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Tham khảo tài liệu 'numerical_methods for nonlinear variable problems episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4 Application to the Solution of Elliptic Problems for Partial Differential Operators 347 then 4.29 has a unique solution in 1 L2 which is also the unique solution of the variational problem Find u e FFifi such that AVw Vi dx aouv dx I fv dx I gv dr Vue H fi . Jo -fl Jil Jr 4.95 Proof. It suffices to prove that the bilinear form occurring in 4.95 is H1 Q -elliptic. This follows directly from Lemma 4.1 and from the fact that if V constant c then an x dx 0 la and 0 x t 2 dx c2 70 v dx 0 la la imply c 0 i.e. V 0. Proposition 4.7. We consider the Neumann problem 4.29 with Q bounded and A still obeying 4.47 if we suppose that a0 0 then 4.29 has a unique solution u in Hfif R20 if and only if fdx i g dr 0 4.96 Jn Jr u is also the unique solution in W iR of the variational problem Find ue H Q such that f AVm Vf dx f fv dx f gv dr V V e 4.97 In In Jr Proof. For clarity we divide the proof into several steps. Step 1. Suppose that aa 0 if u is a solution of 4.29 and if c is a constant it is clear from V u C Vu that I c is also a solution of 4.29 . If u is a solution of 4.29 we can show as in Sec. 4.2.2 that 4.97 holds taking V 1 in 4.97 we obtain 4.96 . Step 2. Consider the bilinear form over H Q X H fO defined by ã y w i Vn Vw dx I I V dx V f w dx j V V w e H ii 4.98 la Jn ln 20 This means that u is determined in O only to within an arbitrary constant. 348 App. I A Brief Introduction to Linear Variational Problems the bilinear form ã - is clearly continuous and from Lemma 4.1 it is il -elliptic it suffices to observe that if V constant c then 0 i f V dxi c2 meas íl 2 c V 0 . Wn From these properties V - ã v v 1 2 defines over a norm equivalent to the usual H1 Q -norm defined by 4.40 . Step 3. We now consider the space Iq VIV e f v x dx 0 I nJ Vi being the kernel of the linear continuous functional V - y x dx Ja is a closed subspace of H ii . If we suppose that has been equipped with the scalar product defined by ã see 4.98 it follows from Step 2 and from the definition of .
