An Intro to the Study of the Elements of the Diff and Int by A. Harnack

By A. Harnack

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This result will be used for the construction of Lp -strong Feller processes in the next chapter. It will be also used for the construction of the boundary local time. 1 Elliptic Regularity up to the Boundary We partially generalize a regularity result of Morrey to the case of local assumptions on the coefficients and data. Morrey’s result applies for Ω being a relatively compact set and coefficients fulfilling certain integrability conditions and bounds on an open set Γ with Ω ⊂ Γ. In particular, it is assumed that the coefficient matrix A is uniformly elliptic.

Define Ft , 0 ≤ t < ∞, by (Ft )Pν Ft := ν∈P(E Δ ) and F := ν∈P(E Δ ) (F )Pν . 8). Note that the path measures (Px )x∈E Δ naturally extend to F. Define M = (Ω, F, (Ft )t≥0 , (Xt )t≥0 , (Px )x∈E Δ ). The path regularity properties are clear. So it is left to show the (strong) Markov property. 2. 4. Let A ∈ A, 0 ≤ t1 ≤ ... , An ) as in the definition of A. PtΔn −tn−1 1An ) (x). Since PtΔ u is B(E Δ )-measurable for u ∈ Bb (E Δ ), we get that the expression on the right-hand side is B(E Δ )-measurable.

Bogachev, Krylov and R¨ ockner (see [BKR97] and [BKR01]) prove regularity results for measures which solve elliptic (or parabolic) equations in distributional form. Although we do not apply these results here directly, we got many ideas from these articles, in particular the iteration sequence used for the proof in the interior case. We have published the results stated in this chapter in [BG13]. 1. Let Ω ⊂ Rd , d ∈ N and d ≥ 2, be open. Let 2 ≤ p < ∞. dp for p < d, or p < q < ∞ for p ≥ d. Let x ∈ Ω and r > 0 Let p < q ≤ d−p such that Br (x) ⊂ Ω if x ∈ Ω and Br (x) ∩ ∂Ω is C 1 -smooth if x ∈ ∂Ω.

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