Suppose $\Omega\subset\mathbb{R}^n$ is a regular open set, $f\in L^2(\Omega)$ and consider the following elliptic problem.
$$-\Delta u + u=f'(u) , \;\;u_{|\partial \Omega}=0,$$
where $f'$ is the derivative of a function $f:\mathbb{R}\to\mathbb{R}$ and $f'(u)\in L^2(\Omega)$ for all $u\in H^1_0(\Omega)$ (choose any assumptions you like on $f'$ to obtain this). Suppose also that that $f''$ exists.
It is well known that $u$ minimizes the energy functional $J:H^1_0(\Omega)\to \mathbb{R}$ given by
$$J(u)=\frac{1}{2}\int_\Omega|\nabla u|^2 +\frac{1}{2}\int_\Omega u^2-\int_\Omega f(u).$$
Now this also means that $u$ is a root of the Frechet (equiv to Gâteaux in this case) derivative $J':H^1_0(\Omega)\to H^{-1}(\Omega)$. In particular, this means that
$$J'(u)v=\int_\Omega\nabla u\cdot\nabla v+\int_\Omega u v-\int_\Omega f'(u)v=0, \;\;\;\text{ for all }v\in H^1_0(\Omega).$$
(Notice that this is nothing but the variational formulation of the pde)
Suppose now that we wish to apply the Newton method in Banach spaces to minimize the function $J'$. I will assume that $J'':H^1_0(\Omega)\to L(H^1_0(\Omega),H^{-1}(\Omega)) $ exists and that $J''(u)$ be identified with the continuous bilinear form $J''(u)(\cdot,\cdot)\in \mathcal{B}(H^1_0(\Omega))$ as
$$J''(u)(v,w)=\int_\Omega\nabla v\cdot\nabla w+\int_\Omega vw-\int_\Omega f''(u)vw, \;\;\;\text{ for all }v,w\in H^1_0(\Omega).$$
Starting with a guess $u_0\in H^1_0(\Omega)$, the usual Newton method iterative procedure to minimize $J'$ is then to solve to following "abstract" equation
$$J''(u_n)(u_{n+1}-u_n)=-J'(u_n). \;\;\;\text{(Equality is in $H^{-1}(\Omega)$)}.\tag{1}\label{1}$$
- I would like to check whether or not \eqref{1} is equivalent to $$J''(u_n)(u_{n+1},v)-J''(u_n)(u_n,v)=-J'(u_n)v, \;\;\;\text{ for all }v\in H^1_0(\Omega).$$
- If the above is true, to apply Newton method, is is necessary that that $$J''(u)^{-1}:L(H^{-1}(\Omega),H^1_0(\Omega))\simeq \mathcal{B}(H^1_0(\Omega))\to H^1_0(\Omega)$$ exists is bounded in the operator norm. How does one even start thinking about such a problem?
Important Fix: Question 2 is wrong. What we need to show is that $J''(u)$ is invertible for every $u$, ie, $J''(u)^{-1}:H^{-1}(\Omega)\to H^1_0(\Omega)$ exists.
Best Answer
Let me write out the equation $J''(u)w = g$ for $g\in H^{-1}$ and $w\in H^1_0(\Omega)$. This is equivalent to $$ \int_\Omega \nabla w \cdot \nabla v - f''(u)wv = g(v) \quad \forall v\in H^1_0(\Omega). $$ This is the weak formulation of $$ -\Delta w - f''(u)w = g$$ plus boundary conditions. To show existence of solutions, you need some assumptions on $f''$ to be able to invoke Lax-Milgram theorem or other existence theorems. The easiest one would be to require $f''(u) \le0$ for all $u$.
In order to apply the convergence theory of Newton's method, you will need that $J''(u)^{-1}$ is a bounded linear operator, and that the inverses $J''(u)^{-1}$ are bounded on a neightborhood of the reference solution.