Elliptic Operator - Elliptic Regularity Theorem

Elliptic Regularity Theorem

Let L be an elliptic operator of order 2k with coefficients having 2k continuous derivatives. The Dirichlet problem for L is to find a function u, given a function f and some appropriate boundary values, such that Lu = f and such that u has the appropriate boundary values and normal derivatives. The existence theory for elliptic operators, using Gårding's inequality and the Lax–Milgram lemma, only guarantees that a weak solution u exists in the Sobolev space Hk.

This situation is ultimately unsatisfactory, as the weak solution u might not have enough derivatives for the expression Lu to even make sense.

The elliptic regularity theorem guarantees that, provided f is square-integrable, u will in fact have 2k square-integrable weak derivatives. In particular, if f is infinitely-often differentiable, then so is u.

Any differential operator exhibiting this property is called a hypoelliptic operator; thus, every elliptic operator is hypoelliptic. The property also means that every fundamental solution of an elliptic operator is infinitely differentiable in any neighborhood not containing 0.

As an application, suppose a function satisfies the Cauchy-Riemann equations. Since the Cauchy-Riemann equations form an elliptic operator, it follows that is smooth.