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A general theory of hypersurface potentials. (English) Zbl 0855.31004

The author investigates hypersurface potentials with general kernels. Particularly, he proves

lim t+0 n-1 ϕ(x)K(x,t)dx=γϕ(0)+ n-1 ϕ(x)K(x,0)dx,lim xx 0 xν x 0 Σ ϕ(y) ν x 0 h(x-y)dσ y =γ(x 0 )ϕ(x 0 )+ Σ ϕ(y) ν x 0 h(x 0 -y)dσ y ,

where ϕ belongs to the Hölder’s class, Lyapunov’s manifold Σ is the boundary of a bounded domain, the values γ and γ(x 0 ) are defined by the kernels, ν x 0 is the inner normal at the point x 0 Σ, K(x,t) and h(x) are kernels of special classes. Potentials of functions of the class L p and potentials of measures are considered as well.

MSC:
31B25Boundary behavior of harmonic functions (higher-dimensional)
35B15Almost and pseudo-almost periodic solutions of PDE
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