What is elliptic bootstrapping?

While reading about elliptic differential operators, I have seen the phrase elliptic bootstrapping in several places, but none of them explain exactly what it means. I know it has something to do with the regularity of elliptic differential equations.

Does elliptic bootstrapping refer to a collection of theorems, a method/technique, or just a vague concept?

Or, more succinctly:

What is elliptic bootstrapping?

I am also happy to be directed to references which discuss elliptic bootstrapping in some detail.

What is elliptic bootstrapping?

It is an inductive argument combining "elliptic regularity" lifting (and/or Sobolev embedding) and "bootstrap" meaning you are exploiting this lifting (or embedding) many times.

A simple example: consider the equation $$ -\Delta u + \lambda u = f,\tag{1} $$ in a smooth domain, with smooth data $f$. A weak solution $u$ of (1) has $H^1$-regularity. Now for $(-\Delta + \lambda I)$ is an elliptic operator, elliptic regularity says the full second derivative's $L^2$-norm can be bounded by only the $\Delta u$'s $L^2$-norm (plus $u$'s $L^2$-norm): $$ \|u\|_{H^2(\Omega)} \leq C\left(\|\Delta u\|_{L^2(\Omega)} + \|u\|_{L^2(\Omega)}\right). $$ This says $u$ will get 2 regularity lifting from whatever $\Delta u$ is, in this case it is $\lambda u-f$. Now that $\lambda u - f$ has a higher regularity, we can say: $$ \|u\|_{H^4(\Omega)} \leq C\left(\|\Delta u\|_{H^2(\Omega)} + \|u\|_{L^2(\Omega)}\right). $$ Hence $$ \|u\|_{H^{s+2}(\Omega)} \leq C\left(\|\Delta u\|_{H^s(\Omega)} + \|u\|_{L^2(\Omega)}\right). $$ This tells us we can exploit the elliptic regularity repeatedly to raise the differentiability of $u$. This is a "bootstrapping argument".

Now for integrability: first $u \in H^1= W^{1,2}$, if the dimension $n$ is greater than 2, then by Sobolev embedding $$ W^{1,2} \hookrightarrow L^{2^*}, \quad \text{where } 2^* = \frac{2n}{n-2} >2. $$ Hence we have $\Delta u \in L^{2^*}$ implying $ u\in W^{1,2^*}$, and we just raise the integrability of $u$ a little bit. Repeating the embedding we can raise the integrability of $u$ again if $2^* < n$. This is another "bootstrapping argument".

Above example is just an artificial example. In real PDE research, I have seen in seminar people always uses this argument to deal with the regularity for nonlinear elliptic problem, say $$ -\Delta u = f(x,u,\nabla u), $$ when $f$ satisfies certain condition.

Ngô Quốc Anh's blog has several more practical examples for the bootstrapping argument.

Caffarelli used the bootstrapping argument a lot in his papers.

See the slide on elliptic bootstrapping here for a succinct answer and brief discussion. Basically, knowing that $\Delta u \in L_k^2$ allows you to conclude $u \in L_{k+2}^2$. (Here they are using subscripts to denote the regularity of $u$.) The harmonic function $u$ "lifts itself up by its own bootstraps" in the sense that knowing $u$ is harmonic with a certain known regularity implies $u$ has a greater regularity because $u$ is harmonic.

That's the rough idea. Here you wind up with $u$ being smooth because $\Delta u = 0$ where $\Delta$ is an elliptic operator and $0$ is smooth. More generally, you can get the Elliptic Regularity Theorem, which says that if $E$ is an elliptic operator of order $2k$ and $f$ an $L^2$ function, a weak solution $u$ to $Eu = f$ will have at least $2k$ weak derivatives that are square-integrable.

For a more rigorous treatment including a proof of the Elliptic Regularity Theorem, see Folland's Real Analysis, in his chapter on Sobolev spaces and the Elliptic Regularity Theorem.