Related papers: Universal conformal weights on Sobolev spaces

Let (M,g) be a compact Riemannian three-dimensional manifold with boundary. We prove the compactness of the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface.…

In prior work, the author has characterized the real numbers $a,b,c$ and $1\leq p,q,r<\infty $ such that the weighted Sobolev space $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{a}{q}}u\in…

We consider conformal homeomorphisms $\varphi$ of generalized Jordan domains $U$ onto planar domains $\Omega$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary…

For appropriate domains $\Omega_{1}, \Omega_{2}$ we consider mappings $\Phi_{\mathbf A}:\Omega_{1}\to\Omega_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(\Omega_{1})$ into finitely many…

We introduce a weighted Sobolev space theory for the non-local elliptic equation $$ \Delta^{\alpha/2}u=f, \quad x\in \mathcal{O}\,; \quad r_{\overline{\mathcal{O}}^c}u=g $$ as well as for the non-local parabolic equation $$…

We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) &…

We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight $\lambda$ depends on the independent variable $z$. We prove that for an…

We prove the existence of infinitely many radial solutions for elliptic systems in Rn with power weights. A key tool for the proof will be a weighted imbedding theorem for fractional-order Sobolev spaces, that could be of independent…

We study the non-compact Sobolev embeddings into the optimal scale of Lorentz spaces, $W_0^mL^{p,q}(\Omega) \to L^{\frac{dp}{d - mp},r}(\Omega)$, where $\Omega \subseteq \mathbb{R}^d$, $1 \le m \le d$ and $0<q<r\le\infty$ with $1<p<\frac…

By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan…

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively…

Given a finitely connected region $\Omega$ of the Riemann sphere whose complement consists of $m$ mutually disjoint closed disks $\bar{U}_j$, the random homeomorphism $h_j$ on the boundary component $\partial U_j$ is constructed using the…

Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study…

We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.

We determine the homeomorphism type of the space of smooth complete nonnegatively curved metrics on surfaces of positive Euler characteristic equipped with the topology of $C^\gamma$ uniform convergence on compact sets, when $\gamma$ is…

We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the…

We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang's density result in $W^{1,…

Sobolev-type embeddings on metric measure spaces encode a subtle interaction between the analytic regularity of functions and the geometry of the underlying domain space. In this paper we develop an embedding theory for variable…

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed…

We establish embeddings on a class of Sobolev spaces with potential weights on unbounded domains. Our results provide embeddings into weighted Lebesgue spaces $L^q_\theta$ with radial power weights and establish the existence and…