A Local Parabolic Monotonicity Formula on Riemannian Manifolds
نویسندگان
چکیده
In the theory of two-phase free boundary problems, it is well established that regularity of the interface is closely related to asymptotic behavior of solution near the free boundary. In 1984 Alt, Caffarelli and Friedman [2] established a monotonicity formula to describe the interaction of the two pieces of the solution on each side of the free boundary. This formula has been extremely powerful in the regularity theory and it reads as follows: Let u1, u2 be two non-negative continuous functions in B1 (the unit ball in R ) such that ∆ui ≥ 0 (i = 1, 2) are satisfied in distribution. Suppose u1 ·u2 = 0 and u1(0) = u2(0) = 0, then
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