Examples of monotone operators in PDEs What is an example of a monotone and pseudomonotone operator in analysis of PDE? Why it is useful to study them, and what is the difference heuristically?
Eg. I am looking for something like "laplacian" or "p-Laplacian" (but those are not treated as monotone problems).
 A: This is a big question which doesn't really have a short answer. However, I will try to give an outline of the big picture concerning monotone operators. The reason monotone operators are important is because they are generally surjective. The reason pseudo-monotone operators are important is that a pseudo-monotone perturbation of a monotone operator is generally surjective. This is outlined by the following important theorem of Debrunner and Flor:
Let $V$ be a reflexive Banach space and $K \subset V$ closed, convex and non-empty. Suppose $A:K \rightrightarrows V^*$ is maximal monotone and $B: K \to V^*$ is demicontinuous, bounded, pseudo-monotone and coercive. Then $A + B$ is surjective.
This theorem has various weaker forms, a famous version being the Browder-Minty Theorem. I won't spell out exactly what the other hypothesis of the theorem mean here because we are focusing on monotonicity. It is also important to notice that one can take $A = 0$ in the theorem above so that also $B$ is surjective under the hypothesis (one cannot take $B = 0$ because of the coercivity requirement...) As we will see, this can be done to treat elliptic equations. One can then use $A$ as a time-derivative operator when treating parabolic equations.
One can use the theorem (for example) to solve non-linear Dirichlet problems of the form
$$-\mbox{div }a(x,\nabla u) + a_0(x,u) = f$$
for $u$ when $a,a_0$ and $f$ are given by verifying the hypothesis of the theorem for $A = 0$ and $B(u) = -\mbox{div }a(x,\nabla u) + a_0(x,u)$ (taken in the weak sense) on the space $W^{1,p}_0(\Omega)$ for appropriately chosen $p$. Of course the functions $a, a_0$ have to satisfy certain growth, coercivity and monotonicity conditions if one wants the resulting operator to be pseudo-monotone. Your example of the $p$-Laplacian falls into this category (take $a(x, u) = |u|^{p-2}u, a_0(x,u) = 0$).
There are of course a lot of technical details to be observed here ranging from choosing appropriate function spaces to well-definedness of the operators to choosing the correct weak formulations. This is a fairly lengthy ordeal that should be left to a textbook or at least some detailed lecture notes. To give you an idea, we spent about 4 weeks working out these ideas in detail in our non-linear PDE class last semester.
As a nice introductory reference which covers this equation (in less generality) I would recommend chapter 10 in An Introduction to Partial Differential Equations by Renardy and Rogers. They focus on working out the details of applying the Browder-Minty Theorem to a non-linear elliptic problem.
This covers elliptic equations. But the true power of the theorem above becomes clear when we don't take $A = 0$. This allows us to solve non-linear parabolic equations of the form
$$ \frac{\partial}{\partial t} u(x,t) - \mbox{div }a(x, t,\nabla u(x,t)) + a_0(x,t,u(x,t)) = f(x,t)$$
which is a very general form of equation covering all sorts of non-linear heat and Schroedinger equations, and after slight modification wave equations. This problem is correctly treated in the context of Bochner Spaces. Basically, the idea is now to consider functions $u(x,t)$ as functions of $t \in \mathbb{R}$ with values in the space $L^p(\Omega)$. In this context, the time derivative $\frac{d}{dt}$ defines a maximal monotone operator from one Bochner space to its dual. So now we can take $A$ to be the time derivative and $B$ the (now time-dependent) operator we considered in the elliptic problem to get solutions. Again there is a huge amount of technical detail to be filled in here. And again, this is too much for an MSE answer. Unfortunately, I am not aware of a good English reference for this. On the off-chance that you can read German, I learned this stuff from these lecture notes (chapters 5-7).
