and generalised notions of solutions. This also concerns their regularity properties. In particular, we aim for optimal ranges of the underlying ellipticity parameters to yield improved regularity statements. The latter applies to both convex and semiconvex problems (such as variants of Morrey's quasiconvexity) which emerge, e.g., from continuum mechanics or material science.
Real and Harmonic Analysis, Function Spaces Many problems I have considered/do consider require bounds on the Lp-norms of differential expressions. Especially in the borderline case p=1, routine techniques from real and harmonic analysis do not apply to such scenarios and require severe refinements. This comes with novel techniques which are of interest in the study of more classical spaces too.
Continuum and Fluid Mechanics and Modelling More recently, I have become more and more interested in problems emerging from applications. This ranges from modelling issues in continuum mechanics, the study of nonlinear boundary conditions in fluid problems to the description of viscoplastic fluids/materials such as sea ice. Moreover, a more recent focus is on the understanding of vortex phenomena.

COAUTHORS

Click name to be forwarded to the single webpages:

Jan Kristensen, University of Oxford
Jean Van Schaftingen, University Cath. of Louvain
Sergio Conti, University of Bonn
Matthias Hieber, TU Darmstadt
Robert Denk, University of Konstanz
Gui-Qiang G. Chen, University of Oxford,
Monica Torres, Purdue University,
Endre Süli, University of Oxford
Ferdinand Eitler, University of Augsburg
Miroslav Bulicek, Charles University of Prague
Lars Diening, University of Bielefeld
Lisa Beck, University of Augsburg
Bogdan Raita, University of Georgetown
Dominic Breit, Technical University of Clausthal
Daniel Spector, OIST
Linus Behn, University of Bielefeld
Stefan Schiffer, University of Bonn/MPI Leipzig
Felix Finster, University of Regensburg
Matthias Bärlin, University of Konstanz
Patrizio Neff, University of Duisburg-Essen
Peter Lewintan, University of Duisburg-Essen
Christopher Irving, Technical University of Dortmund
Alexei Gazca-Orozco, Charles University of Prague
Erika Maringová-Kokavcova, IST Austria
Tabea Tscherpel, TU Darmstadt
Mathias Schäffner, MLU Halle
Paul Stephan, University of Konstanz

Selected Papers

F Gmeineder: The regularity of minima for the Dirichlet problem on BD. (Arch. Ration. Mech. Anal., link to published version). Preprint (Aug 2019, v2, 54 pp.) available here.
F Gmeineder & J Kristensen: Partial regularity for BV minimizers. (Arch. Ration. Mech. Anal., link to published version). Preprint (Mar 2018, 33 pp.) available here.
F Gmeineder & J Kristensen: Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers. (Arch. Ration. Mech. Anal., link to published version). Preprint (Sep 2022, 90 pp.) available here.
L Beck, F Eitler & F Gmeineder: Gradient integrability of bounded BD-minimizers. (Arch. Ration. Mech. Anal., to appear). Preprint (Dec 2024, 32 pp.) available here.
F Gmeineder: Partial regularity for symmetric quasiconvex functionals on BD. (J. Math. Pures Appl., link to published version) Preprint (Mar 2019, v1, Oct 2019, v2, 40 pp.) available here.
D Breit, L Diening & F Gmeineder: On the trace operator for functions of bounded 𝔸-variation. (Analysis & PDE, link to published version). Preprint (Jul 2017, 31 pp.) available here.
F Gmeineder, B Raita & J Van Schaftingen: Limiting L¹-estimates on halfspaces. (Adv. Math., link to published version). Preprint (Nov 2022, v1, Oct 2023, v2, 20 pp.) available here.
G.-Q. Chen, F Gmeineder & M Torres: Curl measure fields, the generalized Stokes theorem and vorticity fluxes. Preprint (Sep 2025, 78 pp.) available here.

Papers by Topic

My research can be broadly divided into three topic areas - note that there are some overlaps between the single topics.

All Preprints & Published Papers

All preprints are available on the ArXiv too. For some of the papers, talk slides are attached for a quick overview.

35. On Hibler's sea ice model in the singular limit. w/ R Denk & M Hieber

Preprint (Nov 2025, 54 pp.) available here.

34. Curl-measure fields, the generalized Stokes theorem and vorticity fluxes. w/ G-Q Chen & M Torres

Preprint (Sep 2025, 78 pp.) available here.

33. On Korn inequalities with lower order trace terms. w/ E Süli & T Tscherpel

Preprint (Sep 2025, 24 pp.) available here.

32. Action-Driven Flows for Causal Variational Principles. w/ F Finster

Preprint (Mar 2024, 27 pp.) available here.

31. Rate-Form Equilibrium for an Isotropic Cauchy-Elastic Formulation. Part I: Modeling. w/ Th. Blesgen, S. Holthausen, N. Husemann & P Neff

(J. Nonlinear Science, link to published version), Preprint (Apr 2025, 32 pp.) available here.

30. On a Nitsche method for incompressible fluids with dynamic boundary conditions. w/ P Gazca, E Maringová & T Tscherpel

Preprint (Feb 2025, 58 pp.) available here.

29. Gradient integrability for bounded BD-minimizers. w/ L Beck & F Eitler

(Arch. Ration. Mech. Anal., to appear), Preprint (Dec 2024, 32 pp.) available here.

28. The Biot stress - right stretch relation for the compressible Neo-Hooke-Ciarlet-Geymonat model and Rivlin's cube problem w/ D Ghiba, S Holthausen, RJ Martin & P Neff

(Comm. Math. Anal., link to published version), Preprint (Aug 2024, 25 pp.) available here.

27. Extensions of divergence-free fields in L¹-based spaces. w/ S Schiffer

Preprint (Aug 2024, 48 pp.) available here.

26. Critical KMS inequalities in all dimensions. w/ P Lewintan & J Van Schaftingen

Preprint (May 2024, 12 pp.) available here.

25. 𝔸-harmonic approximation and partial regularity, revisited. w/ M Bärlin, C Irving & J Kristensen

(U Parma Rev Mat), Preprint (Dec 2022, 18 pp.) available here.

24. Sharp KMS-inequalities for general incompatibilities. w/ P Lewintan & P Neff

(M³AS - Math. Meth. Mod. Appl. Sc., link to published version), Preprint (Dec 2022, 32 pp.) available here.

23. Limiting L¹-estimates on halfspaces. w/ B Raita & J Van Schaftingen

(Adv. Math., link to published version), Preprint (Nov 2022, 22 pp.) available here.

22. Quasiconvex functionals of (p,q)-growth and the partial regularity of relaxed minimizers. w/ J Kristensen

(Arch. Ration. Mech. Anal., link to published version, Sep 2022, 90 pp.) available here.

21. Optimal incompatible Korn-Maxwell-Sobolev inequalities in all dimensions. w/ P Lewintan & P Neff

(Calc. Var. PDE, link to published version), Preprint (Jun 2022, 24 pp.) available here.

20. Natural annihilators and operators of constant rank over ℂ. w/ S Schiffer

(Atti Acc. Rendiconti Lincei (Mat. ed Appl.), link to published version), Preprint (Mar 2022, 18 pp.) available here.

19. On symmetric div-quasiconvex hulls and divsym-free L^∞-truncations. w/ L Behn & S Schiffer

(Ann. Inst. Henri Poincaré (C), link to published version), Preprint (Aug 2021, 40 pp.) available here.

18. Sharp trace and Korn inequalities for differential operators. w/ L Diening

(Potential Analysis, link to published version), Preprint (May 2021, 48 pp.) available here. For an overview talk, click here.

17. On Korn-Maxwell-Sobolev inequalities. w/ D Spector

(J. Math. Anal. Appl., link to published version). Preprint (Oct 2020, 12 pp.) available here.

16. 𝒜-quasiconvexity and partial regularity. w/ S Conti

(Calc. Var. PDE, link to published version). Preprint (Sep 2020, 24 pp.) available here.

15. Riesz potentials for ℂ-elliptic differential operators and continuity points for BV^A maps. w/ L Diening.

(Q.J. Math. (Oxford), link to published version). Preprint (Nov 2019, 13 pp.) available here.

14. The regularity of minima for the Dirichlet problem on BD.

(Arch. Ration. Mech. Anal., link to published version). Preprint (Aug 2019, v2, 54 pp.) available here.

13. Partial regularity for symmetric quasiconvex functionals on BD.

(J. Math. Pures Appl., link to published version) Preprint (Mar 2019, v1, Oct 2019, v2, 40 pp.) available here.

12. The Lipschitz truncation of functions of bounded variation. w/ D Breit & L Diening

(Indiana Univ. Math. J.). Preprint (Aug 2019, 16 pp.) available here.

11. On limiting trace inequalities for vectorial differential operators. w/ B Raita & J Van Schaftingen

(Indiana Univ. Math. J.). Preprint (Mar 2019, 30 pp.) available here.

10. Regularity for higher order quasiconvex problems with linear growth from below. w/ J Kristensen

Preprint (Mar 2019, 5 pp.) available here.

9. Partial Regularity for BV minimizers. w/ J Kristensen

(Arch. Ration. Mech. Anal., link to published version). Preprint (Mar 2018) available here. For an overview talk, click here.

8. Electrorheological fluids under random influences w/ D Breit

(Stochastics and PDEs: Analysis and Computations, link to published version). Preprint (Mar 2018) available here.

7. Critical Lp-differentiability of BD-maps. w/ B Raita

(Rev. Mat. Iberoam., link to published version). Preprint (Feb 2018, title has slightly changed) available here.

6. On a Neumann problem for functionals with linear growth. w/ M Bulicek & L Beck

(Annali della Scuola Normale Superiore (Pisa)). Preprint (Jan 2018) available here. For an overview talk, click here.

5. Limiting embeddings for 𝔸-weakly differentiable functions on domains. w/ B Raita

(Journal of Functional Analysis (JFA), link to published version) . Preprint (Sep 2017) available here.

4. On traces for functions of bounded 𝔸-variation. w/ D Breit & L Diening

(Analysis & PDE, link to published version). Preprint (Jul 2017) available here. For an overview talk, click here.

3. Sobolev regularity for convex functionals on BD. w/ J Kristensen

(Calc. Var. PDE, link to published version). Preprint (Mar 2018) available here.

2. On the singular set of generalised minimisers.

(OxPDE technical report 16.03). Preprint (Oct 2016) available here.

1. Symmetric-convex functionals of linear growth

(J. Ell. Parab. Equations, 2016). Preprint (Sep 2016) available here.

In Preparation

On the convergence of energies principle. /w B Raita & J Kristensen
Evolution by minimality for causal variational principles. w/ F Finster

Theses

[T3] Gmeineder, F.: Regularity Theory for Variational Problems on BD. DPhil Thesis, University of Oxford, 2017. xiv+298 pp. Thesis available at ORA.

[T2] Gmeineder, F.: Approximation Theorems for parabolic BV- and BD-spaces, MSc Thesis, University of Munich, 2013. vi + 87 pp.

[T1] Gmeineder, F.: Functions of bounded Variation and related Problems in the Calculus of Variations, University of Munich, 2012. x + 156 pp.