12 Energy Methods, Weak Solutions and the Dirichlet Principle

The explicit formulas of the previous sections need special geometry: lines, half-spaces, balls, rectangles. For a general region, and for variable coefficients, existence must be proved by other means, and the most powerful are variational. A steady state minimises energy; so look for the minimiser of the energy, and show that it solves the equation. This is the Dirichlet principle. Riemann used it freely in complex analysis; Weierstrass showed in 1870 that the minimum need not exist; and Hilbert rescued it in 1900. The rescue required a new notion of solution, the weak solution, and new function spaces, and it led directly to the finite element method used in engineering today.

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