Publikationen

Forschung in der Arbeitsgruppe

Die Forschungsschwerpunkte der Arbeitsgruppe Analysis liegen in den Bereichen Partielle Differentialgleichungen, Variationsrechnung und Geometrischer Analysis. Ein wichtiger Aspekt dabei ist die Existenz- und Regularitätstheorie für zeitabhängige Probleme. Es wird eine sehr vielfältige Auswahl an Problemstellungen bearbeitet, wie etwa geometrische Flüsse, Hindernisprobleme, Knotenenergien, degenerierte/singuläre parabolische Gleichungen und Systeme, und Probleme mit linearem Wachstum, die zum Beispiel in der Bildverarbeitung Anwendung finden.

Minimalfläche von Costa Bei unserer Forschung legen wir großen Wert auf Teamarbeit und internationale Kooperationen. Die Arbeitsgruppe hat eine Vielzahl von internationalen Kontakten und Kooperationspartnern in Deutschland (Aachen, Duisburg-Essen, Erlangen, Karlsruhe), Finnland (Helsinki, Jyväskylä), Italien (Cagliari, Florenz, Neapel, Pavia), Japan (Tokio), Polen (Warschau) und der Schweiz (Zürich).

2022

S. Blatt. Analyticity for solution of fractional integro-differential operators. Nonlinear Anal., 224, 113071, 2022.
S. Blatt, C. Hopper and N. Vorderobermeier. A minimising movement scheme for the p-elastic energy of curves. J. Evol. Equ. 22, 41, 2022.
S. Blatt, C. Hopper and N. Vorderobermeier. A regularized gradient flow for the p-elastic energy. Adv. Nonlinear Anal. 11(1):1383-1411, 2022.
S. Blatt, A. Ishizeki and T. Nagasawa. A Möbius invariant discretization of O’Hara’s Möbius energy. J. Knot Theory Ramifications 31(03), 2250016, 2022.
S. Blatt, P. Reiter and A. Schikorra. On O’Hara knot energies I: Regularity for critical knots. J. Differential Geom. 121(3):385-424, 2022.
V. Bögelein, F. Duzaar, R. Giova and A. Passarelli di Napoli. Higher regularity in congested traffic dynamics. Math. Ann., 2022.
V. Bögelein, F. Duzaar, N. Liao and L. Schätzler. On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II. Rev. Mat. Iberoam., 2022.
V. Bögelein, F. Duzaar, N. Liao and C. Scheven. Boundary regularity for parabolic systems in convex domains. J. Lond. Math. Soc. 105(3):1702-1751, 2022.
V. Bögelein, F. Duzaar, N. Liao and C. Scheven. Gradient Hölder regularity for degenerate parabolic systems. Nonlinear Anal., 225, 113119, 2022.
V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven. Boundary regularity for elliptic systems with p,q-growth. J. Math. Pures Appl. 159:250-293, 2022.
U. Gianazza and N. Liao. A boundary estimate for singular sub-critical parabolic equations. Int. Math. Res. Not. IMRN, 2022(10):7332-7353, 2022.
U. Gianazza and N. Liao. Continuity of the temperature in a multi-phase transition problem. Math. Ann. 384:1-35, 2022.
K. Moring and R. Rainer. Stability for systems of porous medium type. J. Math. Anal. Appl., 506(1), 125532, 2022.
K. Moring and L. Schätzler. On the Hölder regularity for obstacle problems to porous medium type equations. J. Evol. Equ. 22, 81, 2022.
N. Liao. Local continuity of weak solutions to the Stefan problem involving the singular p-Laplacian. SIAM J. Math. Anal., 54(2):2570-2586, 2022.
N. Liao. On the logarithmic type boundary modulus of continuity for the Stefan problem: To the memory of Emmanuele DiBenedetto. Adv. Math. 408(B),108612, 2022.
N. Liao and L. Schätzler. On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part III. Int. Math. Res. Not. IMRN, 2022(3):2376-2400, 2022.
D. Steenebrügge and N. Vorderobermeier. On the analyticity of critical points of the generalized integral Menger curvature in the Hilbert case. Nonlinear Anal., 221, 112858, 2022.

2021

V. Bögelein, N. Dietrich and M. Vestberg. Existence of solutions to a diffusive shallow medium equation. J. Evol. Equ. 21:845-889, 2021.
V. Bögelein, F. Duzaar and N. Liao. On the Hölder regularity of signed solutions to a doubly nonlinear equation. J. Funct. Anal. 281(9), 2021.
V. Bögelein, A. Herán, L. Schätzler and T. Singer. Harnack’s inequality for doubly nonlinear equations of slow diffusion type. Calc. Var. Partial Differential Equations, 60, 215, 2021.
U. Gianazza and N. Liao. Continuity of the temperature in a multi-phase transition problem. Math. Ann. 1-35, 2021.
N. Liao. Hölder regularity for porous medium systems. Calc. Var. Partial Differential Equations, 60, 156, 2021.
N. Liao. Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior. J. Math. Pures Appl., 147:179–204, 2021.
N. Liao. Remarks on parabolic De Giorgi classes. Ann. di Mat. Pura ed Appl., 2021.
R. Rainer, J. Siltakoski and T. Stanin. An evolutionary Haar-Rado type theorem. Manuscripta Math., 2021.
L. Schätzler. The obstacle problem for degenerate doubly nonlinear equations of porous medium type. Ann. di Mat. Pura ed Appl., 200(2):641-683, 2021.
T. Stanin. Global continuity of variational solutions weakening the one-sided bounded slope condition. Forum Math., 2021.

2020

S. Blatt. The gradient flow of the Möbius energy: $ε$ -regularity and consequences. Anal. PDE, 13(3):901-941, 2020.
V. Bögelein, B. Dacorogna, F. Duzaar, P. Marcellini and C. Scheven. Integral convexity and parabolic systems. SIAM J. Math. Anal., 52(2):1489–1525, 2020.
V. Bögelein, F. Duzaar, J. Kinnunen und C. Scheven. Higher integrability for doubly nonlinear parabolic systems. J. Math. Pures Appl., 143:31–72, 2020.
V. Bögelein, F. Duzaar und C. Scheven. Higher integrability for the singular porous medium system. J. Reine Angew. Math., 2020(767):203–230, 2020.
V. Bögelein and T. Stanin. The one-sided bounded slope condition in evolution problems. Ann. di Mat. Pura ed Appl., 199(2):573–587, 2020.
U. Gianazza and N. Liao. A boundary estimate for degenerate parabolic diffusion equations. Potential Analysis, 53:977-995, 2020.
U. Gianazza and N. Liao. A boundary estimate for singular sub-critical parabolic equations. Int. Math. Res. Not. IMRN, 2020.
N. Liao, I. Skrypnik und V. Vespri. Local regularity for an anisotropic elliptic equation. Calc. Var. Partial Differential Equations, 59, 116, 2020.
N. Liao. A unified approach to the Hölder regularity of solutions to degenerate and singular parabolic equations. J. Differential Equations, 268(10):5704-5750, 2020.
L. Schätzler. The obstacle problem for singular doubly nonlinear equations of porous medium type. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 31(3):503–548, 2020.
N. Vorderobermeier. On the regularity of critical points for O’Hara’s knot energies: From smoothness to analyticity. Commun. Contemp. Math., 1–28, 2020.

2019

S. Blatt. A note on singularities in finite time for the L2-gradient flow of the Helfrich functional, Journal of Evolution Equations 19:463–477, 2019.
S. Blatt. Curves between Lipschitz and C1 and their relation to geometric knot theory, Journal of Geometric Analysis, 29:3270–3292, 2019.
S. Blatt and N. Vorderobermeier. On the analyticity of critical points of the Möbius energy, Calc. Var. Partial Differential Equations, 58(1):58:16, 2019.
V. Bögelein, F. Duzaar, R. Korte and C. Scheven. The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8(1):1004-1034, 2019.
V. Bögelein, F. Duzaar, L. Schätzler and C. Scheven. Existence for evolutionary problems with linear growth by stability methods, J. Differential Equations, 266:7709-7748, 2019.
V. Bögelein, P. Lehtelä and S. Sturm. Regularity of weak solutions and supersolutions to the Porous Medium Equation, Nonlinear Anal., 185:49-67, 2019.
R. Korte, P. Lehtelä and S. Sturm. Lower semicontinuous obstacles for the porous medium equation, J. Differential Equations, 266(4):1851-1864, 2019.
N. Liao. A sufficient condition for the continuity of solutions to a logarithmic diffusion equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XIX(3):1161-1184, 2019.
L. Schätzler. Existence for evolutionary Neumann problems with linear growth by stability results. Ann. Acad. Sci. Fenn. Math., 2019(44):1055–1092, 2019.
L. Schätzler. Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values. J. Elliptic Parabol. Equ., 2019(5):383–421, 2019.

2018

S. Blatt, The gradient flow of O’Hara’s knot energies, Math. Ann. 370(3-4):993–1061, 2018.
V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven. A variational approach to doubly nonlinear equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29:739-772, 2018.
V. Bögelein, F. Duzaar, P. Marcellini and C. Scheven. Doubly Nonlinear Equations of Porous Medium Type, Arch. Ration. Mech. Anal. 229:503–545, 2018.
V. Bögelein, F. Duzaar, C. Scheven and T. Singer. Existence of variational solutions in non-cylindrical domains, SIAM J. Math. Anal., 50(3), 3007-3057, 2018.
U. Gianazza, N. Liao and T. Lukkari. A boundary estimate for singular parabolic diffusion equations. NoDEA Nonlinear Differential Equations Appl. 25, 33, 2018.
C. Klaus and N. Liao. A short proof of Hölder continuity for functions in DeGiorgi classes. Ann. Acad. Sci. Fenn. Math., 43:931-934, 2018.
S. Sturm. Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data, Manuscripta Math. 157:295-322, 2018.

2017

S. Blatt. Monotonicity formulas for extrinsic triharmonic maps and the triharmonic Lane–Emden equation, J. Differential Equations. 262(12):5691-5734, 2017.
V. Bögelein, F. Duzaar and N. Fusco. A quantitative isoperimetric inequality on the sphere, Adv. Calc. Var., 10(3):223-265, 2017.
V. Bögelein, F. Duzaar, P. Marcellini and S. Signoriello. Parabolic equations and the bounded slope condition, Ann. Inst. H. Poincaré, Anal. Non Linéaire. 34(2):355-379, 2017.
V. Bögelein, F. Duzaar and C. Scheven. The obstacle problem for parabolic minimizers, J. Evol. Equ., 17(4):1273-1310, 2017.
V. Bögelein, T. Lukkari and C. Scheven. Hölder regularity for degenerate parabolic obstacle problems, Ark. Mat., 55:1-39, 2017.
V. Bögelein, F. Ragnedda, S. Vernier Piro and V. Vespri. Moser-Nash kernel estimates for degenerate parabolic equations, J. Funct. Anal. 272(7):2956-2986, 2017.
L. Schätzler. Existence of variational solutions for time dependent integrands via minimizing movements. Analysis (Germany), 37(4):199–222, 2017.
S. Signoriello and T. Singer. Hölder continuity of parabolic quasi-minimizers, J. Differential Equations, 263(9):6066-6114, 2017.
S. Sturm. Existence of very weak solutions of doubly nonlinear parabolic equations with measure data, Ann. Acad. Sci. Fenn. Math. 42:931-962, 2017.
S. Sturm. Existence of weak solutions of doubly nonlinear parabolic equations, J. Math. Anal. Appl. 455:842-863, 2017.

2016

S. Blatt, P. Reiter and A. Schikorra. Harmonic analysis meets critical knots. Critical points of the Möbius energy are smooth. Trans. Amer. Math. Soc., 368:6391-6438, 2016.
S. Blatt and M. Struwe. Well-posedness of the supercritical Lane-Emden heat flow in Morrey spaces, ESAIM: Control, Optimisation and Calculus of Variations, 22(4):1370-1381, 2016.
V. Bögelein, F. Duzaar and U. Gianazza. Sharp boundedness and continuity results for the singular porous medium equation, Israel J. Math. 214:259-314, 2016.
V. Bögelein, F. Duzaar, and C. Scheven. The obstacle problem for the total variation flow, Ann. Sci. Éc. Norm. Supér. 49(5):1143–1188, 2016.
V. Bögelein, F. Duzaar, and C. Scheven. The total variation flow with time dependent boundary values, Calc. Var. Partial Differential Equations, 55:108, 2016.
T. Singer. Existence of weak solutions of parabolic systems with p,q-growth, Manuscripta Math. 151, 87–112, 2016.
T. Singer. Local Boundedness of variational solutions to evolutionary problems with non-standard growth, NoDEA Nonlinear Differential Equations Appl., 23:19, 2016.

2015

S. Blatt and P. Reiter. Regularity theory for tangent-point energies: The non-degenerate sub-critical case, Adv. Calc. Var. 8(2):93–116, 2015.
S. Blatt and P. Reiter. Towards a regularity theory for integral Menger curvature, Ann. Acad. Sci. Fenn., Math. 40:149–181, 2015.
S. Blatt and M. Struwe. An analytic framework for the supercritical Lane-Emden equation and its gradient flow, Int. Math. Res. Not. IMRN. 2015 (9): 2342-2385, 2015.
S. Blatt and M. Struwe. Boundary Regularity for the supercritical Lane-Emden heat flow, Calc. Var. Partial Differential Equations. 54(2): 2269-2284, 2015.
V. Bögelein. Global gradient bounds for the parabolic p-Laplacian system, Proc. London Math. Soc. 111(3):633-680, 2015.
V. Bögelein. Partial boundary regularity of non-linear parabolic systems in low dimensions, Analysis (Berlin). 35:1-28, 2015.
V. Bögelein, F. Duzaar and N. Fusco. A sharp quantitative isoperimetric inequality in higher codimension, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(3):309-362, 2015.
V. Bögelein, F. Duzaar and U. Gianazza. Very weak solutions of singular porous medium equations with measure data. Commun. Pure Appl. Anal. 14(1):23-49, 2015.
V. Bögelein, F. Duzaar and P. Marcellini. A time dependent variational approach to image restoration, SIAM J. Imaging Sci. 8(2):968-1006, 2015.
V. Bögelein, F. Duzaar, P. Marcellini, and S. Signoriello. Nonlocal diffusion equations, J. Math. Anal. Appl. 432(1):398-428, 2015.
V. Bögelein, F. Duzaar and C. Scheven. A sharp quantitative isoperimetric inequality in hyperbolic n-space, Calc. Var. Partial Differential Equations, 54(3):3967-4017, 2015.
V. Bögelein, F. Duzaar and C. Scheven. Short-time regularity for the H-surface-flow. Int. Math. Res. Not. IMRN. 12:3694-3750, 2015.
V. Bögelein, T. Lukkari and C. Scheven. The obstacle problem for the porous medium equation, Math. Ann. 363(1), 455-499, 2015.
N. Liao. Existence and nonexistence of solutions to a logarithmic diffusion equation in bounded domains. Manuscripta Math., 147:101-138, 2015.
S. Signoriello and T. Singer. Local Calderón-Zygmund estimates for parabolic minimizers, Nonlinear Anal. 125: 561–581, 2015.
T. Singer. Parabolic equations with p,q-growth: The subquadratic case. Q. J. Math. 66 (2): 707-742, 2015.
S. Sturm. Pointwise estimates for porous medium type equations with low order terms and measure data. Electron. J. Diff. Equ. 2015:1-25, 2015.

2014

P. Baroni and V. Bögelein. Calderón-Zygmund estimates for parabolic p(x,t)-Laplacian systems, Rev. Mat. Iberoam. 30(4):1355-1386, 2014.
S. Blatt and P. Reiter. How nice are critical knots? Regularity theory for knot energies. Journal of Physics: Conference Series, 544, published online, 2014.
S. Blatt and P. Reiter. Modeling repulsive forces on fibres via knot energies, Mol. Based Math. Biol. 2(1):56-72, 2014.
V. Bögelein. Global Calderón & Zygmund theory for nonlinear parabolic systems. Calc. Var. Partial Differential Equations. 51(3-4):555-596, 2014.
V. Bögelein, F. Duzaar and U. Gianazza. Continuity estimates for porous medium type equations with measure data. J. Funct. Anal. 267:3351-3396, 2014.
V. Bögelein and Q. Li. Very weak solutions of degenerate parabolic systems with non-standard p(x,t)-growth. Nonlinear Anal. 98:190-225, 2014.
V. Bögelein, F. Duzaar and P. Marcellini. Existence of evolutionary variational solutions via the calculus of variations. J. Differential Equations. 256(12):3912-3942, 2014.
E. DiBenedetto, U. Gianazza and N. Liao. Two remarks on the local behavior of solutions to logarithmically singular diffusion equations and its porous-medium type approximations. Riv. Mat. Univ. Parma, 5(1):139-182, 2014.

2013

S. Blatt. A note on Integral Menger Curvature. Math. Nachr. 286(2-3):149-159, 2013.
S. Blatt. The Energy space of the Tangent Point Energies. J. Topol. Anal. 5(3):261-270, 2013.
S. Blatt and P. Reiter. Stationary Points of O’Hara’s Knot Energies. Manuscripta Math. 140(1-2):29-50, 2013.
V. Bögelein, F. Duzaar and U. Gianazza. Porous medium type equations with measure data and potential estimates. SIAM J. Math. Anal. 45(6):3283-3330, 2013.
V. Bögelein, F. Duzaar and P. Marcellini. Parabolic equations with p,q-growth. J. Math. Pures Appl. 100(4):535-563, 2013.
V. Bögelein, F. Duzaar and P. Marcellini. Parabolic systems with p,q-growth: A variational approach. Arch. Ration. Mech. Anal. 210(1):219-267, 2013.
V. Bögelein, F. Duzaar and G. Mingione. The regularity of general parabolic systems with degenerate diffusions. Mem. Amer. Math. Soc. 221(1041), 2013.
V. Bögelein, F. Duzaar and C. Scheven. Weak solutions to the heat flow for surfaces of prescribed mean curvature. Trans. Amer. Math. Soc. 365:4633-4677, 2013.
M. Revers. On the asymptotics of polynomial interpolation for |x|α at the Chebyshev nodes. J. Approx. Theory 165; 70-82, 2013.

2012

S. Blatt. Boundedness and Regularizing Effects of O’Hara’s Knot Energies. J. Knot Theory Ramifications. 21(1), 9 pages, 2012.
S. Blatt. The Gradient Flow of the Möbius Energy near local Minimizers. Calc. Var. Partial Differential Equations. 43(3-4):403-439, 2012.
S. Blatt and S. Kolasinski. Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds. Adv. Math. 230(3):839-852, 2012.
V. Bögelein. Partial regularity for minimizers of discontinuous quasiconvex integrals with degeneracy. J. Differential Equations. 252(2):1052-1100, 2012.
V. Bögelein and F. Duzaar. Hölder estimates for parabolic p(x,t)-Laplacian systems. Mathematische Annalen. 354(3):907-938, 2012.
V. Bögelein, F. Duzaar, J. Habermann and C. Scheven. Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients. Adv. Calc. Var. 5(1):1-57, 2012.
V. Bögelein, F. Duzaar and C. Scheven. Global solutions to the heat flow for m-harmonic maps and regularity. Indiana Univ. Math. J. 61(6):2157-2210, 2012.
V. Bögelein, M. Foss and G. Mingione. Regularity in parabolic systems with continuous coefficients. Math. Z. 270(3-4):903-938, 2012.
V. Bögelein and C. Scheven. Higher integrability in parabolic obstacle problems. Forum Math. 24(5):931–972, 2012.
E. DiBenedetto, U. Gianazza and N. Liao. Logarithmically singular parabolic equations as limits of the porous medium equation. Nonlinear Anal., 75(12):4513-4533, 2012.
E. DiBenedetto, U. Gianazza and N. Liao. On the local behavior of non-negative solutions to a logarithmically singular equation. Discrete Contin. Dyn. Syst. Ser. B, 17(6):1841-1858, 2012.

2011

V. Bögelein and F. Duzaar. Higher integrability for parabolic systems with non-standard growth and degenerate diffusions. Publ. Mat. 55(1):201-250, 2011.
V. Bögelein, F. Duzaar, J. Habermann and C. Scheven. Partial Hölder continuity for discontinuous elliptic problems with VMO-coefficients. Proc. London Math. Soc. 103(3):371-404, 2011.
V. Bögelein, F. Duzaar and G. Mingione. Degenerate problems with irregular obstacles. J. Reine Angew. Math. 650:107-160, 2011.

2010

S. Blatt. Loss of convexity and embeddedness for geometric evolution equations of higher order. J. Evol. Equ. 10(1):21-27, 2010.
V. Bögelein, F. Duzaar and G. Mingione. The boundary regularity of non-linear parabolic systems I. Ann. Inst. H. Poincaré, Anal. Non Linéaire 27(1):201-255, 2010.
V. Bögelein, F. Duzaar and G. Mingione. The boundary regularity of non-linear parabolic systems II. Ann. Inst. H. Poincaré, Anal. Non Linéaire 27(1):145-200, 2010.
V. Bögelein and J. Habermann. Gradient estimates via non standard potentials and continuity. Ann. Acad. Sci. Fenn. Math. 35:641-678, 2010.
V. Bögelein and M. Parviainen. Self-improving property of nonlinear higher order parabolic systems near the boundary. Nonlinear Differ. Equ. Appl. 17(1):21-54, 2010.

2009

S. Blatt. A singular example for the Willmore flow. Analysis (Oldenbourg) 29(4):407–430, 2009.
S. Blatt. Chord-arc constants for submanifolds of arbitrary codimension. Adv. Calc. Var. 2(3):271–309, 2009.
S. Blatt, H.-P. Blatt and W. Luh. On a generalization of Jentzsch’s theorem. J. Approx. Theory. 159(1):26–38, 2009.
V. Bögelein. Partial regularity and singular sets of solutions of higher order parabolic systems. Ann. Mat. Pura Appl. 188(1):61-122, 2009.
V. Bögelein. Very weak solutions of higher order degenerate parabolic systems. Advances in Differential Equations. 14(1-2):121-200, 2009.

2008

S. Blatt and P. Reiter. Does finite knot energy lead to differentiability? J. Knot Theory Ramifications. 17(10):1281–1310, 2008.
V. Bögelein. Higher integrability for weak solutions of higher order degenerate parabolic systems. Ann. Acad. Sci. Fenn., Math. 33(2):387-412, 2008.

2007

V. Bögelein and A. Zatorska-Goldstein. Higher integrability of very weak solutions of systems of p(x)-Laplacean type. J. Math. Anal. Appl. 336(1):480-497, 2007.

2004

M. Revers and M. Ganzburg. A note on Lagrange Interpolation for |x|α at equidistant nodes. Bull. Austral. Math. Soc. 70:475-480, 2004.

2001

M. Revers. On Lagrange interpolation for functions of bounded variation. Math. Pannon. 12(1):133-138, 2001.

2000

M. Revers. Approximation constants in equidistant Lagrange interpolation. Period. Math. Hungar. 40(2):167-175, 2000.
M. Revers. On Lagrange interpolation with equally spaced Nodes. Bull. Austral. Math. Soc. 62(3):357-368, 2000.
M. Revers. On Lagrange interpolatory parabolas to | x|α at equally spaced nodes. Arch. Math. 74(5):385-391, 2000.
M. Revers. On the zero-divergence of equidistant Lagrange interpolation. Monatsh. Math. 131(3):215-221, 2000.
M. Revers. The divergence of Lagrange interpolation for |x|α at equidistant nodes. J. Approx. Theory. 103:269-280, 2000.