Existence and uniqueness theorems for variational problems

This article summarizes a list of existence and uniqueness theorems for variational problems from (Monk 2003), which are organized from simple to complex.

Theorem 1 (Riesz representation) Let (mathcal{X}) be a Hilbert space. For each (g in mathcal{X}') there exists a unique (uinmathcal{X}) such that [label{eq:riesz-representation} (u, v)_{mathcal{X}} = g(v) quad forall vinmathcal{X},] and ( orm{u}_{mathcal{X}}= orm{g}_{mathcal{X}'}).

N.B. This is the simplest variational formulation where the left hand side is just an inner product.

Theorem 2 (Lax-Milgram) Let (a(cdot,cdot):mathcal{X} imesmathcal{X} ightarrowmathbb{C}) be a bounded and coercive sesquilinear form, i.e. for boundedness [label{eq:boundedness-in-lax-milgram} abs{a(u,phi)} leq C orm{u}_{mathcal{X}} orm{phi}_{mathcal{X}} quad forall uinmathcal{X},phiinmathcal{X}] and for coercivity, there is exists a constant (alpha>0) independent of (uinmathcal{X}) such that [label{eq:coercivity} abs{a(u,u)} geq alpha orm{u}_{mathcal{X}}^2 quad forall uinmathcal{X}.] Then for all (finmathcal{X}') there exists a unique solution (uinmathcal{X}) to the following variational problem [label{eq:variational-problem} a(u,phi)=f(phi) quad forall phiinmathcal{X}.] Furthermore, the norm of (u) is controlled by the norm of (f) as [label{eq:lax-milgram-error-estimate} orm{u}_{mathcal{X}} leq frac{C}{alpha} orm{f}_{mathcal{X}'}.]

Theorem 3 (Generalized Lax-Milgram) Let (mathcal{X}) and (mathcal{Y}) be Hilbert spaces. Let (a(cdot,cdot):mathcal{X} imesmathcal{Y} ightarrowmathbb{C}) be a bounded sesquilinear form which satisfies the following properties:

1. For all (uinmathcal{X}) and (vinmathcal{Y}), [label{eq:boundedness-in-general-lax-milgram} abs{a(u,v)} leq C orm{u}_{mathcal{X}} orm{v}_{mathcal{Y}}.]

2. There is a constant (alpha) such that [label{eq:inf-sup-condition} inf_{uinmathcal{X}, orm{u}_{mathcal{X}}=1} sup_{vinmathcal{Y}, orm{v}_{mathcal{Y}} leq 1} abs{a(u,v)} geq alpha > 0.]

3. For every (vinmathcal{Y}), (v eq 0) [sup_{uinmathcal{X}} abs{a(u,v)} > 0.]

If (ginmathcal{Y}'), there exists a unique solution (uinmathcal{X}) to the following variational problem [label{eq:general-variational-problem} a(u,phi) = g(phi) quad forall phiinmathcal{Y}.] Furthermore, the norm of (u) is controlled by the norm of (g) as [label{eq:general-lax-milgram-error-estimate} orm{u}_{mathcal{X}} leq frac{C}{alpha} orm{g}_{mathcal{Y}'}.]

Theorem 4 (Existence and uniqueness theorem for mixed variational problem) Let (mathcal{X}) and (S) be Hilbert spaces. Let (a(cdot,cdot): mathcal{X} imesmathcal{X} ightarrowmathbb{C}) and (b(cdot,cdot): mathcal{X} imesmathcal{S} ightarrowmathbb{C}) be bounded sesquilinear forms: [label{eq:boundedness-in-mixed-variational-problem} egin{aligned} abs{a(u,phi)} & leq C orm{u}_{mathcal{X}} orm{phi}_{mathcal{X}} quad forall u,phiinmathcal{X}, \ abs{b(u,xi)} &leq C orm{u}_{mathcal{X}} orm{xi}_{mathcal{S}} quad forall uinmathcal{X}, xiinmathcal{S}. end{aligned}]

(a(cdot,cdot)) satisfies the (mathcal{Z})-coercive condition, i.e. there exists a constant (alpha > 0) independent of (u) such that [abs{a(u,u)} geq alpha orm{u}_{mathcal{X}}^2 quad forall u in mathcal{Z},] where [label{eq:z-coercivity} mathcal{Z} = { uinmathcal{X} vert b(u,xi)=0 ; forall xiinmathcal{S} },] i.e. (mathcal{Z}) is the annihilator of (mathcal{S}).

(b(cdot,cdot)) satisfies the Babuška-Brezzi condition, i.e. there exists a constant (eta>0) independent of (p) such that [label{eq:babuska-brezzi-condition} sup_{winmathcal{X}} frac{abs{b(w,p)}}{ orm{w}_{mathcal{X}}} geq eta orm{p}_{mathcal{S}} quad forall pinmathcal{S}.]

If (finmathcal{X}') and (ginmathcal{S}'), there exists a unique solution ((u,p)inmathcal{X} imesmathcal{S}) to the following variational problem [egin{aligned} label{eq:mixed-variational-problem-a} a(u,phi) + b(phi,p) &= f(phi) quad forall phiinmathcal{X} \ label{eq:mixed-variational-problem-b} b(u,xi) &= g(xi) quad forall xiinmathcal{S}. end{aligned}] Furthermore, the norms of (u) and (p) can be controlled by the norms of (f) and (g) as [label{eq:mixed-variational-problem-error-estimate} orm{u}_{mathcal{X}} + orm{p}_{mathcal{S}} leq C( orm{f}_{mathcal{X}'} + orm{g}_{mathcal{S}'})]

N.B. By fixing an arbitrary (p) in (mathcal{S}), the left hand side of Babuška-Brezzi condition is the norm of the functional (b(cdot,p): mathcal{X} ightarrowmathcal{C}), which has a lower bound determined by ( orm{p}_{mathcal{S}}).

References

Monk, Peter. 2003. Finite Element Methods for Maxwell’s Equations.