# An introduction to mathematical probability by Julian Lowell Coolidge

By Julian Lowell Coolidge

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The maximum principle for the discrete Dirichlet problem is stated in the following way. 24). Then from the condition Λ z(x) ≥ 0 x ∈ Dh , z(x) ≡ const, x ∈ Dh , it follows that z(x) can not take its largest positive value on the nodes of Dh . The maximum principle is often applied in the following form of a comparison lemma. 24), and for the functions v(x) and w(x), x ∈ Dh , let the condition |Λ v(x)| ≤ −Λ w(x) x ∈ Dh , |v(x)| ≤ w(x), x ∈ Γh , be satisfied. Then |v(x)| ≤ w(x), x ∈ Dh . Here w(x), x ∈ Dh is a majorant (barrier) function for v(x), x ∈ Dh .

61c) h∗ > 0 is an arbitrary number. 59) holds and locally almost canonical in x2 , . . 59), for an arbitrary point x∗ ∈ D, one can find a neighborhood Q(x∗ ) and functions ρ∗s (x) and ρ∗s (x), © 2009 by Taylor & Francis Group, LLC Boundary value problems in smooth domains 47 for s = 2, . . 61) hold . 61) involve only the coefficients ask (x), for s, k = 2, . . , the operator involving differentiation in the variables x2 , . .

We denote the right-hand side by f (1) (x), where f (1) ∈ C l−1+α (D)), and we have the inequality |f (1) (x)| ≤ M exp −mε−1 r(x, Γ) , x ∈ D. Taking into account the inequality ∂ϕ(x) ∂U (x) − ≤ M, ∂xj ∂xj V (1) (x) = x ∈ Γ, one can establish the estimate ∂k k k j−1 j+1 ∂x1k1 . . ∂xj−1 ∂xj ∂xj+1 . . ∂xknn V (x) ≤ ≤ M ε−k1 + ε1−k exp −m ε−1 r(x, Γ) , 0 ≤ k ≤ k 0 , j > 1, k = k1 + . . + kj−1 + 1 + kj+1 + . . + kn . In a similar way, we obtain the estimate ∂k V (x) ≤ M ε−k1 + ε2−k exp −mε−1 r(x, Γ) , ∂xk11 .