## The energy method in nonlinear partial differential equationsInstituto de Matemática Pura e Aplicada, Conselho Nacional de Pesquisas, 1969 - Differential equations, Nonlinear - 154 pages |

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assume Au,u Au,v Av,u belongs boundary conditions Browder choose coercive const continuous function continuous linear extension continuous with values continuously embedded converges strongly converges weakly convex subset COROLLARY dE(u defined demicontinuous denotes derivatives differential equation du/dt dual space elliptic elliptic operators embedding theorem energy identity estimate EXERCISE exists u e finite-dimensional follows Hilbert space inequality integral interval Lecture 1.3 Lemma Let f e Lions LP(X monotone operators non-negative norm open set pde's periodic solutions problem Proof of Theorem prove real number reflexive Banach space reflexive spaces satisfies sequence smooth Sobolev Sobolev space Strauss strongly continuous subspace tends to zero term test functions topology u e C(H u e D(A u,v e uniformly unique solution vanishes vector w e D(A weak solution weak topology weakly continuous Wm,p(fi