By Gianni Dal Maso
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Extra resources for An introduction to G-convergence
Un , any function of the form c1 u 1 + c2 u 2 + · · · + cn u n , where c1 , c2 , . . , cn are constants, is called a linear combination of u1 , u2 , . . , un . The following theorem follows immediately from the result of Exercise 12 of the previous section. 1 If u1 , u2 , . . , un are solutions of the linear, homogeneous PDE L[u] = 0, then so is any linear combination of u1 , u2 , . . , un . ) PROOF The fact that u1 , u2 , . . , un are solutions gives us L[u1 ] = L[u2 ] = · · · = L[un ] = 0.
In general, the onedimensional equation of continuity/conservation law in any similar situation is ρt + Φx = 0, § where ρ is the concentration and Φ is the ﬂux of the “substance” involved. ”) Examples abound—the equation of continuity shows up whenever we have something which is diﬀusing or ﬂowing. § In higher dimensions we have ρt + ∇ · Φ = ρt + div Φ = 0. An Introduction to Partial Diﬀerential Equations with MATLAB R 50 Fluid ﬂow Suppose we have a liquid in one-dimensional ﬂow through a pipe with constant cross sectional area A.
Then, for any linear combination c1 u1 + c2 u2 + · · · + cn un , L[c1 u1 + c2 u2 + · · · + cn un ] = c1 L[u1 ] + c2 L[u2 ] + · · · + cn L[un ] = c1 · 0 + c2 · 0 + · · · + cn · 0 = 0. Now, in the theory of ODEs, for an nth -order linear, homogeneous equation, we need only ﬁnd n linearly independent solutions. Then, the general solution consists of all possible (ﬁnite) linear combinations of these solutions. However, life is much more complicated in the realm of PDEs. Often, we will need to ﬁnd inﬁnitely many solutions, u1 , u2 , .