FACULTY OF

MATHEMATICS

Research Interests

• Boundary value problems for Willmore surfaces
• Qualitative properties of solutions of elliptic boundary value problems of higher order
• Semilinear eigenvalue problems involving critical Sobolev exponents
• Biharmonic equations with supercritical growth
• Parabolic systems with critical growth
• Navier-Stokes equations

M.Sc. Boris Gulyak

Seminar on Differential Geometry and Analysis

Monograph

Gazzola, F.; Grunau, H.-Ch.; Sweers, G.:
Polyharmonic boundary value problems,
Positivity preserving and nonlinear higher order elliptic equations in bounded domains.
Springer Lecture Notes in Mathematics 1991, Springer-Verlag: Heidelberg etc., 2010.
[Pdf]
[Errata] (as uncovered so far)
Copyright: Springer-Verlag. The original monograph is available on http://link.springer.com/

Lecture notes

Grunau, H.-Ch.:
Abbildungsgrad und Fixpunktsätze (Degree of mapping and fixed point theorems),
Vorlesungsausarbeitung, basiert auf Vorlesungen von E. Heinz (Göttingen)
Lecture notes, based upon lectures of E. Heinz (Göttingen)
[Ps] [Pdf]

Textbook

Grauert, H., Grunau, H.-Ch.:
Lineare Algebra und analytische Geometrie,
Copyright: Oldenbourg-Verlag, 1999-2009.
Ab 2009: Alle Rechte bei den Autoren.
[PS-Datei] [DVI-Datei] [PDF-Datei]

Articles

Grunau, H.-Ch.; Müller, M.:
A biharmonic analogue of the Alt-Caffarelli problem, Math. Ann. , to appear.
[Pdf]

Grunau, H.-Ch.; Okabe, S.:
Willmore obstacle problems under Dirichlet boundary conditions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 24, 1415-1462 (2023).
[Pdf]
The original article is available on https://journals.sns.it/index.php/annaliscienze/

Grunau, H.-Ch.
Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients, Arch. Math. 117, 95-104 (2021).
[Pdf]
The original article is available on http://link.springer.com/

Deckelnick, K.; Doemeland, M.; Grunau, H.-Ch.
Boundary value problems for the Helfrich functional for surfaces of revolution -- Existence and asymptotic behaviour, Calc. Var. Partial Differ. Equ., 60, Article number 32 (2021).
[Pdf]
The original article is available on http://link.springer.com/

Grunau, H.-Ch.; Miyake, N.; Okabe, S.:
Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations, Adv. Nonlinear Anal. 10, 353-370 (2021).
[Pdf]
The original article is available on https://www.degruyter.com/

Grunau, H.-Ch.; Romani, G., Sweers, G.:
Differences between fundamental solutions of general higher-order elliptic operators and of products of second-order operators, Math. Ann. 381, 1031-1084 (2021).
[Pdf]
The original article is available on http://link.springer.com/

Grunau, H.-Ch.:
Boundary Value Problems for the Willmore Functional,
survey article, RIMS Kokyuroku series, No. 2146 (2020)
Proceedings of the workshop ``Analysis of Shapes of Solutions to Partial Differential Equations'', June 27-29, 2018.
[Pdf]

Eichmann, S.; Grunau, H.-Ch.:
Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions.
Adv. Calc. Var. 12, 333–361 (2019).
[Pdf]
Copyright: de Gruyter. The original article is available on https://www.degruyter.com/view/j/acv

Dipierro, S.; Grunau, H.-Ch.:
Boggio's formula for fractional polyharmonic Dirichlet problems.
Ann. Mat. Pura Appl. (1923-) 196, 1327-1344 (2017).
[Pdf]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Deckelnick, K.; Grunau, H.-Ch.; Röger, M:
Minimising a relaxed Willmore functional for graphs subject to boundary conditions.
Interfaces Free Bound. 19, 109-140 (2017).
[Pdf]
Copyright: EMS - European Mathematical Society Publishing House.
The original article is available on https://www.ems-ph.org/journals/journal.php?jrn=ifb

Grunau, H.-Ch.; Lenor, St.:
Uniform estimates and convexity in capillary surfaces.
Nonlinear Analysis A: T.M.A. 97, 83-93 (2014).
[Pdf]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Grunau, H.-Ch.; Sweers, G.:
In any dimension a "clamped plate" with a uniform weight may change sign.
Nonlinear Analysis A: T.M.A. 97, 119-124 (2014).
[Pdf]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Grunau, H.-Ch.; Sweers, G.:
A clamped plate with a uniform weight may change sign.
Discrete Cont. Dynam. Systems - S (Proceedings etc.) 7, 761 - 766 (2014)
[Pdf]
Copyright: American Institute of Mathematical Sciences, first published with AIMS Press.

Grunau, H.-Ch.; Robert, F.:
Uniform estimates for polyharmonic Green functions in domains with small holes.
In: J. Serrin, E. Mitidieri, V. Radulescu (eds.), Recent Trends in Nonlinear Partial Differential Equations II: Stationary Problems.
Contemporary Mathematics 595, 263-272 (2013).
[Ps] [Pdf] [Dvi]
Copyright: American Mathematical Society. The original article is available on http://www.ams.org/books/conm/595/

Grunau, H.-Ch.:
The asymptotic shape of a boundary layer of symmetric Willmore surfaces of revolution.
In: C. Bandle et al. (eds.), Inequalities and Applications 2010.
International Series of Numerical Mathematics 161, 19-29 (2012).
[Ps] [Pdf]
Copyright: Springer Basel. The original article is available on http://link.springer.com/

Grunau, H.-Ch.; Robert, F.; Sweers, G.:
Optimal estimates from below for biharmonic Green functions.
Proc. Amer. Math. Society 139, 2151-2161 (2011).
[Pdf]
Copyright: American Mathematical Society AMS. The original article is available on http://www.ams.org/journals/proc/

Gazzola, F.; Grunau, H.-Ch.; Sweers, G.:
Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions.
Ann. Mat. Pura Appl. 189, 475-486 (2010).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Dall'Acqua, A., Fröhlich, St., Grunau, H.-Ch., Schieweck, F.:
Symmetric Willmore surfaces of revolution satisfying arbitrary Dirichlet boundary data.
Adv. Calc. Var. 4, 1-81 (2011).
[Ps] [Pdf]
Copyright: de Gruyter. The original article is available on https://www.degruyter.com/view/j/acv

Grunau, H.-Ch.:
Nonlinear questions in clamped plate models (Survey article).
Milan J. Math. 77, 171-204 (2009).
[Ps.gz] [Pdf]
Copyright: Birkhäuser-Verlag. The original article is available on http://link.springer.com/

Deckelnick, K. , Grunau, H.-Ch.:
A Navier boundary value problem for Willmore surfaces of revolution.
Analysis 29, 229-258 (2009).
[Ps] [Pdf]
Copyright: de Gruyter. The original article is available on http://www.degruyter.com/

Gazzola, F., Grunau, H.-Ch.:
Some new properties of biharmonic heat kernels.
Nonlinear Analysis T.M.A. 70, 2965 - 2973 (2009).
[Ps] [Pdf] [Dvi]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Dall'Acqua, A., Deckelnick, K., Grunau, H.-Ch.:
Classical solutions to the Dirichlet problem for Willmore surfaces of revolution.
Adv. Calc. Var. 1, 379 - 397 (2008).
[Ps] [Pdf] [Dvi]
Copyright: de Gruyter. The original article is available on https://www.degruyter.com/view/j/acv

Ferrero, A., Grunau, H.-Ch., Karageorgis, P.:
Supercritical biharmonic equations with power-type nonlinearity.
Ann. Mat. Pura Appl. 188, 171 - 185 (2009).
[Ps] [Pdf] [Dvi], http://arxiv.org/abs/0711.2202Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Ferrero, A., Gazzola, F., Grunau, H.-Ch.:
Decay and eventual local positivity for biharmonic parabolic equations.
Discrete Cont. Dynam. Systems 21, 1129 - 1157 (2008).
[Ps] [Pdf] [Dvi]
Copyright: American Institute of Mathematical Sciences, first published with AIMS Press.

Deckelnick, K., Grunau, H.-Ch.:
Stability and symmetry in the Navier problem for the one-dimensional Willmore equation.
SIAM J. Math. Anal. 40, 2055 - 2076 (2009).
[Ps] [Pdf]
Copyright: SIAM. The original article is available on http://epubs.siam.org/SIMA/sima_toc.html

Grunau, H.-Ch., Robert, F.:
Positivity and almost positivity of biharmonic Green's functions under Dirichlet boundary conditions.
Arch. Rational Mech. Anal., 195, 865-898 (2010).
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Preprint versions:
Stability of the positivity of biharmonic Green's functions under perturbations of the domain.
[Ps] [Pdf] [Dvi],
Positivity issues of biharmonic Green's functions under Dirichlet boundary conditions.
[Ps] [Pdf] [Dvi], http://arxiv.org/abs/0705.3301Research announcement:
Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains.
C. R. Math. Acad. Sci. Paris , Ser. I 347, 163 - 166 (2009).
Copyright: Academie des Sciences / Elsevier Masson SAS.
The article is available on http://www.sciencedirect.com/science/journal/1631073X.

Gazzola, F., Grunau, H.-Ch.:
Local eventual positivity for a biharmonic heat equation in R^n.
Discrete Cont. Dynam. Systems - S (Proceedings etc.) 1, 83 - 87 (2008).
[Ps] [Pdf] [Dvi]
Copyright: American Institute of Mathematical Sciences, first published with AIMS Press.

Grunau, H.-Ch., Ould Ahmedou, M., Reichel, W.:
The Paneitz equation in hyperbolic space.
Annales Inst. H. Poincare (C) Nonlinear Analysis 25, 847 - 864 (2008).
[Ps] [Pdf] [Dvi]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Ferrero, A., Grunau, H.-Ch.:
The Dirichlet problem for supercritical biharmonic equations with power-type nonlinearity.
J. Differ. Equations 234, 582 - 606 (2007).
[Ps] [Pdf] [Dvi]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Arioli, G., Gazzola, F., Grunau, H.-Ch., Sassone, E.:
The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four.
Nonl. Differ. Equ. Appl. NoDEA 15, 69-90 (2008).
[Ps] [Pdf] [Files for computer assisted proof]
Copyright: Birkhäuser-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch., Sweers, G.:
Regions of positivity for polyharmonic Green functions in arbitrary domains.
Proc. Amer. Math. Society 135, 3537-3546 (2007).
[Ps] [Pdf] [Dvi]
Copyright: AMS. The original article is available on http://www.ams.org/journals/proc/

Berchio, E., Grunau, H.-Ch.:
Local regularity of weak solutions of semilinear parabolic systems with critical growth.
J. Evolution equations 7, 177-196 (2007).
[Ps] [Pdf] [Dvi]
Copyright: Birkhäuser-Verlag. The original article is available on http://link.springer.com/

Gazzola, F., Grunau, H.-Ch.:
Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay.
Calc. Var. 30, 389-415 (2007).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Arioli, G., Gazzola, F., Grunau, H.-Ch.:
Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity.
J. Differ. Equations 230, 743 - 770 (2006).
[Ps] [Pdf] [Files for computer assisted proof]
Copyright: Elsevier. The original article is available on http://www.sciencedirect.com/

Deckelnick, K., Grunau, H.-Ch.:
Boundary value problems for the one-dimensional Willmore equation,
Calc. Var. 30, 293-314 (2007).
[Ps] [Pdf]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/
Preliminary and more special version: Technical Report 05-02, University of Magdeburg.
[Ps] [Pdf]

Gazzola, F., Grunau, H.-Ch.:
Radial entire solutions for supercritical biharmonic equations.
Math. Annal. 334, 905 - 936 (2006).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch., Kühnel, M.:
On the existence of Hermitian-harmonic maps from complete Hermitian to complete Riemannian manifolds,
Math. Z. 249, 297-327 (2005).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Arioli, G., Gazzola, F., Grunau, H.-Ch., Mitidieri, E.:
A semilinear fourth order elliptic problem with exponential nonlinearity,
SIAM J. Math. Anal. 36, 1226-1258 (2005).
[Ps] [Pdf] [Files for computer assisted proof]
Copyright: SIAM. The original article is available on http://www.siam.org/journals/sima/sima.htm

Dall'Acqua, A., Grunau, H.-Ch., Sweers, G.:
On a conditioned Brownian motion and a maximum principle on the disk,
J. Anal. 93, 309-329 (2004).
[Ps] [Pdf] [Dvi]

Gazzola, F., Grunau, H.-Ch., Squassina, M.:
Existence and nonexistence results for critical growth biharmonic elliptic equations,
Calc. Var. PDE 18, 117-143 (2003).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/       

Gazzola, F., Grunau, H.-Ch., Mitidieri, E.:
Hardy inequalities with optimal constants and remainder terms,
Transactions Amer. Math. Soc. 356, 2149-2168 (2004).
[Ps] [Pdf] [Dvi]
Copyright: AMS. The original article is available on http://www.ams.org/tran/

Grunau, H.-Ch., Sweers, G.:
Optimal conditions for anti-maximum principles,
Ann. Sc. Norm. Sup. Pisa. Cl. Sci. (4) 30, 499-513 (2001)

Grunau, H.-Ch.:
Positivity, change of sign and buckling eigenvalues in a one-dimensional fourth order model problem,
Adv. Differ. Equations. 7, 177-196 (2002).
[Ps] [Pdf]
Copyright: Khayyam-Publishing http://projecteuclid.org/euclid.ade/

Grunau, H.-Ch., Sweers, G.:
Sharp estimates for iterated Green functions,
Proc. Royal Soc. Edinburgh. 132A, 91-120 (2002).
[Ps] [Pdf] [Dvi]
This article is reproduced by permission of the Royal Society of Edinburgh.
The original article is available on http://ninetta.ingentaselect.com/

Grunau, H.-Ch., Sweers, G.:
Nonexistence of local minima of supersolutions for the circular clamped plate,
Pacific J. Math 198, 437-442 (2001).
[Ps] [Pdf] [Dvi]
The original article is available on http://msp.org/pjm/

Gazzola, F., Grunau, H.-Ch.:
Critical dimensions and higher order Sobolev inequalities with remainder terms,
Nonl. Differ. Equ. Appl. NoDEA 8, 35-44 (2001).
[Ps] [Pdf] [Dvi]
Copyright: Birkhäuser-Verlag. The original article is available on http://link.springer.com/

Gazzola, F., Grunau, H.-Ch.:
On the role of space dimension $n=2+2sqrt{2}$ in the semilinear Brezis-Nirenberg eigenvalue problem,
Analysis 20, 395-399 (2000).
[Ps] [Pdf] [Dvi]
Copyright: de Gruyter http://www.degruyter.com

Grunau, H.-Ch., Sweers, G.:
Sign change for the Green function and for the first eigenfunction of equations of clamped plate type,
Archive Rational Mech. Anal. 150, 179-190 (1999).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch.:
L^p-decay rates for strong solutions of a perturbed Navier-Stokes system in R^3, in: J. G. Heywood, K. Masuda, R. Rautmann, V. A. Solonnikov (eds.),
Theory of the Navier-Stokes Equations, Series Adv. Math. Appl. Sciences 47, 64-71 (1998).

Grunau, H.-Ch., Sweers, G.:
The role of positive boundary data in generalized clamped plate equations,
Z. angew. Math. Phys., 49, 420-435 (1998).
[Ps] [Pdf] [Dvi]
Copyright: Birkhäuser-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch., Sweers, G.:
Maximum principles and positive principal eigenfunctions for polyharmonic equations, in: G. Caristi, E. Mitidieri (eds.),
Reaction Diffusion Systems, Lecture Notes in Pure and Applied Mathematics 194, 163-182 (1998).

Grunau, H.-Ch.:
Uniqueness of small solutions to the Dirichlet problem for the higher dimensional H-system,
Rocky Mountain J. Math. 27, 801-815 (1997).

Grunau, H.-Ch., Sweers, G.:
Positivity properties of elliptic boundary value problems of higher order,
Proc. 2nd World Congress of Nonlinear Analysis, Nonlinear Anal., T. M. A. 30, 5251-5258 (1997).

Grunau, H.-Ch., Wahl, W. von:
Regularity considerations for semilinear parabolic systems,
Rend. Istit. Mat. Univ. Trieste 28 (Suppl.), 221-233 (1997).

Grunau, H.-Ch., Sweers, G.:
Classical solutions for some higher order semilinear elliptic equations under weak growth conditions,
Nonlinear Anal., T.M.A. 28, 799-807 (1997).

Grunau, H.-Ch., Sweers, G.:
Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions,
Math. Ann. 307, 589-626 (1997).
[Ps] [Pdf] [Dvi]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch.:
On a conjecture of P. Pucci and J. Serrin,
Analysis 16, 399-403 (1996).
[Ps] [Pdf] [Dvi]
Copyright: de Gruyter http://www.degruyter.com

Grunau, H.-Ch., Sweers, G.:
Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions,
Math. Nachr. 179, 89-102 (1996).
[Ps] [Pdf] [Dvi]
Copyright: Mathematische Nachrichten, Wiley Verlag http://www.wiley-vch.de/

Grunau, H.-Ch.:
Critical exponents and multiple critical dimensions for polyharmonic operators. II,
Boll. Unione Mat. Ital.(7) 9-B, 815-847 (1995).

Grunau, H.-Ch.:
Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,
Calculus of Variations and PDE 3, 243-252 (1995).
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Bernis, F., Grunau, H.-Ch.:
Critical exponents and multiple critical dimensions for polyharmonic operators,
J. Differ. Equations117, 469-486 (1995).

Grunau, H.-Ch., Wahl, W. von:
Regularity of weak solutions of semilinear parabolic systems of arbitrary order,
Journal d'Analyse 62, 307-322 (1994).
[Ps] [Pdf] [Dvi]

Grunau, H.-Ch.:
L^2-decay rates for weak solutions of a perturbed Navier-Stokes system in R^3,
J. Math. Anal. Appl. 185, 340-349 (1994).

Grunau, H.-Ch.:
The Reynolds number and large time behaviour for weak solutions of the Navier-Stokes equations,
Z. angew. Math. Phys. 44, 587-593 (1993).

Grunau, H.-Ch.:
Boundedness for large |x| of suitable weak solutions of the Navier-Stokes equations with prescribed velocity at infinity,
Commun. Math. Phys. 151, 577-587 (1993).
[Pdf]
Copyright: Springer-Verlag. The original article is available on http://link.springer.com/

Grunau, H.-Ch.:
The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order,
Analysis 11, 83-90 (1991).

These lectures gives an impression of my present research interests: Marburg, Perugia, Nizza .

Boundary value problems for Willmore surfaces

The Willmore equation, i.e. the Euler-Lagrange equation of the Willmore functional, is a particularly challenging and important problem in nonlinear analysis. It is quasilinear and of fourth order. Many methods which are well established in second order problems do not apply any longer. Nevertheless, significant progress could be achieved in the past years e.g. by L. Simon, E. Kuwert, R. Schätzle, T. Riviere and many others. Most of these results, however, concern closed surfaces. As for boundary value problems only little is known so far since very hard compactness difficulties have to be overcome. By means of numerical studies and analytical investigations we want to study first quite symmetric prototype situations in order to outline in which direction one may expect to obtain a-priorily bounded minimising sequences and classical solutions. Our student Stephan Lenor has produced a number of pictures. This is a joint project with my colleagues Klaus Deckelnick und Friedhelm Schieweck who intend to develop numerical algorithms and convergence results also in more general situations, e.g. for graphs over two dimensional domains. Analysis, numerical analysis and numerics will intensively interact.
Supported by Deutsche Forschungsgemeinschaft, 1.10.2008-30.09.2010. Anna Dall'Acqua was working on this project.

Qualitative properties of solutions of elliptic boundary value problems of higher order

Here one may think of the clamped plate equation (biharmonic operator under Dirichlet boundary
conditions) as the most simple prototype for higher order elliptic boundary value problems. Of particular interest are positivity preserving properties of the corresponding solution operators:
Is it possible to find conditions on the domain and the differential operator such that positive data always yield positive solutions? Even for the "simple" example of the clamped plate equation it is up to now not possible to answer this question completely. There are as well examples of domains with positivity preserving as of domains, where change of sign occurs. I am mainly interested in the first class of domains: In two dimensions, e.g., in domains close to the disk, positive right hand sides always give rise to positive solutions. "Upwards pushing yields upwards bending."

Positivity properties of linear boundary value problems are not only of interest on their own, but one may also try to apply them to nonlinear problems. Although first results could be achieved it is still by far not obvious to what extent the (relatively restricted) positivity results will be of use for general nonlinear elliptic boundary value problems of higher order. I hope for some progress in problems from physics (mechanics, hydrodynamics) and differential geometry.

In this field I collaborate with Guido Sweers (Cologne, TU Delft) and Frederic Robert.

Semilinear eigenvalue problems involving critical Sobolev exponents

These equations are closely related to problems in conformal geometry. Here we are basing upon two widely known papers of Brezis-Nirenberg and Pucci-Serrin and try to find out in how far the results kown for second order problems extend to boundary value problems of higher order. Emphasis is laid on the interplay with qualitative properties of solutions. In this connection a modified version of a conjecture of Pucci and Serrin concerning the "critical dimension phenomenon" for semilinear polyharmonic problems could be proved.  A full proof of the original conjecture, however, seems up to now out of reach.
Recently, the connections with the above mentioned positivity properties could be more intensively exploited. Here a decomposition method with respect to pairs of dual cones in higher order Sobolev spaces has proved to be particularly useful. This method replaces the decomposition in positive and negative part, which is no longer admissible in these spaces. In this way, compactness properties of the corresponding variational functionals could be described efficiently.
In this field I am particularly working with Filippo Gazzola.

Biharmonic equations with supercritical growth

Variational techniques are no longer available. Instead, comparison principles, the sub-/supersolution method or in the case of radially symmetric solutions methods from dynamical systems have to be applied. Combining supercritical growth with differential operators of higher order gives rise to subtle technical problems. Also here, I collaborate with Filippo Gazzola, which was 2005 and 2006 supported by the Vigoni-programme of DAAD (Bonn) and CRUI (ROM).

Parabolic systems with critical growth

Under consideration are semilinear systems, whose nonlinear terms grow critically with respect to the canonical energy norm of weak solutions, i.e. we are dealing with controllable growth conditions.
Without further assumptions (like e.g. sign conditions on the nonlinear term) it was shown in a joint paper with Wolf von Wahl that every weak solution is regular. For this result we crucially employ a continuity method with the time as parameter of continuity.
Further, we consider geometric evolution equations in order to construct Hermitian harmonic mappings between noncompact complete manifolds. This semilinear system has quadratic growth with respect to the gradient and is not in divergence form.

Navier-Stokes equations

Here I am interested in instationary problems in exterior domains with nonzero prescribed velocity at infinity. I have been investigating regularity of suitable weak solutions and the asymptotic behaviour, as the time tends to infinity, of perturbations of physically reasonable stationary solutions with small energetic Reynolds number.
 

Meister, Ch.:

L^p - L^q - Theorie und asymptotisches Verhalten der Störungsenergie bei einem Navier-Stokes-System im Außenraum mit Anströmgeschwindigkeit für kleine Reynoldszahlen
Dissertation, Universität Bayreuth, 2001.
[Pdf]

Bräu, T.:

Eine Zerlegungsmethode bzgl. Paaren zueinander dualer Kegel und Anwendung in Sobolevräumen höherer Ordnung
Studienarbeit, Universität Magdeburg, 2005.
[Pdf]
English translation: A decomposition method with respect to dual cones and its application to higher order Sobolev spaces
Seminar paper, University of Magdeburg, 2005.
[Pdf]

Köckritz, Ch.:

Lokale Positivität der polyharmonischen Greenschen Funktion
Studienarbeit, Universität Magdeburg, 2007.
[Pdf]

Jachalski, S.:

Asymptotic behaviour for Willmore surfaces of revolution under natural boundary conditions
Englisches Preprint, Teil der Diplomarbeit, Universität Magdeburg, 2010.
[Pdf]

Pulst, L.:

Ein Analogon eines Satzes von Moser auf der S^4
Diplomarbeit, Universität Magdeburg, 2010.
[Pdf]

Lenor, S.:

Konvexität und gleichmäßige Abschätzungen bei der Kapillaritätsgleichung
Diplomarbeit, Universität Magdeburg, 2012.
[Pdf]

Eichmann, S.:

Nichtperiodische Fortsetzbarkeit von Willmore-Flächen unter Axialsymmetrie
Diplomarbeit, Universität Magdeburg, 2014.
[Pdf]

Gulyak, B.:

Erste a-priori-Abschätzungen für Willmore-Graphen über allgemeinen Gebieten
Projektarbeit, Universität Magdeburg, 2014.
[Pdf]

Doemeland, M.

Verallgemeinerung eines Existenzsatzes für axialsymmetrische Minimierer des Willmore-Funktionals auf das Helfrich-Funktional
Projektarbeit, Universität Magdeburg, 2015.
[Pdf]

Pulst, L.:

Dominance of positivity of the Green's function associated to a perturbed polyharmonic Dirichlet boundary value problem by pointwise estimates
PhD Dissertation, Universität Magdeburg, 2015.
[Pdf]

Barlage, E.:

Positive Lösungen elliptischer Gleichungen zweiter Ordnung mit kritischem Wachstum in kontrahierbaren Gebieten
Masterarbeit, Universität Magdeburg, 2017.
[Pdf]

Eichmann, S.:

Willmore Surfaces of Revolution Satisfying Dirichlet Data
Dissertation, Universität Magdeburg, 2017.
[Pdf]

Doemeland, M.:

Axialsymmetrische Minimierer des Helfrich Funktionals
Masterarbeit, Universität Magdeburg, 2017.
[Pdf]

Himmel, B.:

Die Boggio-Formel für polyharmonische Dirichletprobleme
Bachelorarbeit, Universität Magdeburg, 2019.
[PDF]

Gulyak, B.:

Willmore Boundary Value Problems
PhD-Dissertation, Universität Magdeburg, 2023.
[Pdf]

I disclaim responsibility for the contents of web sites of third parties, which are linked to my websites.
No liability for the information provided on my websites.

Buch mit Hans Grauert "Lineare Algebra und Analytische Geometrie"

Copyright: Oldenbourg-Verlag, München, 1999-2009.
Ab 2009: Alle Rechte bei den Autoren.

Kompletter Download: [ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Vorlesungsmanuskript "Abbildungsgrad und Fixpunktsätze"

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Algorithmische Mathematik: Beispielprogramme

Paragraph 1
[ Additionsmaschine1.c] [ Additionsmaschine2.c] [ Additionsmaschine2a.c] [ Additionsmaschine2b.c]
[ Additionsmaschine2c.c] [ Fakultaet.c] [ Taschenrechner.c]
Paragraph 2
[ ggT.c] [ Primfaktorzerlegung.c]
Paragraph 3
[ Einheiten_in_Z_mod_mZ.c] [ Chin_Restsatz.c]
Paragraph 4
[ Einlesen_strings.c] [ Einlesen_lange_strings.c] [ Liste_einfach.c] [ Liste_doppelt_verkettet.c] [ Addition_lang.c]
Paragraph 5
[ Breitensuche.c] [ Abstandsbestimmung.c]
Paragraph 6
[ Bellman_Ford.c] [ Dijkstra.c]
Paragraph 7
[ Exponentialfunktion1.c] [ Exponentialfunktion2.c] [ Exponentialfunktion3.c] [ Exponentialfunktion4.c]
[ Logarithmus1.c] [ Logarithmus2.c] [ Cordic_Division.c]
Gnuplot-Interface von N. Devillard: [ gnuplot_i.c] [ gnuplot_i.h] [ gnuplot_i.o]
[ Kompilierungskommandos.txt]
Paragraph 8
[ Logarithmus_bisek.c] [ Min_Max_a.c] [ Min_Max_b.c] [ Wurzel1.c] [ Wurzel2.c] [ Kompilierungskommandos.txt]
Paragraph 9
[ Differentiation1.c] [ Differentiation2.c] [ Integration1.c] [ Integration2.c] [ Kompilierungskommandos.txt]


Übungen:
Web-Seite von Herrn Ludwig Pulst

Maple-Worksheets zu Lineare Algebra I

Grundlagen: Matrizenkalkül, Gaußscher Algorithmus
[ Grundlagen.mws]
Trigonalisierung beliebiger komplexer qudratischer Matrizen mittels unitärer Koordinatenwechsel
[ Trigonalisierung.mws]
Vertiefung: Determinanten, Eigenwerte, Eigenvektoren, Diagonalisierung
[ Vertiefung.mws]

Maple-Worksheets zu Analysis I

(erstellt von Frau Stephanie Lehmann und Frau Elke Eisenschmidt)

Zur Konvergenz von Folgen
[ Folgen.mws]
Polarkoordinaten
[ Komplex.mws]
Stetigkeit
[ Stetig.mws]
Kurvendiskussion und Differentialrechnung
[ Differentialrechnung.mws]

Maple-Worksheets zu Analysis II

Unstetigkeitsverhalten von Funktionen zweier Veränderlicher
[ Unstetig.mws]
Zur Integration
[ Integration.mws]
Kurven und Ableitungen von Funktionen mehrerer Veränderlicher
[ PartielleAbleitung.mws]
Implizite Funktionen
[ ImpliziteFunktion.mws]

Maple-Worksheets zu Mathematik für Physiker III/IV

Fourierreihen
[ fourierseries2.mws]

Maple-Worksheets zu Funktionentheorie

Hier gibt es ein aktuelles Lehrbuch, welches die theoretische Entwicklung des Stoffs mit zahlreichen Maple-Worksheets auf sehr gelungene Weise unterstützt und illustriert:
W. Forst, D. Hoffmann,
Funktionentheorie erkunden mit Maple, Springer-Verlag: Berlin etc. 2002.
Die zugehörigen Worksheets können auf der Homepage von Herrn Kollegen W. Forst (Ulm) heruntergeladen werden.

Maple-Worksheets zu Partielle Differentialgleichungen

(Animationen für Laplace-, Wellen- und Wärmeleitungsgleichung mittels Fourierreihenentwicklung)

Laplacegleichung
[ poissonkernel.mws]
Wärmeleitungsgleichung räumlich eindimensional
[ heatfourier.mws]
Wärmeleitungsgleichung räumlich zweidimensional, Quadrat
[ heatfourier2D_1.mws]
Wellengleichung räumlich eindimensional
[ wavefourier.mws]
Wellengleichung räumlich zweidimensional, Quadrat
[ wavefourier2D_1.mws]
Wellengleichung räumlich zweidimensional, Kreis
[ wavefourier2D_2.mws]
Wellengleichung räumlich zweidimensional, Kreis, radialsymmetrisch
[ wavefourier2D_2radial.mws]

Aufgaben Algorithmische Mathematik I (Wintersemester 2012/13)

[ PDF-Datei]

Aufgaben Algorithmische Mathematik II (Sommersemester 2012/13)

[ PDF-Datei]

Aufgaben Analysis I (Wintersemester 2010/11)

[ PDF-Datei]

Aufgaben Analysis II (Sommersemester 2011)

[ PDF-Datei]

Aufgaben Analysis III (Wintersemester 2011/12)

[ PDF-Datei]

Aufgaben zu Funktionentheorie (Sommersemester 2012)

[ PDF-Datei]

Aufgaben Partielle Differentialgleichungen I

[ PDF-Datei]

Aufgaben Partielle Differentialgleichungen II

[ PDF-Datei]

Aufgaben Differentialgeometrie I (Wintersemester 2012/13)

[ PDF-Datei]

Aufgaben Differentialgeometrie II (Sommersemester 2012/13)

[ PDF-Datei]

Aufgaben Dynamische Systeme

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Aufgaben zu "Analysis der Navier-Stokes-Gleichungen"

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Aufgaben zu Abbildungsgrad und Fixpunktsätze

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Lösungen dazu

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Aufgaben zu Lineare Algebra I für Physiker

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Aufgaben zu Mathematik III für Physiker

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]

Aufgaben zu Mathematik IV für Physiker

[ PS-Datei] [ DVI-Datei] [ PDF-Datei]
 
 

Keine Haftung für die Inhalte von Internetseiten Dritter, die mit meinen Internetseiten verlinkt sind.
Alle Angaben auf meinen Internetseiten ohne Gewähr.