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I was speaking with a friend the other day, and he asked me why so many elliptic PDE references (e.g. Gilbarg-Trudinger, Han-Lin) restrict themselves to PDEs of second order.

Annoyingly, I realized I could not come up with a satisfactory answer. So here's the question:

What goes wrong when one try to generalize the theory of elliptic PDEs of second order into elliptic PDEs of higher-order?

Are the problems technical (e.g. there is an elliptic regularity theorem, but its proof is horrifying) or conceptual (e.g. one cannot hope for regularity theory except for some trivial cases)?

NG_
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1 Answers1

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The main issues are:

  1. The maximum principle fails for higher order elliptic equations.

  2. Somewhat by extension, the De Giorgi-Nash-Moser theory also fails.

The failure of the maximum principle can already be shown using simple examples for the biharmonic equation ($\Delta^2 u = 0$). At a technical level one also sees the proof breaks down since you cannot classify a critical point $x_0$ that satisfies $D^4u(x_0)=0$. This breaks a lot of techniques that work for the (scalar) second order case, including viscosity methods which relies on the concept of subsolutions.

A counterexample of the failure of regularity is due to Maz'ya from 1968, and similar counterexamples for elliptic systems were found by De Giorgi around the same time. At a technical level, one can't use test functions of the type $\max\{0,u-k\}$ for $k \in \mathbb R$ that typically feature in the proofs.

Despite the above difficulties however, some aspects of the regularity theory do still apply to general elliptic equations. For instance:

  1. The difference-quotient methods that work in the $H^k$ spaces, found e.g. in Evans' text or Section 8.3 of Gilbarg & Trudinger readily generalise to the higher order setting.

  2. Linear estimates in the Hölder scales (Schauder estimates, Ch. 6 of G&T) and Lebesgue scales (Calderón-Zygmund estimates, Ch. 9 of G&T) also hold for higher order equations. However, the potential methods used in Gilbarg & Trudinger aren't so easy to generalise, as one needs to construct more complicated kernels. While this can be done, there are other approaches like the approach using Morrey-Campanato spaces to prove Schauder estimates, which can be found in Chapter 3 of Han & Lin.

  3. For quasilinear equations of higher order, pertubative methods can be used to obtain partial regularity; namely that the solution $u$ is smooth away from a "small" singular set $\Sigma \subset \Omega$.

These are usually studied in the context of elliptic systems; the case of higher order equations is usually analogous (I think a formal reduction is also possible in some cases). Here a classic reference is:

Giaquinta, Mariano, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, 105. Princeton, New Jersey: Princeton University Press. VII, 296; 21.50 (1983). ZBL0516.49003.

ktoi
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