On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

For a bounded open set $$\Omega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ S ( a + ϵ V ) = inf 0 ≢ u ∈ H 0 1 ( Ω ) ∫ Ω ( | ∇ u | 2 + ( a + ϵ V ) | u | 2 ) d x ( ∫ Ω u 6 d x ) 1 / 3 involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of $$S(a+\epsilon V)-S$$ S ( a + ϵ V ) - S as $$\epsilon \rightarrow 0+$$ ϵ → 0 + , where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $$S(a+\epsilon V)<S$$ S ( a + ϵ V ) < S for all sufficiently small $$\epsilon >0$$ ϵ > 0 .

The Robin function and conformal welding – A new proof of the existence

Green’s function of the mixed boundary value problem for harmonic functions is sometimes named the Robin function {R(z,\zeta\/)} after the French mathematical physicist Gustave Robin (1855–1897). The aim of this paper is to provide a new proof of the existence of the Robin function for planar n-fold connected domains using a special version of the well-known Koebe’s uniformization theorem and a conformal mapping which is closely related to the Robin function in the simply connected case.

PROFILE AND EXISTENCE OF SIGN-CHANGING SOLUTIONS TO AN ELLIPTIC SUBCRITICAL EQUATION

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.