Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems partial-differential-equations-by-evans · Problem 3
Nonlinear First-Order PDE
Mathematics — Calculus · mean-value-formula-poisson-equation
Problem Modify the proof of the mean value formulas to show for n ≥ 3 n \geq 3 n ≥ 3 u ( 0 ) = ∫ ∂ B ( 0 , r ) g d S + 1 n ( n − 2 ) α ( n ) ∫ B ( 0 , r ) ( 1 ∣ x ∣ n − 2 − 1 r n − 2 ) f d x , u(0) = \int_{\partial B(0,r)} g\, dS + \frac{1}{n(n-2)\alpha(n)} \int_{B(0,r)} \left(\frac{1}{|x|^{n-2}} - \frac{1}{r^{n-2}}\right) f\, dx, u ( 0 ) = ∫ ∂ B ( 0 , r ) g d S + n ( n − 2 ) α ( n ) 1 ∫ B ( 0 , r ) ( ∣ x ∣ n − 2 1 − r n − 2 1 ) f d x , { − Δ u = f in B 0 ( 0 , r ) u = g on ∂ B ( 0 , r ) . \begin{cases} -\Delta u = f & \text{in } B^0(0,r) \\ u = g & \text{on } \partial B(0,r). \end{cases} { − Δ u = f u = g in B 0 ( 0 , r ) on ∂ B ( 0 , r ) . Browse the problem library