1. The Dirichlet problem for the exterior of a circle is =0 on {(x,y) E R?...

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1. The Dirichlet problem for the exterior of a circle is =0 on {(x,y) E R? : x² + y² > a²} u =h on {(x, y) E R² : x² + y² = a²} Uz1 + Uy,y lim sup |u(x, y) CC < ∞ Derive the Poisson formula in polar coordinates for this problem: p2 – a? 27 ũ(r, 0) a? – 2ar cos(0 – 4) + r2 dø - where h(0) = h(a cos 0, a sin 0). Hint: Follow the derivation of the Poisson formula on the disk Ba(0). 1. The Dirichlet problem for the exterior of a circle is =0 on {(x,y) E R? : x² + y² > a²} u =h on {(x, y) E R² : x² + y² = a²} Uz1 + Uy,y lim sup |u(x, y) CC < ∞ Derive the Poisson formula in polar coordinates for this problem: p2 – a? 27 ũ(r, 0) a? – 2ar cos(0 – 4) + r2 dø - where h(0) = h(a cos 0, a sin 0). Hint: Follow the derivation of the Poisson formula on the disk Ba(0).