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Extra info for A new table of seven-place logarithms
It follows that, since V is complete under (•, •)» it is also complete under a(-, •)• The next point is the following: If I is a bounded linear functional on V, this means that there exists a constant M > 0 such that In this case, since 44 Chapter 2. The weak form of a BVP the inequality also holds. Therefore, if t is bounded with respect to the original norm on V, it is also bounded with respect to the equivalent energy norm on V. The converse is also true: If «(-,-) is bounded and V-elliptic, then I e V* if t is bounded with respect to the energy norm (see Exercise 10).
26) can hold for all v e Hl(£l) only if du/dn is zero on d£l (the reader should bear in mind that K is strictly positive). Since Dirichlet conditions must be explicitly imposed in the weak form, while Neumann conditions are implied even though not explicitly imposed, Dirichlet conditions are often called essential boundary conditions, while Neumann conditions are called natural boundary conditions. 3. 2 31 Mixed boundary conditions A BVP can have both Dirichlet and Neumann boundary conditions, with the two different conditions applied to different parts of the boundary.
15), since these reasons are quite instructive. 13a) models a mechanical system in which u is the displacement and / is an external body force, the total potential energy of the system is 22 Chapter 2. The weak form of a BVP where £Q is some constant. The state of equilibrium of the system corresponds to the displacement u that minimizes the potential energy. 13)). 13b) are admissible. If u and w are two admissible displacements (that is, if u, w e C2D(Q)) and v — w — u (so that w = u + u), then v is also in C2D(£t).