Hans Knüpfer
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Hans Knüpfer has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Laurent Bétermin
Institut Camille Jordan Université Claude Bernard Lyon 1 43 Bd du 11 novembre 1918 69622 Villeurbanne Cedex France
6 shared publications - Wenhui Shi
College of Materials Science and Engineering College of Chemical Engineering State Key Laboratory of Advanced Separation Membrane Materials Zhejiang University of Technology Hangzhou China
4 shared publications - Denis Brazke
Institute of Applied Mathematics, University of Heidelberg (Germany), Heidelberg, Germany
2 shared publications Florian Nolte
Universität Heidelberg
2 shared publications- Florian Nolte
Institute of Applied Mathematics and IWR, University of Heidelberg, Heidelberg, Germany
2 shared publications - Juan J. L. Velazquez
Institute for Applied Mathematics, University of Bonn, Bonn, Germany
2 shared publications
Latest Publications
- Open Access
- 16 July 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (8) , 178
Correction to: Second Order Expansion for the Nonlocal Perimeter Functional
- 16 July 2026
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 407 (8) , 178
Open AccessCorrection to: Second Order Expansion for the Nonlocal Perimeter Functional
- Open Access
Solutions of the thin film equation obtained in the limit of vanishing slip
We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation.
Open AccessSolutions of the thin film equation obtained in the limit of vanishing slip
We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation.
- Open Access
- 9 October 2025
- Published by Elsevier in Nonlinear Analysis
- Vol. 263
Γ -convergence of higher-order phase transition models
- 9 October 2025
- Published by Elsevier in Nonlinear Analysis
- Vol. 263
Open AccessΓ -convergence of higher-order phase transition models
- Open Access
Gamma-Convergence of Higher-Order Phase Transition Models
We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities.Open AccessGamma-Convergence of Higher-Order Phase Transition Models
We investigate the asymptotic behavior as $\varepsilon \to 0$ of singularly perturbed phase transition models of order $n \geq 2$, given by \begin{align} G_\varepsilon^{λ,n}[u] := \int_I \frac 1\varepsilon W(u) -λ\varepsilon^{2n-3} (u^{(n-1)})^2 + \varepsilon^{2n-1} (u^{(n)})^2 \ dx, \quad u \in W^{n,2}(I), \end{align} where $λ>0$ is fixed, $I \subset \mathbb{R}$ is an open bounded interval, and $W \in C^0(\mathbb{R})$ is a suitable double-well potential. We find that there exists a positive critical parameter depending on $W$ and $n$, such that the $Γ$-limit of $G_\varepsilon^{λ,n}$ with respect to the $L^1$-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo-Nirenberg type inequalities. - Open Access
Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions
We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.
Open AccessWell-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions
We consider the incompressible and stationary Stokes equations on an infinite two-dimensional wedge with non-scaling invariant Navier-slip boundary conditions. We prove well-posedness and higher regularity of the Stokes problem in a certain class of weighted Sobolev spaces. The novelty of this work is the occurrence of two different scalings in the boundary condition, which is not treated so far for the Stokes system in unbounded wedge-type domains. These difficulties are overcome by first constructing a variational solution in a second order weighted Sobolev space and subsequently proving higher regularity up to the tip of the wedge by employing an iterative scheme. We believe that this method can be used for other problems with variational structure and multiple scales.
- Open Access
- 12 May 2023
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 154 (3) , 1-24
Minimal energy for geometrically nonlinear elastic inclusions in two dimensions
We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from Conti and Schweizer (Commun. Pure Appl. Math. 59 (2006), 830–868) and Knüpfer and Kohn (Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011), 695–717), we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.
- 12 May 2023
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Vol. 154 (3) , 1-24
Open AccessMinimal energy for geometrically nonlinear elastic inclusions in two dimensions
We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal scaling of the energy of an elastic inclusion of a fixed volume for which the energy is determined by a surface and an (anisotropic) elastic contribution. Following ideas from Conti and Schweizer (Commun. Pure Appl. Math. 59 (2006), 830–868) and Knüpfer and Kohn (Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 467 (2011), 695–717), we derive the lower scaling bound by invoking a two-well rigidity argument and a covering result. The upper bound follows from a well-known construction for a lens-shaped elastic inclusion.
- Open Access
- 6 April 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (4) , 1-22
Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $$\Omega _t:= \Omega \times [0,t]$$ for $$\Omega \Subset \mathbb {R}^2$$ with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by $$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$ describing the energy difference for a given magnetization $$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$ with $$|M| = \chi _{\Omega _t}$$ and the uniform magnetization $$e_3 \chi _{\Omega _t}$$ . For $$t \ll d$$ , we derive the scaling of the minimal energy and a BV-bound in the critical regime, where the base area of the film has size of order $$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$ . We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film.- 6 April 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (4) , 1-22
Open AccessOnset of pattern formation in thin ferromagnetic films with perpendicular anisotropy
We consider the onset of pattern formation in an ultrathin ferromagnetic film of the form $$\Omega _t:= \Omega \times [0,t]$$ for $$\Omega \Subset \mathbb {R}^2$$ with preferred perpendicular magnetization direction. The relative micromagnetic energy is given by $$\begin{aligned} \mathcal {E}[M]= \int _{\Omega _t} d^2 |\nabla M|^2+ Q \int _{\Omega _t} (M_1^2+M_2^2) + \int _{\mathbb {R}^3} |\mathcal {H}(M)|^2 - \int _{\mathbb {R}^3} |\mathcal {H}(e_3 \chi _{\Omega _t})|^2, \end{aligned}$$ describing the energy difference for a given magnetization $$M: \mathbb {R}^3 \rightarrow \mathbb {R}^3$$ with $$|M| = \chi _{\Omega _t}$$ and the uniform magnetization $$e_3 \chi _{\Omega _t}$$ . For $$t \ll d$$ , we derive the scaling of the minimal energy and a BV-bound in the critical regime, where the base area of the film has size of order $$|\Omega |^{{\frac{1}{2}}} \sim (Q-1)^{{-\frac{1}{2}}} d e^{\frac{2\pi d}{t} \sqrt{Q-1}}$$ . We furthermore investigate the onset of non-trivial pattern formation in the critical regime depending on the size of the rescaled film. - Open Access
- 27 January 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (3) , 1-18
$\Gamma $–limit for a sharp interface model related to pattern formation on biomembranesWe derive a macroscopic limit for a sharp interface version of a model proposed in Komura et al. (Langmuir 22:6771–6774, 2006) to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercritical parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.
- 27 January 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (3) , 1-18
Open Access$\Gamma $–limit for a sharp interface model related to pattern formation on biomembranesWe derive a macroscopic limit for a sharp interface version of a model proposed in Komura et al. (Langmuir 22:6771–6774, 2006) to investigate pattern formation due to competition of chemical and mechanical forces in biomembranes. We identify sub- and supercritical parameter regimes and show with the introduction of the autocorrelation function that the ground state energy leads to the isoperimetric problem in the subcritical regime, which is interpreted to not form fine scale patterns.
- Open Access
- 19 November 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 398 (3) , 1371-1402
Second Order Expansion for the Nonlocal Perimeter Functional
The seminal results of Bourgain et al. (Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001) and Dávila (Calc Var Partial Differ Equ 15(4):519–527, 2002) show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the $$\Gamma $$ –limit of these functionals as $$\epsilon \rightarrow 0$$ . We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function.- 19 November 2022
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 398 (3) , 1371-1402
Open AccessSecond Order Expansion for the Nonlocal Perimeter Functional
The seminal results of Bourgain et al. (Optimal Control and Partial Differential Equations, IOS, Amsterdam, 2001) and Dávila (Calc Var Partial Differ Equ 15(4):519–527, 2002) show that the classical perimeter can be approximated by a family of nonlocal perimeter functionals. We consider a corresponding second order expansion for the nonlocal perimeter functional. In a special case, the considered family of energies is also relevant for a variational model for thin ferromagnetic films. We derive the $$\Gamma $$ –limit of these functionals as $$\epsilon \rightarrow 0$$ . We also show existence for minimizers with prescribed volume fraction. For small volume fraction, the unique, up to translation, minimizer of the limit energy is given by the ball. The analysis is based on a systematic exploitation of the associated symmetrized autocorrelation function. - 4 February 2021
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 239 (3) , 1875-1923
$\Gamma $-Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls- 4 February 2021
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 239 (3) , 1875-1923
$\Gamma $-Limit for Two-Dimensional Charged Magnetic Zigzag Domain Walls