American Institute of Mathematical Sciences

$Y_{\lambda}^p$:=$\sum_{n=0}^{\infty}\pm \lambda ^n$

where the "$+$" and "$-$" signs are chosen independently with probability $p$ and $1-p$. Let $\nu_\lambda^p$ be the distribution of the random sum $\nu_\lambda^p(E)$:=$Prob(Y_{\lambda}^p \in E)$ for every set $E$. The conjecture is that for every $p \in (0,1)$ the measure $\nu_\lambda^p$ is absolutely continuous w.r.t. Lebesgue measure and with the density in $L^2(R)$ for almost every $\lambda\in (p^p\cdot(1-p)^{(1-p)},1).$
    B. Solomyak and Y. Peres [3, [Corollary 1.4] proved that for every $p \in (\frac{1}{3},\frac{2}{3})$ the distribution $\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for almost every $\lambda \in (p^2+(1-p)^2,1).$ In this paper we extend the parameter interval where a weakened version of the conjecture still holds. Namely, we prove Corollary 3 that for every $p \in (0,\frac{1}{3}]$ the measure $\nu_\lambda^p$ is absolutely continuous with $L^2(R)$ density for almost every $\lambda\in(F(p),1)$, where $F(p)=(1-2p)^{2-\log41/\log 9}$, see Figure 3.

$\F(u) = \int_\O [f(x,Du(x))+h(x,u(x))]dx$

defined on Sobolev functions subject to Dirichlet boundary conditions. We give an existence result for minimum points, based on regularity assumptions on the minimizers of the relaxed functional, applying the method of extremization of the integral.