Variational approach to Laplace's problem

[fu^2]

We consider the following problem (*)

[tex]\begin{cases}
-\Delta u=f\\
\left[u\right]_{\partial \Omega}=h
\end{cases}[/tex]

with [tex]\Omega[/tex] a boundary set of [tex]\mathbb{R}^n[/tex] with regular edge, [tex]f\in C^0(\Omega)[/tex] and [tex]h\in C^0(\partial\Omega)[/tex].

Show that [tex]u[/tex] is a solution of (*) iff [tex]\displaystyle\min_{w\in C_h^1(\Omega)}E(w)=E(u)[/tex], where

[tex]\begin{center}E(w)=\dfrac{1}{2}\displaystyle\int_{\Omega}\left|\nabla w(x)\right|^2dx -\displaystyle\int_{\Omega}w(x)f(x)dx \end{center}[/tex]

and

[tex]\begin{center}C_h^1(\Omega)=\left\{v\mid v\in C^2(\Omega)\cap C^1(\bar{\Omega}), \left[v\right]_{\partial \Omega}=h\right\}\end{center}[/tex]


-------------------------------------------------------------

P.s. We obatian the Laplace problem if we put $f=0$ (I call this situation with this name...).

[gugo82]

Nonono, my friend... I didn't mean to write anything else but kinda.

[j18eos]

"gugo82":...kinda...

Gugo82 maybe did you want to write "kind a"?

[fu^2]

hint: to prove the proof, you can use the following fact:

If $u\in C_h^1(\Omega)$, then $u+\epsilon v\in C_h^1(\Omega)$, with $v\inC_0^1(\Omega)$ and $\epsilon$ a real constant.

[gugo82]

[OT, on terminology]

"fu^2":We consider the following problem (*)

[tex]\begin{cases}
-\Delta u=f\\
\left[u\right]_{\partial \Omega}=h
\end{cases}[/tex]

with [tex]\Omega[/tex] a boundary set of [tex]\mathbb{R}^n[/tex] with regular edge, [tex]f\in C^0(\Omega)[/tex] and [tex]h\in C^0(\partial\Omega)[/tex].

I used to call a problem of this kind inhomogeneous Dirichlet problem for the Poisson equation.

Moreover, the period following the formulas seems kinda messed up.
I suppose you wrote boundary instead of bounded (for "limitato") and edge meaning "frontiera", while in this case the proper translation is boundary.

[/OT]