The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: $C_{\nu }\left(E₁\times E₂\right)=min(C_{\nu ₁}\left(E₁\right),C_{\nu ₂}\left(E₂\right))$, where $E_j$ and ${\nu }_j$ are respectively a compact set and a norm in ${ℂ}^{N_j}$ (j = 1,2), and ν is a norm in ${ℂ}^{N₁+N₂}$, ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of ${ℂ}^N$, denote by C(E) the standard L-capacity and by ${\omega }_E$ the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$). ...

Let $\Omega$ be a bounded domain of class $C^2$ in $ℝ$N and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/\left(N-1\right)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\text{'}}}$ and prove that $U_K$ is $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\text{'}}}$.

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

The aim of this paper is to show that for every Banach space $(X,\parallel ·\parallel )$ containing asymptotically isometric copy of the space $c_0$ there is a bounded, closed and convex set $C\subset X$ with the Chebyshev radius $r\left(C\right)=1$ such that for every $k\ge 1$ there exists a $k$-contractive mapping $T:C\to C$ with $\parallel x-Tx\parallel >1-1/k$ for any $x\in C$.

We consider the quasilinear equation ${\Delta }_{p}u+K\left(\right|x\left|\right)u^q=0$, and present the proof of the local existence of positive radial solutions near $0$ under suitable conditions on $K$. Moreover, we provide a priori estimates of positive radial solutions near $\infty$ when $r^{-\ell }K\left(r\right)$ for $\ell \ge -p$ is bounded near $\infty$.

We consider linear elliptic equations $-\Delta u+q\left(x\right)u=\lambda u+f$ in bounded Lipschitz domains $D\subset {ℝ}^N$ with mixed boundary conditions $\partial u/\partial n=\sigma \left(x\right)\lambda u+g$ on $\partial D$. The main feature of this boundary value problem is the appearance of $\lambda$ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient $\sigma \left(x\right)$. We study positivity principles and anti-maximum principles. One of our main results states that if $\sigma$ is somewhere negative, $q\ge 0$ and ${\int }_{D}q\left(x\right)dx>0$ then there exist two eigenvalues ${\lambda }_{-1}$, ${\lambda }_1$ such the positivity...

Let 1 < p < ∞, q = p/(p-1) and for $f\in L^p(0,\infty )$ define $F\left(x\right)=(1/x){ʃ}_0^{x}f\left(t\right)dt$, x > 0. Moser’s Inequality states that there is a constant $C_p$ such that $su{p}_{a\le 1}su{p}_{f\in B_p}{ʃ}_0^{\infty }exp[a{x}^q{\left|F\left(x\right)\right|}^q-x]dx=C_p$ where $B_p$ is the unit ball of $L^p$. Moreover, the value a = 1 is sharp. We observe that $F=K_1$ f where the integral operator $K_1$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...