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](https://www.ams.org/journals/proc/1955-006-05/S0002-9939-1955-0071730-1/S0002-9939-1955-0071730-1.pdf) 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
$$J''(u\_n)(u\_{n+1}-u\_n)=-J'(u\_n). \;\;\;\text{(Equality is in $H^{-1}(\Omega)$)}.\tag{1}\label{1}$$


1. I would like to check whether or not \eqref{1} is equivalent to $$J''(u\_n)(u\_{n+1},v)-J''(u\_n)(u\_n,v)=-J'(u\_n)v, \;\;\;\text{ for all }v\in H^1\_0(\Omega).$$
2. 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.



Semi-linear elliptic problem, energy functionals, Fréchet derivatives and the Newton method in Banach spaces
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.


