Spinodal decomposition

In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška, Peter Lax and Arthur Milgram.

Background

In the modern, functional-analytic approach to the study of partial differential equations, one does not attempt to solve a given partial differential equation directly, but by using the structure of the vector space of possible solutions, e.g. a Sobolev space W^k,p. Abstractly, consider two real normed spaces U and V with their continuous dual spaces U^∗ and V^∗ respectively. In many applications, U is the space of possible solutions; given some partial differential operator Λ : U → V^∗ and a specified element f ∈ V^∗, the objective is to find a u ∈ U such that

$\mathrm{\Lambda }u=f.$

However, in the weak formulation, this equation is only required to hold when "tested" against all other possible elements of V. This "testing" is accomplished by means of a bilinear function B : U × V → R which encodes the differential operator Λ; a weak solution to the problem is to find a u ∈ U such that

$B(u,v)=⟨f,v⟩\text{ for all }v\in V.$

The achievement of Lax and Milgram in their 1954 result was to specify sufficient conditions for this weak formulation to have a unique solution that depends continuously upon the specified datum f ∈ V^∗: it suffices that U = V is a Hilbert space, that B is continuous, and that B is strongly coercive, i.e.

$|B(u,u)|\ge c\Vert u{\Vert }^2$

for some constant c > 0 and all u ∈ U.

For example, in the solution of the Poisson equation on a bounded, open domain Ω ⊂ Rⁿ,

$\{\begin{array}{ll}-\mathrm{\Delta }u(x)=f(x),& x\in \mathrm{\Omega };\\ u(x)=0,& x\in \partial \mathrm{\Omega };\end{array}$

the space U could be taken to be the Sobolev space H₀¹(Ω) with dual H⁻¹(Ω); the former is a subspace of the L^p space V = L²(Ω); the bilinear form B associated to −Δ is the L²(Ω) inner product of the derivatives:

$B(u,v)=\underset{\mathrm{\Omega }}{\int }\mathrm{\nabla }u(x)\cdot \mathrm{\nabla }v(x)\mathrm{d}x.$

Hence, the weak formulation of the Poisson equation, given f ∈ L²(Ω), is to find u_f such that

$\underset{\mathrm{\Omega }}{\int }\mathrm{\nabla }{u}_f(x)\cdot \mathrm{\nabla }v(x)\mathrm{d}x=\underset{\mathrm{\Omega }}{\int }f(x)v(x)\mathrm{d}x\text{ for all }v\in H_0^1(\mathrm{\Omega }).$

Statement of the theorem

In 1971, Babuška provided the following generalization of Lax and Milgram's earlier result, which begins by dispensing with the requirement that U and V be the same space. Let U and V be two real Hilbert spaces and let B : U × V → R be a continuous bilinear functional. Suppose also that B is weakly coercive: for some constant c > 0 and all u ∈ U,

${\sup }_{\Vert v\Vert =1}|B(u,v)|\ge c\Vert u\Vert$

and, for 0 ≠ v ∈ V, for some positive constant k,

${\sup }_{\Vert u\Vert =1}|B(u,v)|\ge k\Vert v\Vert$

Then, for all f ∈ V^∗, there exists a unique solution u = u_f ∈ U to the weak problem

$B(u_f,v)=⟨f,v⟩\text{ for all }v\in V.$

Moreover, the solution depends continuously on the given datum:

$\Vert u_f\Vert \le \frac{1}{c}\Vert f\Vert .$

See also

• Lions–Lax–Milgram theorem

References

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