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Roger Moser

Publications and source records attributed to Roger Moser.

At least 19 recordsLinked to original sources
arXiv2026

The Sesquiharmonic Map Flow from Riemannian Surfaces

Let $M$ be a two-dimensional compact manifold without boundary and let $N$ be a compact manifold without boundary. We study the $L^2$-gradient flow of an energy functional that interpolates between the harmonic map energy and the intrinsic biharmonic map energy. The critical points of this functional are called sesqui-harmonic maps. We investigate regularity properties of this flow, generalizing Struwe's classical regularity result for harmonic maps.

math.DG↗
arXiv2026

A characterisation of $\infty$-harmonic maps in terms of $1$-currents

We consider maps between two Riemannian manifolds and study a functional given in terms of the $L^\infty$-norm of the derivative. This functional is not differentiable, but we can define critical points with the help of a subdifferential. The resulting notion includes, for example, minimisers in a given homotopy class. We derive a geometric condition equivalent to criticality in this sense. The condition is formulated in terms of a vector-valued $1$-current on the domain manifold, which encapsulates some of the key properties of the critical point. Moreover, this $1$-current is itself a critical point of a generalised mass functional.

math.AP↗
arXiv2026

Energy maximum principle for vectorial higher order absolute minimisers in $L^\infty$ and $L^p$

We show that vectorial absolute minimisers of general $k$-th order supremal functionals in $W^{k,\infty}(\Omega,\mathbb R^N)$ satisfy a maximum principle of the form $$ \max_{\overline U} \mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big)=\max_{\partial U}\mathrm{H} \big(\cdot, u, \mathrm D u, ..., \mathrm D^{k}u\big), \qquad\forall\ U\subseteq\Omega \mbox{ open}, $$ suitably interpreted. This is only necessary for absolute minimisers, whilst it characterises a relevant weaker notion of absolute minimality involving compactly supported variations. Further, we obtain an existence result to the Dirichlet problem for such weaker absolute minimisers, as an application of the Baire Category method. Finally, via different methods, we supplement our results by establish a gradient maximum principle for $p$-harmonic maps for $p<\infty$.

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arXiv2025

An $L^\infty$-variational problem involving the Fractional Laplacian

For $s\in(0,1)$ and an open bounded set $\Omega\subset\mathbb R^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_\infty(u)=\|(-\Delta)^s u\|_{L^\infty(\mathbb R^n)},$$ where $(-\Delta)^s$ is the Fractional Laplacian of order $s$ and $u$ has prescribed Dirichlet data in the complement of $\Omega$. We further show that the minimiser $u_\infty$ satisfies the (fractional) PDE $$ (-\Delta)^s u_\infty=E_\infty(u_\infty)\,\mathrm{sgn}f_\infty \qquad\mbox{in }\Omega, $$ for some analytic function $f_\infty\in L^1(\Omega)$ obtained as the restriction of an $s$-harmonic measure $\mu$ in $\Omega$.

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arXiv2025

Asymptotic minimality of one-dimensional transition profiles in Aviles-Giga type models: an approach via 1-currents

For vector fields on a two-dimensional domain, we study the asymptotic behaviour of Modica-Mortola (or Allen-Cahn) type functionals under the assumption that the divergence converges to $0$ at a certain rate, which effectively produces a model of Aviles-Giga type. This problem will typically give rise to transition layers, which degenerate into discontinuities in the limit. We analyse the energy concentration at these discontinuities and the corresponding transition profiles. We derive an estimate for the energy concentration in terms of a novel geometric variational problem involving the notion of $\mathbb{R}^2$-valued $1$-currents from geometric measure theory. This in turn leads to criteria, under which the energetically favourable transition profiles are essentially one-dimensional.

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arXiv2025

Existence, uniqueness and characterisation of vector-valued absolute minimisers for a second order $L^\infty$-variational problem

We study a vectorial $L^\infty$-variational problem of second order, where the supremal functional depends on the vector function $u$ through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser $u_\infty$ under prescribed Dirichlet boundary conditions, together with a characterisation of $u_\infty$ as solution of a specific system of PDEs. Our result can be seen as a twofold extension of the one in Katzourakis-Moser (ARMA 2019): we generalise it to the vectorial setting and, at the same time, we consider more general elliptic operators in place of the Laplacian.

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arXiv2024

Existence, uniqueness and characterisation of local minimisers in higher order Calculus of Variations in $\mathrm L^{\infty}$

We study variational problems for second order supremal functionals $\mathrm F_\infty(u)= \|F(\cdot,u,\mathrm D u,\mathrm{A}\!:\!\mathrm D^2u)\|_{\mathrm L^{\infty}(\Omega)}$, where $F$ satisfies certain natural assumptions, $\mathrm A$ is a positive matrix, and $\Omega \Subset \mathbb R^n$. Higher order problems are very novel in the Calculus of Variations in $\mathrm L^{\infty}$, and exhibit a strikingly different behaviour compared to first order problems, for which there exists an established theory, pioneered by Aronsson in 1960s. The aim of this paper is to develop a complete theory for $\mathrm F_\infty$. We prove that, under appropriate conditions, ``localised" minimisers can be characterised as solutions to a nonlinear system of PDEs, which is different from the corresponding Aronsson equation for $\mathrm F_\infty$; the latter is only a necessary, but not a sufficient condition for minimality. We also establish the existence and uniqueness of localised minimisers subject to Dirichlet conditions on $\partial \Omega$, and also their partial regularity outside a singular set of codimension one, which may be non-empty even if $n=1$.

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arXiv2024

Minimisers of supremal functionals and mass-minimising 1-currents

We study vector-valued functions that minimise the $L^\infty$-norm of their derivatives for prescribed boundary data. We construct a vector-valued, mass minimising $1$-current (i.e., a generalised geodesic) in the domain such that all solutions of the problem coincide on its support. Furthermore, this current can be interpreted as a streamline of the solutions. The construction relies on a $p$-harmonic approximation. In the case of scalar-valued functions, it is closely related to a construction of Evans and Yu. We therefore obtain an extension of their theory.

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arXiv2023

Variational problems in $L^\infty$ 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.

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arXiv2022

Weighted $\infty$-Willmore Spheres

On the two-sphere $\Sigma$, we consider the problem of minimising among suitable immersions $f \,\colon \Sigma \rightarrow \mathbb{R}^3$ the weighted $L^\infty$ norm of the mean curvature $H$, with weighting given by a prescribed ambient function $\xi$, subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of ``pseudo-minimiser'' surfaces must satisfy a second-order PDE system obtained as the limit as $p \rightarrow \infty$ of the Euler-Lagrange equations for the approximating $L^p$ problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: $H \in \{ \pm \vert \vert \xi H \vert \vert_{L^\infty} \}$ away from the nodal set of the PDE system, and $H = 0$ on the nodal set (if it is non-empty).

math.DG↗
arXiv2022

The $\infty$-elastica problem on a Riemannian manifold

We consider the following problem: on any given complete Riemannian manifold $(M,g)$, among all curves which have fixed length as well as fixed end-points and tangents at the end-points, minimise the $L^\infty$ norm of the curvature. We show that the solutions of this problem, as well as a wider class of curves, must satisfy a second order ODE system. From this system we obtain some geometric information about the behaviour of the curves.

math.DG↗
arXiv2022

Regularity of gradient vector fields giving rise to finite Caccioppoli partitions

For a finite set $A \subseteq \mathbb{R}^n$, consider a function $u \in \mathrm{BV}_{\mathrm{loc}}^2(\mathbb{R}^n)$ such that $\nabla u \in A$ almost everywhere. If $A$ is convex independent, then it follows that $u$ is piecewise affine away from a closed, countably $\mathcal{H}^{n - 1}$-rectifiable set. If $A$ is affinely independent, then $u$ is piecewise affine away from a closed $\mathcal{H}^{n - 1}$-null set.

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arXiv2021

The streamlines of $\infty$-harmonic functions obey the inverse mean curvature flow

Given an $\infty$-harmonic function $u_\infty$ on a domain $\Omega \subseteq \mathbb{R}^2$, consider the function $w = -\log |\nabla u_\infty|$. If $u_\infty \in C^2(\Omega)$ with $\nabla u_\infty \neq 0$ and $\nabla |\nabla u_\infty| \neq 0$, then it is easy to check that (1) the streamlines of $u_\infty$ are the level sets of $w$ and (2) $w$ solves the level set formulation of the inverse mean curvature flow. For less regular solutions, neither statement is true in general, but even so, $w$ is still a weak solution of the inverse mean curvature flow under far weaker assumptions. This is proved through an approximation of $u_\infty$ by $p$-harmonic functions, the use of conjugate $p'$-harmonic functions, and the known connection of the latter with the inverse mean curvature flow. A statement about the regularity of $|\nabla u_\infty|$ arises as a by-product.

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arXiv2020

Separation of domain walls with nonlocal interaction and their renormalised energy by $\Gamma$-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 $\epsilon > 0$. The model gives rise to transition layers, called N\'eel walls, and we study their behaviour in the limit $\epsilon \to 0$. 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 N\'eel 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 N\'eel 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 $\Gamma$-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.

math.AP↗
arXiv2019

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.

math.DG↗
arXiv2018

Energy minimisers of prescribed winding number in an $\mathbb{S}^1$-valued nonlocal Allen-Cahn type model

We study a variational model for transition layers in thin ferromagnetic films with an underlying functional that combines an Allen-Cahn type structure with an additional nonlocal interaction term. The model represents the magnetisation by a map from $\mathbb{R}$ to $\mathbb{S}^1$. Thus it has a topological invariant in the form of a winding number, and we study minimisers subject to a prescribed winding number. As shown in our previous paper Ignat-Moser (JDE 2017), the nonlocal term gives rise to solutions that would not be present for a functional including only the (local) Allen-Cahn terms. We complete the picture here by proving existence of minimisers in all cases where it has been conjectured. In addition, we prove non-existence in some other cases.

math.AP↗
arXiv2017

Existence, Uniqueness and Structure of Second Order absolute minimisers

Let $\Omega \subseteq \mathbb{R}^n$ be a bounded open $C^{1,1}$ set. In this paper we prove the existence of a unique second order absolute minimiser $u_\infty$ of the functional \[ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm{F}(\cdot, \Delta u) \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq \Omega \text{ measurable}, \] with prescribed boundary conditions for $u$ and $\mathrm{D} u$ on $\partial \Omega$ and under natural assumptions on $\mathrm{F}$. We also show that $u_\infty$ is partially smooth and there exists a harmonic function $f_\infty \in L^1(\Omega)$ such that \[ \mathrm{F}(x, \Delta u_\infty(x)) \, =\, e_\infty\, \mathrm{sgn}\big(f_\infty(x)\big) \] for all $x \in \{f_\infty \neq 0\}$, where $e_\infty$ is the infimum of the global energy.

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