Related papers: Limits of $\alpha$-harmonic maps

We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic…

Motivated by a class of near BPS Skyrme models introduced by Adam, S\'anchez-Guill\'en and Wereszczy\'nski, the following variant of the harmonic map problem is introduced: a map $\phi:(M,g)\rightarrow (N,h)$ between Riemannian manifolds is…

In this article, we show that sequences of $(n+\alpha)$-harmonic maps with a free boundary in $\mathbb S^{d-1}$, where $\alpha$ is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the…

For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian…

We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the…

Let $A \subset \mathbb{R} ^2 $ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus…

We extend the celebrated theorem of Kellogg for conformal diffeomorphisms to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimiser of Dirichlet energy of Sobolev mappings between doubly connected Riemanian…

The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace…

The Teichm\"uller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to…

4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very…

We consider the question of quantitative stability of minimisers for a well-known variational problem for which the infimum of the energy is not achieved in the classical sense, namely for the Dirichlet energy of degree $1$ maps from closed…

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic,…

In this paper, we establish the uniqueness of heat flow of harmonic maps into (N, h) that have sufficiently small renormalized energies, provided that N is either a unit sphere $S^{k-1}$ or a compact Riemannian homogeneous manifold without…

We prove the energy identity for min-max sequences of the Sacks-Uhlenbeck and the biharmonic approximation of harmonic maps from surfaces into general target manifolds. The proof relies on Hopf-differential type estimates for the two…

We study the asymptotic Dirichlet problem for A-harmonic equations and for the minimal graph equation on a Cartan-Hadamard manifold M whose sectional curvatures are bounded from below and above by certain functions depending on the distance…

We introduce (integro-differential) harmonic maps into spheres, which are defined as critical points of the Besov-Slobodeckij energy $\int\limits_{\Omega}\int\limits_{\Omega} \frac{|v(x)-v(y)|^{p_s}}{|x-y|^{n+sp_s}}\ dx\ dy$. For $p_s = 2$…

In this article we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that…

We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the…

We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling…

We study Hamiltonian diffeomorphisms on symplectic Euclidean spaces that are equal to non-degenerate linear maps at infinity. Under the assumption that there exists an isolated homologically nontrivial fixed point satisfying the twist…