Why isn't there a general comparison principle for higher order equations?

Question

I am studying elliptic equations. For second order equations (linear and nonlinear), comparison principle has many applications, e.g. to show uniqueness of the (weak) solutions, to construct super- and subsolutions, ... But I read somewhere that there is not a general comparison principle for elliptic equations of higher-order. Why is that?

Answer 1

Lost1 pointed out the reason in a comment: the validity of maximum and comparison principles relies on the PDE controlling the second derivative of the function, because this derivative matters for the classification of stationary points. Lower order terms in the PDE may or may not invalidate the maximum principle: e.g., solutions of $u'' + u =0$ do not satisfy the maximum principle while solutions of $u''-u=0$ do.

For higher order equations we don't get such a control on the second derivative. To give a concrete example: $u(x,y)= 1-x^2-y^2 $ solves the biharmonic equation $\Delta\Delta u=0$ in the unit disk, is zero on the boundary, but positive inside.

This example is notable because some form of maximum principle holds for biharmonic functions in a disk: if $u $ and its inward normal derivative $u_\nu$ are nonpositive on the boundary, then $u\le 0$ inside. Hadamard conjectured that this property holds for other convex domains, which turned out to be totally wrong: it is specific to the disk. See An Hadamard Maximum Principle for Biharmonic Operators by Hedenmalm, Jakobsson, and Shimorin, which approaches the subject by studying the positivity of the Green function for the biLaplacian.