| There are many problems in physics and engineering whose models can be expressedas partial di?erential equations with appropriate boundary conditions and initial conditions.This thesis summarizes the development and applications of the Lax-Milgram theorem.The Lax-Milgram theorem extends the inner product in the Riesz representation theoremto a continuous conjugate bilinear form for which the following coercive condition holds:a(u, u)≥δu 2, ?u∈H, where H is Hilbert space. This"little"extension has very impor-tant applications in the study of well-posedness of partial di?erential equations( PDE's).Lax-Milgram Theorem Let a be a continuous coercive bilinear form on a Hilbertspace H and T be a continuous linear form on H. Then there is a unique vector u in H suchthatThere are many problems in physics and engineering whose models can be expressedas partial di?erential equations with appropriate boundary conditions and initial conditions.This thesis summarizes the development and applications of the Lax-Milgram theorem.The Lax-Milgram theorem extends the inner product in the Riesz representation theoremto a continuous conjugate bilinear form for which the following coercive condition holds:a(u, u)≥δu 2, ?u∈H, where H is Hilbert space. This"little"extension has very impor-tant applications in the study of well-posedness of partial di?erential equations( PDE's).Lax-Milgram Theorem Let a be a continuous coercive bilinear form on a Hilbertspace H and T be a continuous linear form on H. Then there is a unique vector u in H suchthatApplying this theorem, we obtain that the following Dirichlet problemhas a week solution.Changing the bilinear form on Hilbert space H in the Lax-Milgram theorem to a bilinearform on H×V, where V ? H is the dense subspace of H , a linear version of the Lax-Milgram theorem can be obtained . This linear version is used to prove the existence of weeksolutions of boundary value problem for parabolic partial di?erential equations.The Lax-Milgram theorem requests that the given space is the Hilbert space. In prac-tical applications many cases are not the Hilbert spaces, while many problems request thatthe given space is the re?exive Banach space. To overcome this limitation in application,it is necessary to extend the Lax-Milgram theorem to re?exive Banach Spaces. Let X be are?exive Banach space over R and {Xn}n∈N be an increasing sequence of closed subspaces of X and Suppose thatis a real-valued function on X×V for which the following conditions hold:(1) An = A is a bounded bilinear form, for all n∈N;(2) A(·, v) is a bounded linear functional on X for all v∈V;(3) A is coercive on V, that is, there exists a c > 0 such thatwe can get a generalization of the Lax-Milgram theorem by applying the Lagrange multipliertheorem, that is, for each bounded linear functional v? on V, there exists an x∈X such thatA linear extension of the generalization of the Lax-Milgram can be obtained by sub-stituting the coercivity condition as meintioned above for a more general inf-sup condition:there exists a c > 0 such that for allλ∈Λ,where X is a re?exive Banach space , Y is a Banach space,Λis a directed set, {Xλ}λ∈Λis afamily of closed subspaces of X , {Yλ}λ∈Λis an upwards directed family of closed subspaceof Y and V =λ∈ΛYλ. Then , for each bounded linear functional v? on V, there exists anx∈X such thatsuppose that function A satisfies the following conditions:(1) A is a type M respect to V;(2)(3) for allλ∈Λand x∈Xλ, where Aλis the restriction of A on Xλ×Xλ; (4) the operator Tλ: Xλ→Xλ, defined by Tλx, y = Aλ(x, y) for all x, y∈Xλ, ismonotone and hemicontinuous for allλ∈Λ.we get a nonlinear extension of the generalization of the Lax-Milgram: that is , foreach v∈V , there exists an x∈X such thatThis extension and the above linear extension both have important applications to singulardi erential equations.In application to partial di erential equations, a(u, v), T(v) in the above Lax-Milgramtheorems are usually denoted by are derivatives of some f and g.Let E be a re exive Banach space. Let K be a closed subspace of E and f be a classA(E). Denote VE, f by V. Assume that(1) f(K) is bounded from below and coercive on V∩K; that means f(u) tends to∞when u is in V∩K and u converges to∞;(2) f is weakly lower semi-continuous on V∩K;(3) f is strongly V-di erentiable at any point x∈V∩K.Let g be a real function on E having the following properties.(4) If {un} is a sequence in V∩K and weakly converges to u∈E, then(5) g is V-di erentiable on V∩K and Dg(u)(u) 0 for every u∈(V∩K) \ {0}.(6) There exists a real number r such that M = {v∈V∩K : g(v)≥r} is not empty anddoes not contain 0.(7) There are two positive real functionsψandθon (0,∞) such that (θ1ψ)((0,∞)) =(0,∞), Df(tx)v≤ψ(t)Df(x)v and Dg(tx)v≥θ(t)Dg(x)v for any x∈M, v∈V∩K andt > 0.Applying the Lagrange multiplier theorem, a nonlinear version of the Lax-Milgram theorem can be obtained, that is , there existγ∈{0, 1} and uˉin K \ {0} such thatActually there are much more complex physical processes expressed by variational in-equalities, which are important nonlinear problems. Noting that the condition (7): thehomogeneous inequality can not directly be applied to the above nonlinear version of theLax-Milgram theorem , only when the Lagrange multiplier be 0 or 1. Then, the followingtheorem can removeλ, and we establish the form of inequality, which is an extension ofthe above nonlinear version of the Lax-Milgram theorem.a nonlinear version of Stampacchia theorem Let E be a re?exive Banach space.Let K be a closed subspace of E and f be a class A(E). Denote VE, f by V. Suppose thatconditions in the above theorem (1), (2), (3), (4), (5), (6) hold, then(i) there exists u∈M such thatwhere(ii) Assume thatλis positive and we have the additional following assumption:(7) there are two positive real functionsψandθon (0,∞) such that (θ?1ψ)((0,∞)) =Then ther such thatThe above nonlinear versions of the Lax-Milgram and Stampacchia Theorems have manyapplications in p-Laplace elliptic problems. |
