Related papers: Orlicz-Besov extension and Ahlfors $n$-regular dom…

Let $n\ge 2$ and $s\in (n-2,n)$. Assume that $\Omega\subset \mathbb{R}^n$ is a one-sided bounded non-tangentially accessible domain with $s$-Ahlfors regular boundary and $\sigma$ is the surface measure on the boundary of $\Omega$, denoted…

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the…

In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=\Delta \big(u\phi(v)\big), v_t=\Delta v-uv, \end{array} \right. \] is considered under the…

Let $u(x,y)$ be a harmonic function in the halfspace $\mathbb{R}^n\times\mathbb{R}_+$ that grows near the boundary not faster than some fixed majorant $w(y)$. Recently it was proven that an appropriate weighted average along the vertical…

We provide comprehensive regularity results and optimal conditions for a general class of functionals involving Orlicz multi-phase of the type \begin{align} \label{abst:1} v\mapsto \int_{\Omega} F(x,v,Dv)\,dx, \end{align} exhibiting…

We prove the existence of nontrivial unbounded exceptional domains in the Euclidean space $\R^N$, $N\geq4$. These domains arise as perturbations of complements of straight cylinders in $\R^N$, and by definition they support a positive…

Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and a one-parameter family of Young functions $\{\Psi_q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in…

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right.…

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $\psi$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic…

We study the qualitative properties of functions belonging to the corresponding De Giorgi classes \begin{equation*} \int\limits_{B_{r(1-\sigma)}(x_{0})}\,\varPhi(x, |\nabla(u-k)_{\pm}|)\,dx \leqslant…

Let $D$ be a connected bounded domain in $\R^2$, $S$ be its boundary which is closed, connected and smooth. Let $\Phi(z)=\frac 1 {2\pi i}\int_S\frac{f(s)ds}{s-z}$, $f\in L^1(S)$, $z=x+iy$. Boundary values of $\Phi(z)$ on $S$ are studied.…

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $\phi_{k+1}(n)\geqslant cn$. Comparing with the upper bound which…

For the real part of the Cauchy-type integral that is known to be the logarithmic potential of the double layer, a necessary and sufficient condition for the continuous extension to the Ahlfors-regular boundary is established.

We show an extention of a theorem of Kaczynski to boundary functions in n-dimensional space. Let $H$ denote the upper half-plane, and let $X$ denote its frontier, the $x$-axis. Suppose that $f$ is a function mapping $H$ into some metric…

Let $A = \{a_{1},a_{2},\dots{}\}$ $(a_{1} < a_{2} < \dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\in A)$. P. Erd\H{o}s, A. S\'ark\"ozy and V.…

We study the existence of non-trivial unbounded domains of $\Omega \subset \mathbb{R}^2$ where the equation \begin{align} - \lambda u_{xx} -u_{tt} &= u \qquad \text{in $\Omega$,}\nonumber u &=0 \qquad \text{on $\partial \Omega$,}\nonumber…

We prove that a trace inequality holds for John domains $\Omega$ satisfying $$ \mathcal H^{n-1}(\partial \Omega\setminus \partial_*\Omega)=0,$$ where $\partial_*\Omega$ denotes the measure-theoretic boundary, together with an upper density…

A behavior of homeomorphisms of Orlicz--Sobolev classes in a closure of a domain is investigated. There are obtained theorems about equicontinuity of classes mentioned above in terms of prime ends of regular domains. In particular, it is…

We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_\Omega…

We extend the existence theorems in [Barchiesi, Henao \& Mora-Corral; ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a…