Editing Elliptic operator (section)

==Elliptic regularity theorems==

Let ''L'' be an elliptic operator of order 2''k'' with coefficients having 2''k'' 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]], [[Lax–Milgram lemma]] and [[Fredholm alternative]], states the sufficient condition for a [[weak solution]] ''u'' to exist in the [[Sobolev space]] ''H''<sup>''k''</sup>.

For example, for a Second-order Elliptic operator as in '''Example 2''', 
* There is a number ''γ>0'' such that for each ''μ>γ'', each <math>f\in L^2(U)</math>, there exists a unique solution <math>u\in H_{0}^{1}(U)</math> of the boundary value problem<br /><math>Lu+\mu u=f \text{ in }U, u=0\text{ on }\partial  U</math>, which is based on [[Lax-Milgram lemma]]. 
* Either (a) for any <math>f\in L^2(U)</math>, <math>Lu=f \text{ in }U, u=0\text{ on }\partial  U</math> (1) has a unique solution, or (b)<math>Lu=0 \text{ in }U, u=0\text{ on }\partial  U</math> has a solution <math>u\not\equiv 0</math>, which is based on the property of [[compact operator]]s and [[Fredholm alternative]].

This situation is ultimately unsatisfactory, as the weak solution ''u'' might not have enough derivatives for the expression ''Lu'' to be well-defined in the classical 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''.

For ''L'' as in '''Example 2''', 
* '''Interior regularity''': If ''m'' is a natural number, <math>a^{ij},b^{j},c \in C^{m+1}(U), f\in H^{m}(U)</math> (2) , <math>u\in H_{0}^{1}(U)</math> is a weak solution to (1), then for any open set ''V'' in ''U'' with compact closure, <math>\|u\|_{H^{m+2}(V)}\le C(\|f\|_{H^{m}(U)}+\|u\|_{L^2(U)})</math>(3), where ''C'' depends on ''U, V, L, m'', per se <math>u\in H_{loc}^{m+2}(U)</math>, which also holds if ''m'' is infinity by [[Sobolev inequality|Sobolev embedding theorem]].
* '''Boundary regularity''': (2) together with the assumption that <math>\partial U</math> is <math>C^{m+2}</math> indicates that (3) still holds after replacing ''V'' with ''U,'' i.e. <math>u\in H^{m+2}(U)</math>, which also holds if ''m'' is infinity.

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 <math>f</math> satisfies the [[Cauchy–Riemann equations]]. Since the Cauchy-Riemann equations form an elliptic operator, it follows that <math>f</math> is smooth.