Theorems for existence and uniqueness of variational problem

Introduction

Among simulation engineers, it is well accepted that the solution of a PDE can be envisioned as the following three general steps (actually, this was also my premature understanding during the early era of my study on numerical simulation).

1. Expand the unknown function to be solved by a set of basis functions.
2. Multiply both sides of the equation by a set of test functions and integrate the product over the solution domain.
3. With the application of integration by parts, the space dimension of the integral is reduced by 1 and the appeared boundary integral can be used to apply predefined boundary conditions.

The formulation thus obtained, which discretizes the original continuous problem, is called weak form or variational problem, from which the weak solution results. At first glance, the above envisioned procedures could be applicable to any PDEs, at least, discretized system of equations can be constructed and system matrix can be filled. However, without a careful and crystal clear proof about the existence and uniqueness of the solution for the weak form or variational problem, the results can never be relied on - after all, any operation on the computer can produce something, usually huge amount of data, which is either truth or rubbish. What kind of meaning will be assigned to it and how much value we can extract from it depend on the wisdom, rationality and rigorousness of the human operator.

In this post, the proof for the existence and uniqueness of the solution of the following variational problem will be presented, which is the corner stone of numerical schemes such as the finite element method and boundary element method (BEM).

Let (H_1) and (H_2) be two Hilbert spaces, (a(cdot, cdot): H_1 imes H_2 ightarrow mathbb{K}) be a sesquilinear form with (mathbb{K} in {mathbb{R}, mathbb{C}}) (Note: when (mathbb{K} = mathbb{R}), (a(cdot, cdot)) is a bilinear form), (l(cdot): H_2 ightarrow mathbb{K}) be a bounded linear functional on (H_2). The solution (u in H_1) of the equation below will be sought under an arbitrarily given (v in H_2):

[
egin{equation}
label{eq:variational-problem}
a(u, v) = l(v) quad (forall v in H_2)
end{equation}
]

In this post, we'll first show that the existence and uniqueness of (u) along with a priori estimate of its norm can be obtained thanks to the inf-sup condition. Secondly, by introducing the famous Lax-Milgram Lemma, the inf-sup condition can be relaxed to H-ellipticity condition which still preserves the solvability of the variational problem.

inf-sup condition

Definition (inf-sup condition) The continuous sesquilinear form (a(cdot, cdot)) satisfies the in-sup condition if there exists a constant (gamma > 0), such that

[
egin{align}
label{eq:inf-sup-condition-a}
inf_{u in H_1 ackslash {0}} sup_{v in H_2 ackslash {0}} frac{abs{a(u, v)}}{ orm{u}_{H_1} orm{v}_{H_2}} & geq gamma > 0, ag{a} \
label{eq:inf-sup-condition-b}
forall v in H_2 ackslash {0}: sup_{u in H_1 ackslash {0}} abs{a(u, v)} &> 0. ag{b}
end{align}
]

Remark

1. For (a), first fix (u in H_1 ackslash {0}), then vary (v in H_2 ackslash {0}) and take the supremum of (frac{abs{a(u, v)}}{ orm{u}_{H_1} orm{v}_{H_2}}). Let (A in L(H_1, H_2')) be the associated operator of (a(cdot, cdot)) satisfying ((Au, v) = a(u, v) ; (forall u in H_1, v in H_2)). Let (phi_u := Au in H_2'). Because (u) is already fixed so is its norm ( orm{u}_{H_1}), the supremum can be considered as a measure of the norm ( orm{phi_u}_{H_2'}).
2. For (b), (v) is firstly fixed and (u) is then varied. Because the supremum of the absolute value of the sesquilinear form should be strictly larger than 0, (phi_u) cannot be a zero operator.

Theorem (Existence and uniqueness) The condition that the sesquilinear form (a(cdot, cdot)) satisfies the inf-sup condition is equivalent to the following condition: for all (l in H_2'), the variational problem eqref{eq:variational-problem} has a unique solution (u in H_1), which satisfies the priori estimate

[
egin{equation}
label{eq:priori-estimate}
orm{u}_{H_1} leq frac{1}{gamma} orm{l}_{H_2'}.
end{equation}
]

Proof: A. Given the inf-sup condition, we prove the existence and uniqueness of the solution and the priori estimate.

1. We'll show the associated operator (A: H_1 ightarrow H_2') of (a(cdot, cdot)) is continuous.

   Because (a(cdot, cdot)) is continuous, we have

   [
   abs{a(u, v)} leq orm{a} orm{u}_{H_1} orm{v}_{H_2} quad (forall u in H_1, v in H_2).
   ]

   Let (phi_u := Au = a(u, cdot)), then

   [
   abs{phi_u(v)} = abs{a(u, v)} leq orm{a} orm{u}_{H_1} orm{v}_{H_2} quad (forall u in H_1, v in H_2).
   ]

   Because (u) is given and fixed in (phi_u), we define the constant (C(a, u) := orm{a} orm{u}_{H_1}), therefore (phi_u) is bounded:

   [
   abs{phi_u(v)} leq C(a, u) orm{v}_{H_2} quad (forall v in H_2).
   ]

   It should be noted that when (phi_u) is applied to (v in H_2), a complex conjugate operation must be applied first to (v) due to the definition of (a(cdot, cdot)) which is complex conjugate linear with respect to its second argument. Therefore, (phi_u) is a bounded complex conjugate linear operator from (H_1) to (H_2^*), where (H_2^*) is the anti-dual space of (H_2). Because the only difference between (H_2^*) and (H_2') is a complex conjugate, the two spaces can be identified isometrically (H_2^* cong H_2'). In the following, we use (H_2') replacing (H_2^*) and let (phi_u) inherently includes a complex conjugate operation, which makes (phi_u) a bounded linear operator in (H_2'). According to this analysis, we know the operator (A) really maps (u) to an element in (H_2').

   Then the norm of (phi_u) in (H_2') is

   [
   orm{phi_u}_{H_2'} = orm{Au}_{H_2'} = sup_{v in H_2 ackslash {0}} frac{abs{phi_u(v)}}{ orm{v}_{H_2}} leq orm{a} orm{u}_{H_1} < infty quad (forall u in H_1),
   ]

   based on which the operator (A: H_1 ightarrow H_2') is continuous.

2. Prove the image of (H_1) under (A) is closed in (H_2'). The basic idea is that the closeness of (A(H_1)) in (H_2') can be proved by showing that any Cauchy sequence in (A(H_1)) is convergent in (A(H_1)).

   According to (a) of the inf-sup condition

   [
   forall u in H_1 ackslash {0}: sup_{v in H_2 ackslash {0}} frac{abs{a(u, v)}}{ orm{u}_{H_1} orm{v}_{H_2}} geq gamma > 0,
   ]

   we have the equivalent

   [
   forall u in H_1 ackslash {0}: orm{Au}_{H_2'} = sup_{v in H_2 ackslash {0}} frac{abs{Au(v)}}{ orm{v}_{H_2}} geq orm{u}_{H_1} gamma.
   ]

   When (u = 0), the equality in the above holds. Therefore,

   [
   forall u in H_1: orm{Au}_{H_2'} = sup_{v in H_2 ackslash {0}} frac{abs{Au(v)}}{ orm{v}_{H_2}} geq orm{u}_{H_1} gamma.
   ]

   Let ((y_n)_{n in mathbb{N}}) be a Cauchy sequence in (A(H_1)) and ((x_n)_{n in mathbb{N}}) be the corresponding sequence in (H_1) such that (A(x_n) = y_n). For all (varepsilon > 0), there exists a larger enough (N_0 in mathbb{N}) such that when (m, n > N_0),

   [
   varepsilon > orm{y_m - y_n}_{H_2'} = orm{A(x_m - x_n)}_{H_2'} geq orm{x_m - x_n}_{H_1} gamma.
   ]

   From this we know that ((x_n)_{n in mathbb{N}}) is also a Cauchy sequence in (H_1). Because (H_1) is a Hilbert space, there exists an (x in H_1) such that

   [
   lim_{n ightarrow infty} orm{x_n - x}_{H_1} = 0.
   ]

   According to step 1, (A) is a continuous linear operator, so we have

   [
   lim_{n ightarrow infty} orm{A(x_n) - A(x)}_{H_2'} leq lim_{n ightarrow infty} orm{A}_{H_2' leftarrow H_1} orm{x_n - x}_{H_1} = 0.
   ]

   Because (A(x) in A(H_1)), any Cauchy sequence in (A(H_1)) is also convergent in (A(H_1)) and (A(H_1)) is closed.

3. Prove (A(H_1) = H_2') and the solution for the variational problem eqref{eq:variational-problem} exists.

   Assume (A(H_1)) is a proper subset of (H_2'). Then there exists a non-zero (y_0 in A(H_1)^{perp}). Due to Riesz representation theorem, there exists (y_0' in (A(H_1)')) such that

   [
   forall y in A(H_1): langle y_0', y angle_{A(H_1)' imes A(H_1)} = (y_0, y)_{H_2'} ; ext{and} ; orm{y_0'}_{H_2''} = orm{y_0}_{H_2'},
   ]

   where (langle cdot, cdot angle_{A(H_1)' imes A(H_1)}) is the dual pairing. It can be seen that (y_0') is a non-zero functional, which operates on (A(H_1)) and evaluates to zero. In addition, because (A(H_1)) is closed in (H_2') according to the proof in step 2, Hahn-Banach theorem can be used to extend the domain of (y_0') from (A(H_1)) to the whole space (H_2'), i.e. there exists a non-zero ( ilde{y}_0' in H_2'') such that ( ilde{y}_0'(y) = 0) for all (y in A(H_1)).

   Further because (H_2) is a Hilbert space, it is reflexive: (H_2 cong H_2''), then ( ilde{y}_0' in H_2) and for all (y in A(H_1))
   [
   ilde{y}_0' (y) = y( ilde{y}_0') = (Au) ( ilde{y}_0') = a(u, ilde{y}_0') = 0 quad (u in H_1, Au = y).
   ]
   Because (y) is arbitrarily selected in the image of (A), (u) can also vary arbitrarily in (H_1). Hence we can conclude that there exists a non-zero ( ilde{y}_0' in H_2) such that
   [
   sup_{u in H_1 ackslash {0}} abs{a(u, ilde{y}_0')} = 0,
   ]
   which contradicts (b) of the inf-sup condition. So we've proved (A(H_1) = H_2') and the solution of the variational problem eqref{eq:variational-problem} exists.

4. Prove (A in L(H_1, H_2')) is injective and the variational problem eqref{eq:variational-problem} has a unique solution for all (l in H_2').
   For all (l in H_2'), there exists a (u in H_1) such that (Au = l) according to the proof in step 3. Assume there are two such solutions, namely, (u_1) and (u_2) being different, we have the following according to step 2
   [
   orm{Au_1 - Au_2}_{H_2'} = orm{A(u_1 - u_2)}_{H_2'} geq orm{u_1 - u_2}_{H_1} gamma quad (u_1 - u_2 in H_1 ackslash {0}),
   ]
   which contradicts ( orm{Au_1 - Au_2}_{H_2'} = 0). Therefore, (A in L(H_1, H_2')) is injective and the variational problem eqref{eq:variational-problem} has a unique solution for all (l in H_2').

5. Prove the priori estimate.
   For all (l in H_2'), there exists a unique (u in H_1) such that (Au = l). According to step 2,
   [
   orm{l}_{H_2'} = orm{Au}_{H_2'} geq orm{u}_{H_1} gamma,
   ]
   which proves the priori estimate.

B. Given the existence and uniqueness of the solution and prove the inf-sup condition eqref{eq:inf-sup-condition-a} and eqref{eq:inf-sup-condition-b}.

1. Prove eqref{eq:inf-sup-condition-b} of the inf-sup condition.

   If the variational problem eqref{eq:variational-problem} has a unique solution for all (l in H_2), associated operator (A in L(H_1, H_2')) of (a(cdot, cdot)) is bijective.

   If (b) of the inf-sup condition does not hold, there must exists (y_0 in H_2 ackslash {0}) such that
   [
   sup_{u in H_1 ackslash {0}} = abs{a(u, y_0)} = 0 Leftrightarrow forall u in H_1, a(u, y_0) = (Au)(y_0) = 0.
   ]
   Because (H_2) is reflexive, (y_0) can be considered in (H_2''):
   [
   (Au)(y_0) = y_0(Au) = 0 quad (forall u in H_1).
   ]
   Then from Riesz representation theorem, there exists ( ilde{y}_0 in H_2') corresponding to (y_0 in H_2'') such that
   [
   y_0(Au) = ( ilde{y}_0, Au)_{H_2'} = 0 quad (forall u in H_1).
   ]
   Therefore, (A(H_1)^{perp} eq {0}), which contradicts the fact that (A) is bijective.

2. Prove eqref{eq:inf-sup-condition-a} of the inf-sup condition.
   [
   egin{aligned}
   inf_{u in H_1 ackslash {0}} sup_{v in H_2 ackslash {0}} frac{abs{a(u, v)}}{ orm{u}_{H_1} orm{v}_{H_2}} &= inf_{u in H_1 ackslash {0}} sup_{v in H_2 ackslash {0}} frac{langle Au, v angle_{H_2' imes H_2}}{ orm{u}_{H_1} orm{v}_{H_2}} \
   &= inf_{w in H_2' ackslash {0}} sup_{v in H_2 ackslash {0}} frac{langle w, v angle_{H_2' imes H_2}}{ orm{A^{-1} w}_{H_1} orm{v}_{H_2}} quad (w in H_2', w = Au) \
   &geq inf_{w in H_2' ackslash {0}} sup_{v in H_2 ackslash {0}} frac{langle w, v angle_{H_2' imes H_2}}{ orm{A^{-1}}_{H_2 leftarrow H_2'} orm{w}_{H_2'} orm{v}_{H_2}}
   end{aligned}
   ]
   For the Hilbert space (H_2), there exists an isometry (J_{H_2}: H_2 ightarrow H_2'). Let ( ilde{w} in H_2) and (J_{H_2} ( ilde{w}) = w), we further have
   [
   egin{aligned}
   inf_{u in H_1 ackslash {0}} sup_{v in H_2 ackslash {0}} frac{abs{a(u, v)}}{ orm{u}_{H_1} orm{v}_{H_2}} &geq gamma inf_{ ilde{w} in H_2 ackslash {0}} sup_{v in H_2 ackslash {0}} frac{langle J_{H_2} ilde{w}, v angle_{H_2' imes H_2}}{ orm{J_{H_2} ilde{w}}_{H_2'} orm{v}_{H_2}} quad ( ext{Let $ orm{A^{-1}}_{H_2 leftarrow H_2'} = gamma^{-1}$.}) \
   &= gamma inf_{ ilde{w} in H_2 ackslash {0}} frac{1}{ orm{ ilde{w}}_{H_2}} sup_{v in H_2 ackslash {0}} frac{langle J_{H_2} ilde{w}, v angle_{H_2' imes H_2}}{ orm{v}_{H_2}} quad (ecause orm{J_{H_2} ilde{w}}_{H_2'} = orm{ ilde{w}}_{H_2}) \
   &= gamma inf_{ ilde{w} in H_2 ackslash {0}} frac{ orm{J_{H_2} ilde{w}}_{H_2'}}{ orm{ ilde{w}}_{H_2}} \
   &= gamma
   end{aligned}.
   ]

H-ellipticity condition and Lax-Milgram Lemma

Definition (H-ellipticity) Let (H_1 = H_2 = H) is reflexive Banach space, (a: H imes H ightarrow mathbb{C}) be a sesquilinear form. (a(cdot, cdot)) is H-elliptic if there exits (gamma > 0) and (sigma in mathbb{C}) with (abs{sigma} = 1), such that
[
forall u in H: Re(sigma a(u, u)) geq gamma orm{u}_H^2.
]
Lemma (Lax-Milgram) Let (H) be a Hilbert space. The sesquilinear form (a: H imes H ightarrow mathbb{C}) is H-elliptic. Then the inf-sup condition holds.

Proof: From the H-ellipticity condition for (a(cdot, cdot)), we have
[
gamma orm{u}_H^2 leq Re(sigma a(u, u)) leq abs{Re(sigma a(u, u))} leq abs{sigma a(u, u)} = abs{sigma} abs{a(u, u)} = abs{a(u, u)}.
]
Substitute this inequality into the LHS of eqref{eq:inf-sup-condition-a} of the inf-sup condition while selecting (v) to be equal to (u),
[
inf_{u in H ackslash {0}} sup_{v in H ackslash {0}} frac{abs{a(u, v)}}{ orm{u}_H orm{v}_H} geq inf_{u in H ackslash {0}} frac{gamma orm{u}_H^2}{ orm{u}_H orm{u}_H} = gamma > 0.
]
This proves eqref{eq:inf-sup-condition-a} of the inf-sup condition.

To prove eqref{eq:inf-sup-condition-b} of the inf-sup condition, given an arbitrary (v in H ackslash {0}) and let (u = v), we have
[
sup_{u in H ackslash {0}} abs{a(u, v)} geq abs{a(v, v)} geq gamma orm{v}_H > 0.
]
According to Lax-Milgram Lemma, we can still have the existence, uniqueness and priori estimate for the solution of the variation problem from the H-elliptic condition on (a(cdot, cdot)).

Summary

In this post, we present conditions and theorems along with their proofs, which ensures the existence and uniqueness for the solution of the general variational problem (a(u, v) = l(v) ; (forall l in H_2)). The underlying condition is the inf-sup condition. During the proof, the application of Hahn-Banach theorem is a key step for proving that the associated operator (A in L(H_1, H_2')) of (a(cdot, cdot)) is surjective. Because of Lax-Milgram Lemma, the inf-sup condition can be relaxed to H-ellipticity condition.