POLAR BOUNDARY SETS FOR SUPERDIFFUSIONS AND REMOVABLE LATERAL SINGULARITIES FOR NONLINEAR PARABOLIC PDES

Suppose L is a second-order elliptic differential operator in R-d and D is a bounded, smooth domain in R-d. Let 1 < alpha less than or equal to 2 and let Gamma be a closed subset of partial derivative D. It is known [13] that the following three properties are equivalent: (alpha) Gamma is partial derivative-polar; that is, Gamma is not hit by the r ange of the corresponding (L, alpha)-superdiffusion in D; (beta) the P oisson capacity of Gamma is equal to 0; that is, the integral integral (D) rho(x)dx[integral(Gamma) k(x, y)nu(dy)](alpha) is equal to 0 or in finity for every measure nu, where rho(x) is the distance to the bound ary and k(x, y) is the corresponding Poisson kernel; and (gamma) Gamma is a removable boundary singularity for the equation Lu = u(alpha) in D; that is, if u greater than or equal to 0 and Lu = u(alpha) in D an d if u = 0 on partial derivative D\Gamma, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder Q = Rx D. Let Gamma be a compact set on the lateral boundary of Q. We show that the following three properties are equivalent: (a) Gamma is G-pol ar; that is, Gamma is not hit by the graph of the corresponding (L, al pha)-superdiffusion in Q; (b) the Poisson capacity of Gamma is equal t o 0; that is, the integral integral(Q) rho(x)dr dx[integral(Gamma) k(r , x; t, y)nu(dt, dy)](alpha) is equal to 0 or infinity for every measu re nu, where k(r, x; t, y) is the corresponding (parabolic) Poisson ke rnel; and (c) Gamma is a removable lateral singularity for the equatio n (u) over dot + Lu = u(alpha) in Q; that is, if u greater than or equ al to 0 and (u) over dot + Lu = u(alpha) in Q and if u = 0 on partial derivative Q\Gamma and on {infinity} x D, then u = 0. (C) 1998 John Wi ley & Sons, Inc.