Babuška–Lax–Milgram theorem

Babuška–Lax–Milgram theorem 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.

Contents 1 Background 2 Statement of the theorem 3 See also 4 References 5 External links 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 {displaystyle 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 {displaystyle B(u,v)=langle f,vrangle {mbox{ for all }}vin 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.

{displaystyle |B(u,u)|geq c|u|^{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, {displaystyle {begin{cases}-Delta u(x)=f(x),&xin Omega ;\u(x)=0,&xin partial Omega ;end{cases}}} 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 Lp space V = L2(Ω); the bilinear form B associated to −Δ is the L2(Ω) inner product of the derivatives: {displaystyle B(u,v)=int _{Omega }nabla u(x)cdot nabla v(x),mathrm {d} x.} Hence, the weak formulation of the Poisson equation, given f ∈ L2(Ω), is to find uf such that {displaystyle int _{Omega }nabla u_{f}(x)cdot nabla v(x),mathrm {d} x=int _{Omega }f(x)v(x),mathrm {d} x{mbox{ for all }}vin H_{0}^{1}(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, {displaystyle sup _{|v|=1}|B(u,v)|geq c|u|} and, for all 0 ≠ v ∈ V, {displaystyle sup _{|u|=1}|B(u,v)|>0} Then, for all f ∈ V∗, there exists a unique solution u = uf ∈ U to the weak problem {displaystyle B(u_{f},v)=langle f,vrangle {mbox{ for all }}vin V.} Moreover, the solution depends continuously on the given data: {displaystyle |u_{f}|leq {frac {1}{c}}|f|.} See also Lions–Lax–Milgram theorem References Babuška, Ivo (1970–1971). "Error-bounds for finite element method". Numerische Mathematik. 16 (4): 322–333. doi:10.1007/BF02165003. hdl:10338.dmlcz/103498. ISSN 0029-599X. MR 0288971. Zbl 0214.42001. Lax, Peter D.; Milgram, Arthur N. (1954), "Parabolic equations", Contributions to the theory of partial differential equations, Annals of Mathematics Studies, vol. 33, Princeton, N. J.: Princeton University Press, pp. 167–190, MR 0067317, Zbl 0058.08703 – via De Gruyter External links Roşca, Ioan (2001) [1994], "Babuška–Lax–Milgram theorem", Encyclopedia of Mathematics, EMS Press Categories: Theorems in analysisPartial differential equations