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Extra resources for Partial Differential Equations with Minimal Smoothness and Applications

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H,k". 2. Let l:18U(X, Yi h, k) We de:6ne: = u(x + h, y + k) D8 ( u u(x + h, y - k) - u(x - h, y + k) + u(x - h, y - k). U(X,Yi h,k) X,y - dr:!. 4hk . h,k". If ! E C 2(R2), then both of these agree with the mixed partial derivative. (X"y) for all (x,y). However, these need not always coincide. Our concern in this paper is to whether one can still conclude that u( x, y) = A( x) + B(lI) if either Du == 0 or D·u == O. A theorem due to B8ge1(3) says that if the unsymmetric generalized mixed partial Du( x, y) = 0 for all (x, y) E R 2 26 then u(x, y) = A(x)+B(y).

Since Theorem A holds for any planar domain of finite area we have that (IU) is strictly stronger than the lifetime estimate. The connection between (IU) and conditioned Brownian motion seem to have been first explicitly noticed in Banuelos and Davis [4]. In this paper, which was inspired by the results of [14] and [16], it is proved that even though an arbitrary planar domain of finite area may not be (IU) it is what one may call "one half (IU)" in the following sense: Fix y ED. £ ml' ymxE D .

Fenn. A L Math 14 (1989), pp. 47-55. MANFREDI, J. , On the Fatou Theorem for p-harmonic functions, Comm. in PDE 13(6) (1988), pp. 651-688. LIFETIME AND HEAT KERNEL ESTIMATES IN NON-SMOOTH DOMAINS RODRIGO BANUELOS* o. INTRODUCTION Let D be a domain in Rn, n ~ 2, and let B t be Brownian motion in D with lifetime TD. The transition probabilities for this motion are given by the Dirichlet heat kernel PP(x, y) for t~ in D. If h is a positive harmonic function in D the Doob h-process, the Brownian motion conditioned by h, is determined by the following transition functions: h 1 D Pt (x,y) = h(x{t (x,y)h(y).

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