Related papers: n/p-Harmonic maps: regularity for the sphere case

By virtue of barrier arguments we prove $C^\alpha$-regularity up to the boundary for the weak solutions of a non-local nonlinear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the…

We classify low-energy $\alpha$-harmonic maps from a closed non-spherical Riemannian surface $\Sigma$ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as $1<\alpha\downarrow 1$ and…

We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the…

The aim of this note is to discuss in more detail the Pohozaev-type identities that have been recently obtained by the author, Paul Laurain and Tristan Rivi\`ere in the framework of half-harmonic maps defined either on $R$ or on the sphere…

Biharmonic maps are the critical points of the bienergy functional and, from this point of view, generalise harmonic maps. We consider the Hopf map $\psi:\s^3\to \s^2$ and modify it into a nonharmonic biharmonic map $\phi:\s^3\to \s^3$. We…

In the joint work of the author with Da Lio and Rivi\`ere (Morse Index Stability for Sequences of Sacks-Uhlenbeck Maps into a Sphere) we studied the stability of the Morse index for Sacks-Uhlenbeck sequences into spheres as $p\searrow2$.…

We study a second order ordinary differential equation corresponding to rotationally symmetric $p$-harmonic maps. We show unique continuation and Liouville's type theorems for positive solutions. We discuss the existence of bounded positive…

A harmonic mapping $f=h+\overline{g}$ in $\mathbb{D}$ is $\varphi$-normal if $f^{\#}(z)=\mathcal{O}(|\varphi(z)|), \text{ as } |z|\to 1^-,$ where $f^{\#}(z)={(|h'(z)|+|g'(z)|)}/{(1+|f(z)|^2)}.$ In this paper, we establish several sufficient…

We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular…

Let $(g^{\alpha\beta}(x))$ and $(h_{ij}(u))$ be uniformly elliptic symmetric matrices, and assume that $h_{ij}(u)$ and $p(x) \, (\, \geq 2)$ are sufficiently smooth. We prove partial regularity of minimizers for the functional [ {\mathcal…

Adapting \cite{strz3}, we define generalized $p$-harmonic maps into Riemannian homogeneous targets, a notion of solutions not belonging to the energy space. Restricting our attention to the subcritical range $p$ greater than the domain…

In this paper, we are devoted to define p symphonic morphism and characterize it partially as in the case of harmonic morphism.

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $\alpha$-harmonic maps}, $\alpha\ge 1/2$, with respect to such a distribution. These are maps $u\in H^{\alpha}({{\mathbb R}}^k,{{\mathbb R}}^m)$…

Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural generalization of harmonic and biharmonic maps. These maps are defined as the critical points of suitable higher order functionals which extend the classical energy…

Let $M$ be a $C^2$-smooth Riemannian manifold with boundary and $N$ a complete $C^2$-smooth Riemannian manifold. We show that each stationary $p$-harmonic mapping $u\colon M\to N$, whose image lies in a compact subset of $N$, is locally…

We study fourth-order quasilinear elliptic problems that involve the p-biharmonic operator and Navier boundary conditions. The nonlinear term grows at the critical Sobolev rate. Starting from a Hamiltonian system of two second-order…

Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric…

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as…

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…