Interior regularity results (local)
The math formulas do not work!!!
Let [math]\Omega[/math] be an open domain and [math] u [/math] a solution of an elliptic equation in [math] \Omega [/math]. The following theorems say that [math] u [/math] satisfies some regularity estimates in the interior of [math] \Omega [/math] (but not necessarily up the the boundary).
[math] div A Du + b Du = 0 [/math] then [math]u[/math] is Holder continuous if [math]A[/math] is just uniformly elliptic and [math]b[/math] is in [math]L^n[/math] (or [math]BMO^{-1}[/math] if [math]div b=0[/math]).
[math]a_ij(x) u_ij + b Du = f[/math] with [math]a_{ij}[/math] unif elliptic (L^infty), b in L^n and f in L^n, then the solution is C^alpha
a_ij(x) u_ij = f with a_ij close enough to the identity and f in L^p, then w is in W^{2,p}. I am not sure how lower order terms affect the result. It should be somewhere in [GT]
a_ij(x) u_ij = f with a_ij close enough to the identity uniformly and f in L^infty, then w is in C^{1,alpha} I am pretty sure you can add first order terms as long as the coefficients are small enough in L^infyt
- Cordes-Nirenberg improved (corollary of work of Caffarelli for nonlinear equations)
[math]a_ij(x) u_ij = f[/math] with a_ij close enough to the identity in a scale invariant Morrey norm in terms of L^n and f in L^n, then w is in C^{1,alpha} (VMO is a particular case of this)
[math]a_ij(x) u_ij = f[/math] with a_ij in C^alpha and f in C^alpha, then u is in C^{2,alpha} I don't remember the hypothesis for the first order terms, but this is in [GT] for sure.
