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討論串[分析] PDE
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3. 1. Let u be a solution of the wave equation in all of R X R.. suppose a > 0 and u(x,0) = u (x,0) = 0 for |x|>a.. t. Show that there is C > 0 s.t..
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如果你可以找到u使得△u + λu = 0，那麼你應該有regularity thm可以用。. △是elliptic operator。這個叫rayleigh quotient，你可以從Euler Lagrange. 的角度去思考(Lagrange Multiplier):. 假設|u|_L^2(Ω
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n 2 __. Let Ω in R be open. Show that if there exists a function u ε C ( Ω ). vanishing on (boundary)Ω for which the quotient. 2. ∫ |▽u|. Ω. ---------
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-φ" + K^2 φ = 0 whenever x≠0. and. -φ'(0+) + φ'(0-) = Q/D. then. φ(x) = a e^{-Kx} when x>0. φ(x) = b e^{Kx} when x<0. then. a=b. and. aK+bK=Q/D. - ▽^2
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