721 research outputs found
Structure and classification results for the â-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 Lâ-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<sup>p</sup> 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 -elastica problem
Consider the following variational problem: among all curves in
of fixed length with prescribed end points and prescribed
tangents at the end points, minimise the -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 ΩâRn be a bounded open C1,1 set. In this paper we prove the existence of a unique second order absolute minimiser uâ of the functionalEâ(u,O):=â„F(â
,Îu)â„Lâ(O),OâΩmeasurable,with prescribed boundary conditions for u and Du on âΩ and under natural assumptions on F. We also show that uâ is partially smooth and there exists a harmonic function fââL1(Ω) such thatF(x,Îuâ(x))=eâsgn(fâ(x))for all xâ{fââ 0} , where eâ is the infimum of the global energy.<br/
Variational problems in involving semilinear second order differential operators
For an elliptic, semilinear differential operator of the form , consider the functional . We study minimisers of 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 Î-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 Ï”>0. The model gives rise to transition layers, called NĂ©el walls, and we study their behaviour in the limit Ï”â0. 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 NĂ©el 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 NĂ©el 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 Î-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 âentropy methodâ. Under certain conditions, the two results yield matching upper and lower estimates for the asymptotic energy. The combination of these then gives a Î-convergence result
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