My advisor in grad school used to say
“Nonlinear” is not a hypothesis but the lack of a hypothesis.
To say something positive about nonlinear equations, you have to replace linearity with some specific property. You want to partially remove the restriction of linearity without letting just anything in.
In partial differential equations, one pattern of nonlinearity is to replace linear with monotone.
We say a function on the real line is monotone if x ≥ y implies f(x) ≥ f(y). Strictly speaking this is the definition of monotone non-decreasing, but in this context the “non-decreasing” qualifier is dropped. Now suppose f is a linear transformation on Rn. What could it mean for f to be monotone when statements like x > y don’t make sense? We could rewrite the one dimensional definition as saying
(f(x) – f(y))(x – y) ≥ 0
for all x and y. This form generalizes to linear transformations if we interpret the multiplication above as inner product. More generally we say that an operator A from a Banach space V to its dual V* is monotone if
(A(x) – A(y))(x – y) ≥ 0
where now instead of an inner product we have more generally the action of the linear functional A(x) – A(y) on the vector x – y. If the space V is a Hilbert space, then this is just an inner product, but in general it doesn’t have to be.
In applications to PDEs, the operator A would represent an operator between function spaces so that the PDE has the form Au = f where u is a solution in V and the right hand side f is in V*. The operator A represents some weak form of the PDE and the space V is some sort of Sobolev space with the necessary boundary value assumptions baked in.
Monotonicity alone isn’t enough to prove existence or uniqueness. We need a few other properties.
We say the operator A from V to V* is coercive if Au(u) / ||u|| goes to infinity as ||u|| goes to infinity.
We say A is Type M if whenever un converges weakly to u, Aun converges weakly to f, and the lim sup of Aun ≤ f(u) all apply, then Au = f.
Here’s an example of the kind of theorems you can prove with these definitions.
If A is Type M, bounded, and coercive on a separable reflexive Banach space V to its dual V*, then A is surjective.
In application, the Banach space in the theorem is some sort of Sobolev space, functions in some Lp space whose generalized derivatives are in the same space, along with some boundary conditions. (You might wonder how boundary conditions can be defined for functions in such a space. They can’t directly, but they can indirectly via trace operators. Generalized boundary values for generalized functions. It’s all very generalized!)
Saying that A is surjective means the equation Au = f has a solution for any f in V*. So we reduce the problem of showing that the equation has a solution to verifying that A is Type M, bounded, and coercive. Type M is a form of continuity; bounded and coercive follow from a priori estimates.
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