Related papers: Rigidity and regularity of co-dimension one Sobole…

We prove the $C^{1}$ regularity and developability of $W^{2,p}$ isometric immersions of $n$-dimensional flat domains into ${\mathbb R}^{n+k}$ where $p\ge \min\{2k, n\}$. Another parallel consequence of our methods is a similar regularity…

We discuss $C^1$ regularity and developability of isometric immersions of flat domains into $\mathbb R^3$ enjoying a local fractional Sobolev $W^{1+s, \frac2s}$ regularity for $2/3 \le s< 1 $, generalizing the known results on Sobolev and…

We develop a framework for characterizing isometric immersions of simply connected, bounded, planar regions with piecewise smooth boundaries into three-dimensional space. Each immersion is associated with a framed curve along the boundary…

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little H\"older regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha}$ regularity with $\alpha…

We prove the existence, uniqueness, and $W^{1,2}$-regularity for the solution to the Pfaff system with antisymmetric $L^2$-coefficient matrix in arbitrary dimensions. Hence, we establish the equivalence between the existence of…

A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $\omega\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into…

We study the problem of constructing $C^{1,\theta}$ isometric immersions of Riemannian metrics on $n$-dimensional domains into $\mathbb{R}^{n+1}$. While the classical Nash--Kuiper theorem established the flexibility of $C^1$ isometries,…

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper says that any short embedding in codimension one can be uniformly approximated by $C^1$ isometric embeddings. This…

The approximation of Sobolev homeomorphisms by smooth diffeomorphisms is well understood in first-order spaces $W^{1,p}$, but remains largely open in the second-order space $W^{2,1}$ due to a fundamental tension between curvature control…

We prove smoothness of $W^{2,2}$ isometric immersions of surfaces endowed with a smooth Riemannian metric of positive Gauss curvature. We then derive the $\Gamma$-limit of three dimensional nonlinear shells with inhomogeneous energy…

This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations extending our findings in [22] to domains of polyhedral type. In particular, we study the smoothness in the specific scale…

We prove that if $\Om \subseteq \R^2$ is bounded and $\R^2 \setminus \Om$ satisfies suitable structural assumptions (for example it has a countable number of connected components), then $W^{1,2}(\Om)$ is dense in $W^{1,p}(\Om)$ for every…

We are concerned with two interrelated problems: smoothability of connection 1-forms with low regularity on bundles with prescribed smooth curvature 2-forms, and existence of isometric immersions with low regularity. We first show that if…

We study quantitative stability results for different classes of Sobolev inequalities on general compact Riemannian manifolds. We prove that, up to constants depending on the manifold, a function that nearly saturates a critical Sobolev…

Let $\Omega$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. It is shown that for $0\leq q\leq n$, $s\geq 0$, the embedding $j_{q}: dom(\overline{\partial})\cap dom(\overline{\partial}^{*}) \hookrightarrow…

In this paper we consider the existence and regularity problem for Coulomb frames in the normal bundle of two-dimensional surfaces with higher codimension in Euclidean spaces. While the case of two codimensions can be approached directly by…

This paper aims to establish counterparts of fundamental regularity statements for solutions to elliptic equations in the setting of low-dimensional structures such as, for instance, glued manifolds or CW-complexes. The main result proves…

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.