By Gazzini M., Serra E.

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**Additional info for The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues**

**Sample text**

1 u and all its (n − 1)th ﬁrst derivatives converge strongly to u and its correspondent derivatives. More precisely, we have for x ∈ Ωi | (u − u) (i) 1 3 h exp (h A m ); 0 ≤ i ≤ n − 1 (35) 3! stands for a matrix norm and A is a constant (x)| ≤ C where h = xi+1 − xi , matrix to be identiﬁed. m 44 D. 1. Set w = u − u and set X to be a nth dimensional vector such that X(1) = X1 = w and X(i) = Xi = wi−1 ; 2 ≤ i ≤ n. We notice in ﬁrst place that ak (x) − ak (x) = (x − xi )2 a ˜k (x); 2! 0≤k≤n (x − xi )2 ˜ f (x) 2!

The condition (15) normalizes the Fd and ensures that the order of series AiN increases as i and d increase in value. There are thus 3(2 + d) undetermined coeﬃcients bij in the expression (15). The requirement (16) reduces the problem to a system of N linear equations for the unknown coeﬃcients of Fd . The entries of the underlying matrix depend only on the N given coeﬃcients ai . Henceforth, we shall take N = 3 (2 + d ), (17) Modelling the Thermal Operation in a Catalytic Converter 57 so that the number of equations equals the number of unknowns.

3. 1. For any initial condition f0 ∈ X , f0 ≥ 0, the initial value problem (19)–(65) for the fully discrete kinetic equation has a unique solution f ∈ C([0, ∞), X ), and f (t) ≥ 0. The result is valid for both nearest neighbor and (next)i -nearest neighbor interactions. Here is a rich source of open problems. Fully discrete models are systems of ﬁrst order ordinary diﬀerential equations. This presents the possibility of numerical computation using elementary codes. Almost nothing is known in this venue.