### A Cantor regular set which does not have Markov's property

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This paper is devoted to the formulation and solution of a free boundary problem for the Poisson equation in the plane. The object is to seek a domain $\Omega $ and a function $u$ defined in $\Omega $ satisfying the given differential equation together with both Dirichlet and Neumann type data on the boundary of $\Omega $. The Neumann data are given in a manner which permits reformulation of the problem as a variational inequality. Under suitable hypotheses about the given data, it is shown that there exists a unique solution...

A real-valued Hardy space $H{\xb9}_{\surd}\left(\right)\subseteq L\xb9\left(\right)$ related to the square root of the Poisson kernel in the unit disc is defined. The space is shown to be strictly larger than its classical counterpart H¹(). A decreasing function is in $H{\xb9}_{\surd}\left(\right)$ if and only if the function is in the Orlicz space LloglogL(). In contrast to the case of H¹(), there is no such characterization for general positive functions: every Orlicz space strictly larger than L log L() contains positive functions which do not belong to $H{\xb9}_{\surd}\left(\right)$, and no Orlicz space...

The Fourier problem on planar domains with time variable boundary is considered using integral equations. A simple numerical method for the integral equation is described and the convergence of the method is proved. It is shown how to approximate the solution of the Fourier problem and how to estimate the error. A numerical example is given.

We consider a transient Hunt process in which the potential density $u$ satisfies the conditions: (a) for each $x$, $u(x,y{)}^{-1}$ is finite continuous in $y$; (b) $u(x,y)=+\infty $ iff $x=y$. In earlier papers Chung established an equilibrium principle, and Rao obtained a Riesz of decomposition for excessive functions. We now begin a deeper study under these conditions, including the uniqueness of the decomposition and Hunt’s hypothesis (B).