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Fachbereich Mathematik
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Foto: UHH/MIN/Christina Brandt

Prof. Dr. Thomas Schmidt

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Professur für Geometrische Partielle Differentialgleichungen

Anschrift

Universität Hamburg
Fakultät für Mathematik, Informatik und Naturwissenschaften
Fachbereich Mathematik
AM – Angewandte Mathematik
Bundesstraße 55
20146 Hamburg

Büro

Raum: 107

Sprechzeiten

Sprechstunden im Sommersemester 2025:
nach Vereinbarung

Sprechstunden in der vorlesungsfreien Zeit:
nach Vereinbarung

Kontakt

Tel.: +49 40 42838-5988

Schwerpunkte

  • Variationsrechnung
  • geometrische Maßtheorie
  • nicht-lineare partielle Differentialgleichungen
  • Regularitätstheorie

Website der Arbeitsgruppe

  • Existence theory for linear-growth variational integrals with signed measure data
    E. Ficola, T. Schmidt
    Preprint, 32 S., 2025, zur Veröffentlichung eingereicht
    Links: CVGMT arXiv
  • Lower semicontinuity and existence results for anisotropic TV functionals with signed measure data
    E. Ficola, T. Schmidt
    Preprint, 52 S., 2024, zur Veröffentlichung eingereicht
    Links: CVGMT arXiv

Publikationen

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Seite 1: Ergebnisse 1 bis 10 von 22

Seite 1: Ergebnisse 1 bis 10 von 22

@article{d930fa6c1dc2435ab1ff03c10d10131e,
title = "Partial regularity for variational integrals with Morrey-H{\"o}lder zero-order terms, and the limit exponent in Massari{\textquoteright}s regularity theorem",
abstract = "We revisit the partial C1,α regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the H{\"o}lder exponent α on structural assumptions for general zero-order terms. A particular case of our conclusions carries over to the parametric setting of Massari{\textquoteright}s regularity theorem for prescribed-mean-curvature hypersurfaces and there confirms optimal regularity up to the limit exponent.",
author = "Thomas Schmidt and Sch{\"u}tt, {Jule Helena}",
year = "2025",
month = apr,
day = "1",
doi = "10.1112/jlms.70139",
language = "English",
volume = "111",
journal = "J. Lond. Math. Soc. (2)",
issn = "0024-6107",
publisher = "John Wiley and Sons Ltd",
number = "4",

}

@article{0e15842ca5e14ca4ada25e4e4994e2c1,
title = "Isoperimetric conditions, lower semicontinuity, and existence results for perimeter functionals with measure data",
abstract = "We establish lower semicontinuity results for perimeter functionals with measure data on Rn and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In other words, we lay foundations of a perimeter-based variational approach to mean curvature measures on Rn capable of proving existence in various prescribed-mean-curvature problems with measure data. As crucial and essentially optimal assumption on the measure data we identify a new condition, called small-volume isoperimetric condition, which sharply captures cancellation effects and comes with surprisingly many properties and reformulations in itself. In particular, we show that the small-volume isoperimetric condition is satisfied for a wide class of (n−1)-dimensional measures, which are thus admissible in our theory. Our analysis includes infinite measures and semicontinuity results on very general domains. ",
author = "Thomas Schmidt",
year = "2025",
month = apr,
doi = "10.1007/s00208-024-03025-1",
language = "English",
volume = "391",
pages = "5729--5807",
journal = "Mathematische Annalen",
issn = "0025-5831",
publisher = "Springer New York",
number = "4",

}

@article{1928139520554ca4be4ff52dc8f5fb5d,
title = "The optimal H{\"o}lder exponent in Massari's regularity theorem",
abstract = "We determine the optimal H{\"o}lder exponent in Massari{\textquoteright}s regularity theorem for sets with variational mean curvature in Lp. In fact, we obtain regularity with improved exponents and at the same time provide sharp counterexamples.",
author = "Thomas Schmidt and Sch{\"u}tt, {Jule Helena}",
year = "2023",
month = jun,
doi = "10.1007/s00526-023-02495-6",
language = "English",
volume = "62",
journal = "Calc. Var. Partial Differential Equations",
issn = "0944-2669",
publisher = "Springer New York",
number = "5",

}

@article{8d3b7b303ac4415e9b7cd7b254f53c80,
title = "Higher integrability for the gradient of Mumford-Shah almost-minimizers",
abstract = "We extend a recent higher-integrability result for the gradient of minimizers of the Mumford-Shah functional to a suitable class of almost-minimizers. The extension crucially depends on an L∞ gradient estimate up to regular portions of the discontinuity set of an almost-minimizer.",
keywords = "Mumford-Shah energy, SBV functions, almost-minimizers, discontinuity set, excess decay, higher integrability, regularity theory",
author = "Sebastian Piontek and Thomas Schmidt",
year = "2020",
month = feb,
day = "10",
doi = "10.1051/cocv/2019063",
language = "English",
volume = "26",
journal = "ESAIM Control Optim. Calc. Var.",
issn = "1292-8119",
publisher = "EDP Sciences",

}

@article{db0c4080279146b09525654a893091f4,
title = "On the dual formulation of obstacle problems for the total variation and the area functional",
abstract = "We investigate the Dirichlet minimization problem for the total variation and the area functional with a one-sided obstacle. Relying on techniques of convex analysis, we identify certain dual maximization problems for bounded divergence-measure fields, and we establish duality formulas and pointwise relations between (generalized) BV minimizers and dual maximizers. As a particular case, these considerations yield a full characterization of BV minimizers in terms of Euler equations with a measure datum. Notably, our results apply to very general obstacles such as BV obstacles, thin obstacles, and boundary obstacles, and they include information on exceptional sets and up to the boundary. As a side benefit, in some cases we also obtain assertions on the limit behavior of p-Laplace type obstacle problems for p↘1.On the technical side, the statements and proofs of our results crucially depend on new versions of Anzellotti type pairings which involve general divergence-measure fields and specific representatives of BV functions. In addition, in the proofs we employ several fine results on (BV) capacities and one-sided approximation.",
keywords = "(Thin) obstacle problem, Anzellotti pairing, BV capacity, Convex duality, Optimality conditions, Total variation",
author = "Christoph Scheven and Thomas Schmidt",
year = "2018",
month = aug,
doi = "10.1016/j.anihpc.2017.10.003",
language = "English",
volume = "35",
pages = "1175--1207",
journal = "Annales de l'Institut Henri Poincare (C) Non Linear Analysis",
issn = "0294-1449",
publisher = "European Mathematical Society Publishing House",
number = "5",

}

@article{c75c6fe819d44729a116263d390118fe,
title = "Partial regularity for mass-minimizing currents in Hilbert spaces",
abstract = "Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1-80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121-1142], [Adv. Calc. Var. 7 (2014), 227-240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415-434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.",
author = "Luigi Ambrosio and {De Lellis}, Camillo and Thomas Schmidt",
year = "2018",
month = jan,
day = "1",
doi = "10.1515/crelle-2015-0011",
language = "English",
volume = "2018",
pages = "99--144",
journal = "J. Reine Angew. Math.",
issn = "1435-5345",
publisher = "Walter de Gruyter GmbH",
number = "734",

}

@article{21e5c0c8e94143ebb8bda3b6717aa0dc,
title = "BV supersolutions to equations of 1-Laplace and minimal surface type",
abstract = "We propose notions of BV supersolutions to (the Dirichlet problem for) the 1-Laplace equation, the minimal surface equation, and equations of similar type. We then establish some related compactness and consistency results.Our main technical tool is a generalized product of L∞ divergence-measure fields and gradient measures of BV functions. This product crucially depends on the choice of a representative of the BV function, and the proofs of its basic properties involve results on one-sided approximation and fine (semi)continuity in the BV context.",
author = "Christoph Scheven and Thomas Schmidt",
year = "2016",
month = aug,
day = "5",
doi = "10.1016/j.jde.2016.04.015",
language = "English",
volume = "261",
pages = "1904--1932",
journal = "J. Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",
number = "3",

}

@article{84fc58a31e7f46998109d9297102b9a6,
title = "Interior gradient regularity for BV minimizers of singular variational problems",
abstract = "We consider a class of vectorial integrals with linear growth, where, as a key feature, some degenerate/singular behavior is allowed. For generalized minimizers of these integrals in BV, we establish interior gradient regularity and-as a consequence-uniqueness up to constants. In particular, these results apply, for 1Ω (1+|∇w(x)|p)1/p dx.",
keywords = "Functions of bounded variation, Generalized minimizers, Regularity, Singular elliptic problems",
author = "Lisa Beck and Thomas Schmidt",
year = "2015",
month = jun,
doi = "10.1016/j.na.2015.02.011",
language = "English",
volume = "120",
pages = "86--106",
journal = "Nonlinear Analysis: Theory, Methods & Applications",
issn = "1873-5215",
publisher = "Elsevier",

}

@article{9366ca2e74e94801bfc7dd4ef9c88b4c,
title = "Convex duality and uniqueness for BV-minimizers",
abstract = "There are two different approaches to the Dirichlet minimization problem for variational integrals with linear growth. On the one hand, one commonly considers a generalized formulation in the space of functions of bounded variation. On the other hand, there is a closely related maximization problem in the space of divergence-free bounded vector fields, namely the dual problem in the sense of convex analysis.In this paper, we extend previous results on the duality correspondence between the generalized and the dual problem to a full characterization of their extremals via pointwise extremality relations. Furthermore, we discuss related uniqueness issues for both kinds of solutions and their relevance in the regularity theory of generalized minimizers.Our approach is sufficiently general to cover arbitrary dimensions, non-smooth integrands, and unbounded, irregular domains.",
keywords = "Convex duality, Functions of bounded variation, Generalized minimizers, Primary, Secondary, Variational integrals",
author = "Lisa Beck and Thomas Schmidt",
year = "2015",
month = may,
day = "15",
doi = "10.1016/j.jfa.2015.03.006",
language = "English",
volume = "268",
pages = "3061--3107",
journal = "J. Funct. Anal.",
issn = "1096-0783",
publisher = "Academic Press Inc.",
number = "10",

}

@article{5e3dd73331e545118d54b6894824b12f,
title = "Strict interior approximation of sets of finite perimeter and functions of bounded variation",
abstract = " It is well known that sets of finite perimeter can be strictly approximated by smooth sets, while, in general, one cannot hope to approximate an open set Ω of finite perimeter in ℝⁿ strictly from within. In this note we show that, nevertheless, the latter type of approximation is possible under the mild hypothesis that the (n−1)-dimensional Hausdorff measure of the topological boundary ∂Ω equals the perimeter of Ω. We also discuss an optimality property of this hypothesis, and we establish a corresponding result on strict approximation of BV-functions from a prescribed Dirichlet class.",
author = "Thomas Schmidt",
year = "2015",
month = may,
doi = "10.1090/s0002-9939-2014-12381-1",
language = "English",
volume = "143",
pages = "2069--2084",
journal = "Proc. Am. Math. Soc.",
issn = "0002-9939",
publisher = "American Mathematical Society",
number = "5",

}

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Alle Lehrveranstaltungen an der Universität Hamburg

Frühere Vorlesungen in Erlangen (2009–2015):

  • Gewöhnliche Differentialgleichungen (Bachelor-Vorlesung, 2+2 SWS)
  • BV-Funktionen (Bachelor-/Master-Vorlesung, 2+1 SWS)
  • Minimalflächen und das Plateau-Problem (Master-Vorlesung, 4+2 SWS)
  • Partielle Differentialgleichungen II (Master-Vorlesung, 4+2 SWS)
  • Partielle Differentialgleichungen I (Bachelor-/Master-Vorlesung, 4+2 SWS)
  • Maßtheorie (Analysis IV) (Bachelor-Vorlesung, 2+2 SWS)
  • Variationsprobleme in BV (Master-Vorlesung, 4+1 SWS)

seit 02/2016: Professor (W2), Fachbereich Mathematik, Universität Hamburg

04/2008–01/2016: Postdoc/akad. Rat a. Z., Department Mathematik, FAU Erlangen-Nürnberg

  • 2011–2013 für Auslandsaufenhalt in Pisa und Zürich beurlaubt
  • 04/2015: Habilitation in Mathematik bei F. Duzaar

09/2011–08/2013: Forschungsstipendiat, SNS Pisa und Institut für Mathematik, Universität Zürich

10/2004–03/2008: Doktorand/Postdoc, Mathematisches Institut, HHU Düsseldorf

  • 12/2006: Promotion in Mathematik bei K. Steffen

10/2000–08/2004: Studium der Mathematik und Physik (Nebenfach Informatik), HHU Düsseldorf

  • 09/2002: Vordiplom in Mathematik
  • 03/2003: Vordiplom in Physik
  • 08/2004: Diplom in Mathematik, Diplomarbeit bei K. Steffen

Kontakt

Katrin Kopp
Fachbereich Mathematik
Team-Assistentin
AM – Angewandte Mathematik
Bundesstraße 55, Raum 134
20146 Hamburg
Tel.: +49 40 42838-4941
Vertretung: Für die Angewandte Mathematik: Monika Jampert, Isabel Schmara

    Mo. - Fr. Gleitzeit 08:00-16:00 Uhr
    (Home Office Di u. Fr)