Related papers: The Sobolev extension problem on trees and in the …
In this paper, we study the spectral problem on a compact Finsler manifold with or without boundary. More precisely, given a certain collection of sets in Sobolev space $H^{1,2}(M)$ and a dimension-like function, we can define a…
Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function $f$ with only two critical values $\pm 1$ and no asymptotic values such that $f^{-1}([-1,1])$ is ambiently homeomorphic to the…
We construct and investigate $(1, p)$-Sobolev space, $p$-energy, and the corresponding $p$-energy measures on the planar Sierpi\'{n}ski carpet for all $p \in (1, \infty)$. Our method is based on the idea of Kusuoka and Zhou [Probab. Theory…
The longest common extension (LCE) of two indices in a string is the length of the longest identical substrings starting at these two indices. The LCE problem asks to preprocess a string into a compact data structure that supports fast LCE…
We give direct proofs and constructions of the trace and extension theorems for Sobolev mappings in $W^{1, 1} (M, N)$, where $M$ is Riemannian manifold with compact boundary $\partial M$ and $N$ is a complete Riemannian manifold. The…
Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev…
Seo and Shin showed that the number of rooted trees on $[n+1]$ such that the maximal decreasing subtree with the same root has $k+1$ vertices is equal to the number of functions $f:[n]\to[n]$ such that the image of $f$ contains $[k]$. We…
We establish local well-posedness in Sobolev spaces $H^s(\mathbb{T})$, with $s\geq -1/2$, for the initial value problem issues of the equation $$ u_t + u_{xxx}+\eta Lu + uu_x=0;\; x\in \mathbb{T},\; t\geq0, $$ where $\eta >0$,…
We prove the well posedness in weighted Sobolev spaces of certain linear and nonlinear elliptic boundary value problems posed on convex domains and under singular forcing. It is assumed that the weights belong to the Muckenhoupt class $A_p$…
We prove trace and extension results for Sobolev-type function spaces that are well suited for nonlocal Dirichlet and Neumann problems including those for the fractional $p$-Laplacian. Our results are robust with respect to the order of…
Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant…
In this paper we consider $L_p$-approximation, $p \in \{2,\infty\}$, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of…
First and second-order inequalities of Friedrichs type for Sobolev functions in arbitrary domains are offered. The relevant inequalities involve optimal norms and constants that are independent of the geometry of the domain. Parallel…
We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space $W^{1,1}$ with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately…
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it…
In a refined Sobolev scale, we investigate an elliptic boundary-value problem with additional unknown functions in boundary conditions for which the maximum of orders of boundary operators is grater than or equal to the order of the…
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which…
Chapter 1 deals with the problem of the existence of an upper/lower envelope from a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space…
In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, then there exists a…
The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic H\"ormander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=<\xi>^{s}\varphi(<\xi>)$ for…