Regularity of Elliptic Equations

PDEsFunctionalAnalysis The fact that the differential operators of Elliptic PDEs are Fredholm Operators guarantees that solutions of are as regular as (and ). Using integration by parts, it is easy to see that in the case where there is an equivalence of the norms of and the gradient of . Heuristically, this shows that weak solutions have two more derivatives than . There are two types of regularity results, some for only the interior of and those that include the boundary:

1. Interior regularity: Given then with for any . In particular, the result holds for .

2. Boundary regularity: Given with then with and when is the unique weak solution

   In particular, the result holds for .