- #1
stunner5000pt
- 1,461
- 2
Let D be a domain inside a simple closed curve C in R2. Consider the boundary value problem
[tex] (\Delta u)(x,y) = 0, \ (x,y) \in D, \\ \frac{\partial u}{\partial n} (x,y) = 0 , \ (x,y) \in C. [/tex]
where n is the outward unit normal on C. Use Green's Theorem to prove taht every solution u is a constant.
and yes its DELTA u not nabla u .
i know that Green's theorem is something lik this
[tex] \int \int _{D} u \nabla^2 u dx dy = \oint_{C} u \frac{\partial u}{\partial n} ds - \int \int_{D} |grad u|^2 dxdy [/tex]
so cna i do this
[tex] \int \int_{D} u \Delta u dx dy = \oint_{C} u \frac{\partial u}{\partial n} ds - \int \int_{D} |\nabla u|^2 dx dy [/tex]
i mean isn't [tex] \Delta u = \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} [/tex] isn't that true??
if that s true then moving on
[tex] \oint_{C} u \frac{\partial u}{\partial n} ds = 0 [/tex] because du/dn = 0 and sinve it is definite integration of a constant the answer is zero
so all that we're left with is
[tex] \int \int_{D} u \Delta u dx dy = - \int \int_{D} |\nabla u|^2 dx dy [/tex]
if i differentiate on both sides it yields
[tex] u \Delta u \displaystyle{|_{D}} = |\nabla^2 u| [/tex]
i m not sure how to proceed from here...
how would one figure out whether u is constnat?
Please help! Thank you.
[tex] (\Delta u)(x,y) = 0, \ (x,y) \in D, \\ \frac{\partial u}{\partial n} (x,y) = 0 , \ (x,y) \in C. [/tex]
where n is the outward unit normal on C. Use Green's Theorem to prove taht every solution u is a constant.
and yes its DELTA u not nabla u .
i know that Green's theorem is something lik this
[tex] \int \int _{D} u \nabla^2 u dx dy = \oint_{C} u \frac{\partial u}{\partial n} ds - \int \int_{D} |grad u|^2 dxdy [/tex]
so cna i do this
[tex] \int \int_{D} u \Delta u dx dy = \oint_{C} u \frac{\partial u}{\partial n} ds - \int \int_{D} |\nabla u|^2 dx dy [/tex]
i mean isn't [tex] \Delta u = \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} [/tex] isn't that true??
if that s true then moving on
[tex] \oint_{C} u \frac{\partial u}{\partial n} ds = 0 [/tex] because du/dn = 0 and sinve it is definite integration of a constant the answer is zero
so all that we're left with is
[tex] \int \int_{D} u \Delta u dx dy = - \int \int_{D} |\nabla u|^2 dx dy [/tex]
if i differentiate on both sides it yields
[tex] u \Delta u \displaystyle{|_{D}} = |\nabla^2 u| [/tex]
i m not sure how to proceed from here...
how would one figure out whether u is constnat?
Please help! Thank you.