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The Multi-Index Notation
Published in M. W. Wong, Partial Differential Equations, 2022
We begin with the standard multi-index notation in the modern theory of partial differential equations. Let x=(x1,x2,…,xn) and y=(y1,y2,…,yn) be points in the Euclidean space ℝn. Then the dot product x·y of x and y is defined by x·y=∑j=1nxjyj
Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
In this book we shall adopt the multi-index notation for the partial derivatives of higher order of the following form: ADαf=∂|α|f∂xα $$ A\boldsymbol{D}^{\boldsymbol{\alpha }}\boldsymbol{f}= \frac{\partial ^{|\boldsymbol{\alpha }|}\boldsymbol{f}}{\partial \boldsymbol{x}^{\boldsymbol{\alpha }}} $$
Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Exercise 9.4. Assume xj ≥ 0 and aj > 0 for 1 ≤ j ≤ n. Find the maximum of ∏j=1nxjaj given that ∑xj=1. Comment: In two dimensions, the maximum of xa yb on x + y = 1 is aabb(a+b)a+b. The answer is αα|α||α| in multi-index notation, which we do not discuss. See page 4 of [T], page 227 of [F1], page 64 of [D1], or many other sources for multi-index notation.
Constrained variational problems governed by second-order Lagrangians
Published in Applicable Analysis, 2020
According to Lagrange theory, an extreme point of (P2) is found among the extreme points of (P2). In order to formulate the necessary conditions of optimality, we shall introduce the Saunders's multi-index notation [12]. A multi-index is an m-tuple I of natural numbers. The components of I are denoted , where α is an ordinary index, . The multi-index is defined by for . The addition and the substraction of the multi-indexes are defined componentwise (although the result of a substraction might not be a multi-index): . The length of a multi-index is , and its factorial is . The number of distinct indices represented by , , is Particular cases.
Continuity of the solution set to a linear PDE with constant coefficients
Published in International Journal of Control, 2022
A multi-index is an r-tuple of nonnegative integers, i.e. an element of . If k is a nonnegative integer, we let denote the set of all multi-indices of order less than or equal to k. (The order of is defined to be .) Write and for every multi-index .
Numerical solution of nonlinear differential boundary value problems using adaptive non-overlapping domain decomposition method
Published in Applicable Analysis, 2022
Nahed Naceur, Moez Khenissi, Jean R. Roche
Fix , and let k a non-negative integer. The Sobolev space is the set of all the functions f defined on Ω such that for every multi-index with , the mixed partial derivative exists in the weak sense and belongs to .