An Introduction to Variational Inequalities and Their Applications
This unabridged republication of the 1980 text, an established classic in the field, is a resource for many important topics in elliptic equations and systems and is the first modern treatment of free boundary problems. Variational inequalities (equilibrium or evolution problems typically with convex constraints) are carefully explained in An Introduction to Variational Inequalities and Their Applications. They are shown to be extremely useful across a wide variety of subjects, ranging from linear programming to free boundary problems in partial differential equations. Exciting new areas like finance and phase transformations along with more historical ones like contact problems have begun to rely on variational inequalities, making this book a necessity once again.
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analytic apply assume bilinear form boundary conditions boundary value problem Brezis Chapter choose coercive vector ﬁeld coincidence set compact conformal mapping consider const converges convex function convex set Corollary deﬁned Deﬁnition denote the solution derivatives differential equations Dirichlet problem dx dy elliptic equations exists a unique f e H f in Q ﬁnd ﬁrst ﬁxed point ﬂuid follows forall free boundary problem given grad Hence Hilbert space hodograph implies Kinderlehrer Legendre transform Lemma Let F Let Q Let u denote linear Lipschitz function locally coercive Math max(u maximum principle meas neighborhood nonlinear nonnegative obstacle problem Problem 1.2 proﬁle proof of Theorem prove satisﬁes satisfy Section sequence smooth boundary Sobolev spaces solution to Problem Stampacchia Stefan problem strongly coercive supersolution Suppose Theorem 2.1 u e H uelK unique solution vanishes variables variational inequality weakly in H