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Then by known regularity results, $f|_{M^\circ}:M^\circ \to N^\circ$ is smooth. Now suppose that $f|_{M^\circ}:\partial M \to \partial N$ is $C^k$. Is $f:M \to N \in C^k$ up to the boundary? …

$\newcommand{\R}{\mathbb R}$ $\newcommand{\N}{\mathbb N}$ $\newcommand{\de}{\delta}$ $\newcommand{\sig}{\sigma}$ $\newcommand{\Average}[1]{\left\langle#1\right\rangle} $ $\newcommand{\IP}[2]{\Average …

I am looking for a reference for the regularity of harmonic forms which belong to $L^1(M)$. Explicitly, let $\M$ be a smooth oriented Riemannian manifold. …

A proof sketch for regularity on $\Omega_0$: (for the full details see theorems 1.1 and 1.2 here). …

$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for som …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly harmonic … I am trying to use rather standard linear elliptic regularity results; the classical regularity proofs for harmonic maps are more involved. …

In codimension $d-k=1$ the answer is positive, since $q$ is determined uniquely by $F$ (in a way that preserve the regularity). In higher codimension some choices have to be made. …

It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$. Is there a $W^{2,2}$ isometric embedding? (i.e an isometric …

Well, for conformal maps equation $(1)$ is merely $d$-harmonicity in disguise:) The equation is $$ \delta\big((\det df)^{1-\frac{2}{d}} df\big)=0. \tag{1} $$ Since for conformal maps, $\det df=\|df\ …

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\Hom}{\operatorname{Hom}}$ $\newcommand{\tr}{\operatorname{tr}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\op …

Is there an example of a pair $M,N$ of connected, oriented equidimensional Riemannian manifolds with the following properties: $M$ is everywhere non-flat, $N$ is flat. There exist a map $f:M \to N$ …

I posted this question in MSE but got no response (even after giving a bounty), so I am trying here. Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. Suppose $f:M \to N$ is a differentiable …