An introduction to G-convergence by Gianni Dal Maso

An introduction to G-convergence by Gianni Dal Maso

By Gianni Dal Maso

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Extra resources for An introduction to G-convergence

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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 , .

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