Hans Knüpfer

On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained.

A Free Boundary Problem of Fourth Order: Classical Solutions in Weighted Hölder Spaces

We prove short-time existence and uniqueness of classical solutions in weighted Hölder spaces for the thin-film equation with linear mobility, zero contact angle, and compactly supported initial data. We furthermore show regularity of the free boundary and optimal regularity of the solution in terms of the regularity of the initial data. Our approach relies on Schauder estimates for the operator linearized at the free boundary, obtained through a variant of Safonovs method that is solely based on energy estimates.

Low density phases in a uniformly charged liquid

This paper is concerned with the macroscopic behavior of global energy minimizers in the three-dimensional sharp interface unscreened Ohta–Kawasaki model of diblock copolymer melts. This model is also referred to as the nuclear liquid drop model in the studies of the structure of highly compressed nuclear matter found in the crust of neutron stars, and, more broadly, is a paradigm for energy-driven pattern forming systems in which spatial order arises as a result of the competition of short-range attractive and long-range repulsive forces. Here we investigate the large volume behavior of minimizers in the low volume fraction regime, in which one expects the formation of a periodic lattice of small droplets of the minority phase in a sea of the majority phase. Under periodic boundary conditions, we prove that the considered energy

Minimal energy for elastic inclusions

{\Gamma}

Optimal lattice configurations for interacting spatially extended particles

Γ-converges to an energy functional of the limit “homogenized” measure associated with the minority phase consisting of a local linear term and a non-local quadratic term mediated by the Coulomb kernel. As a consequence, asymptotically the mass of the minority phase in a minimizer spreads uniformly across the domain. Similarly, the energy spreads uniformly across the domain as well, with the limit energy density minimizing the energy of a single droplet per unit volume. Finally, we prove that in the macroscopic limit the connected components of the minimizers have volumes and diameters that are bounded above and below by universal constants, and that most of them converge to the minimizers of the energy divided by volume for the whole space problem.

On Born's conjecture about optimal distribution of charges for an infinite ionic crystal

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is in the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a multidomain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp interface energy, giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field.

A sharp interface evolutionary model for shape memory alloys

We consider a variant of the isoperimetric problem with a non-local term representing elastic energy. More precisely, our aim is to analyse the optimal energy of an inclusion of a fixed volume the energy of which is determined by surface and elastic energies. This problem has been studied extensively in the physical/metallurgical literature; however, the analysis has mainly been either (i) numerical, or (ii) restricted to a specific set of inclusion shapes, e.g. ellipsoids. In this article, we prove a lower bound for the energy, with no a priori hypothesis on the shape (or even number) of the inclusions.

Self-Similar Lifting and Persistent Touch-Down Points in the Thin-Film Equation

We study the minimum energy configuration of a uniform distribution of negative charge subject to Coulomb repulsive self-interaction and attractive interaction with a fixed positively charged domain. After having established existence and uniqueness of a minimizing configuration, we prove charge neutrality and the complete screening of the Coulomb potential exerted by the positive charge, and we discuss the regularity properties of the solution. We also determine, in the variational sense of