Structure and classification results for the ∞-elastica problem

Consider the following variational problem: among all curves in Rn of fixed length with prescribed end points and prescribed tangents at the end points, minimise the L∞-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of differential equations. Furthermore, we have a lot of information about the structure of solutions, which allows a classification

An L^p regularity theory for harmonic maps

Motivated by the harmonic map heat flow, we consider maps between Riemannian manifolds such that the tension field belongs to an L p L^p -space. Under an appropriate smallness condition, a certain degree of regularity follows. For suitable solutions of the harmonic map heat flow, we have a partial regularity result as a consequence.</p

Structure and classification results for the ∞\infty-elastica problem

Consider the following variational problem: among all curves in $\mathbb{R}^n$ of fixed length with prescribed end points and prescribed tangents at the end points, minimise the $L^\infty$-norm of the curvature. We show that the solutions of this problem, and of a generalised version, are characterised by a system of differential equations. Furthermore, we have a lot of information about the structure of solutions, which allows a classification

Existence, uniqueness and structure of second order absolute minimisers

Let Ω⊆Rn be a bounded open C1,1 set. In this paper we prove the existence of a unique second order absolute minimiser u∞ of the functionalE∞(u,O):=∥F(⋅,Δu)∥L∞(O),O⊆Ωmeasurable,with prescribed boundary conditions for u and Du on ∂Ω and under natural assumptions on F. We also show that u∞ is partially smooth and there exists a harmonic function f∞∈L1(Ω) such thatF(x,Δu∞(x))=e∞sgn(f∞(x))for all x∈{f∞≠0} , where e∞ is the infimum of the global energy.<br/

Variational problems in L∞L^\infty involving semilinear second order differential operators

For an elliptic, semilinear differential operator of the form $S(u) = A : D^2 u + b(x, u , Du)$, consider the functional $E_\infty(u) = \mathop{\mathrm{ess \, sup}}_\Omega |S(u)|$. We study minimisers of $E_\infty$ for prescribed boundary data. Because the functional is not differentiable, this problem does not give rise to a conventional Euler-Lagrange equation. Under certain conditions, we can nevertheless give a system of partial differential equations that all minimisers must satisfy. Moreover, the condition is equivalent to a weaker version of the variational problem

Separation of domain walls with nonlocal interaction and their renormalised energy by Γ-convergence in thin ferromagnetic films

We analyse two variants of a nonconvex variational model from micromagnetics with a nonlocal energy functional, depending on a small parameter ϵ&gt;0. The model gives rise to transition layers, called Néel walls, and we study their behaviour in the limit ϵ→0. The analysis has some similarity to the theory of Ginzburg-Landau vortices. In particular, it gives rise to a renormalised energy that determines the interaction (attraction or repulsion) between Néel walls to leading order. But while Ginzburg-Landau vortices show attraction for degrees of the same sign and repulsion for degrees of opposite signs, the pattern is reversed in this model. In a previous paper, we determined the renormalised energy for one of the models studied here under the assumption that the Néel walls stay separated from each other. In this paper, we present a deeper analysis that in particular removes this assumption. The theory gives rise to an effective variational problem for the positions of the walls, encapsulated in a Γ-convergence result. In the second part of the paper, we turn our attention to another, more physical model, including an anisotropy term. We show that it permits a similar theory, but the anisotropy changes the renormalised energy in unexpected ways and requires different methods to find it.</p

A zigzag pattern in micromagnetics

AbstractWe study a simplified model for the micromagnetic energy functional in a specific asymptotic regime. The analysis includes a construction of domain walls with an internal zigzag pattern and a lower bound for the energy of a domain wall based on an “entropy method”. Under certain conditions, the two results yield matching upper and lower estimates for the asymptotic energy. The combination of these then gives a Γ-convergence result