Related papers: Maz'ya's $\Phi$-inequalities on domains

For $\Omega\subseteq\mathbb{R}^{n}$ an open and bounded region we consider solutions $u\in W_{\text{loc}}^{1,p(x)}\big(\Omega;\mathbb{R}^{N}\big)$, with $N>1$, of the $p(x)$-Laplacian system \begin{equation}…

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u+\phi(|u|)u=f(u)~\text{in} ~\Omega_{\lambda}, u(x)>0 ~\text{in}~\Omega_{\lambda}, u=0~ \mbox{on}…

Let $f$ be a function on a bounded domain $\Omega \subseteq \mathbb{R}^n$ and $\delta$ be a positive function on $\Omega$ such that $B(x,\delta(x))\subseteq \Omega$. Let $\sigma(f)(x)$ be the average of $f$ over the ball $B(x,\delta(x))$.…

The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(\Omega)$ ($\Omega \subset \mathbb R^n$ a bounded domain) with $\int_\Omega |\nabla u|^ndx \le 1$ one has $\int_\Omega (e^{\alpha_n|u|^{\frac n{n-1}}}-1)dx \le c…

We consider the zeta function $\zeta\_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve.We prove non-negativeness and growth properties for…

We consider the minimization problem of $\phi$-divergences between a given probability measure $P$ and subsets $\Omega$ of the vector space $\mathcal{M}_\mathcal{F}$ of all signed finite measures which integrate a given class $\mathcal{F}$…

Let $\Phi$ be a concave function on $(0,\infty)$ of strictly lower type $p_{\Phi}\in(0,1]$ and $\omega\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$. We introduce the weighted local Orlicz-Hardy space $h^{\Phi}_{\omega}(\mathbb{R}^n)$…

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$,…

We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (\phi(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} \Omega$…

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to…

We consider the functional $$I_\Omega(v) = \int_\Omega [f(|Dv|) - v] dx,$$ where $\Omega$ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, G. Crasta [Cr1] has shown that if $I_\Omega$ admits a minimizer in…

Let $\Phi$ be a family of functions analytic in some neighborhood of a complex domain $\Omega$, and let $T$ be a Hilbert space operator whose spectrum is contained in $\overline\Omega$. Our typical result shows that under some extra…

In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in $\RR^n$,\,$(n\geq2)$, and we could obtain several inequalities associated with the Kakeya inequalities. We will show…

We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge…

Let $\Omega $ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u\_n$ in $BV(\Omega)$, to the problem $$(P\_n) \begin{cases} -{\rm div} \sigma +2n (\int\_…

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $\Omega= \{ z: |f(z)| > R\}$ and $f|_\Omega$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model…

Let $0<p<\infty$, $\Gamma$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of analytic functions $F$ satisfying…

Let $ \Omega \subset R^2$ be a bounded piecewise smooth domain and $\phi_\lambda$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $\lambda^2$ and nodal set ${ N}_{\phi_{\lambda}} = {x \in \Omega; \phi_{\lambda}(x) = 0}.$ Let $H…

Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$${\!,}$ $d_\Gamma$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz…

We present some properties of (not necessarily linear) positive maps between $C^*$-algebras. We first extend the notion of Lieb functions to that of Lieb positive maps between $C^*$-algebras. Then we give some basic properties and…