Preprint
Differentiability of Implicit
Functions
Beyond the Implicit Function Theorem
Gerd Wachsmuth∗
November 19, 2012
Research Group
Numerical Mathematics
(Partial Differential Equations)
Abstract
Typically, the implicit function theorem can be used to deduce the differentiability
of an implicit mapping S : u 7→ y given by the equation e(y, u) = 0. However, the
implicit function theorem is not applicable if different norms are necessary for the
differentiation of e w.r.t. y and the invertibility of the partial derivative ey (y, u). We
prove theorems ensuring the (twice) differentiability of the mapping S which can be
applied in this case. We highlight the particular application to quasilinear partial
differential equations whose principal part depends nonlinearly on the gradient of
the state ∇y.
Keywords: implicit function theorem, differentiability, quasilinear partial differential
equations
MSC: 47J07, 35J62, 49J50
∗
Chemnitz University of Technology, Faculty of Mathematics, D–09107 Chemnitz, Germany, [email protected], http://www.tu-chemnitz.de/mathematik/
part_dgl/wachsmuth
Differentiability of Implicit Functions
Wachsmuth
1 Intro
We deal with the differentiability of a mapping S : u 7→ y which is implicitly given by
the equation
e(y, u) = 0.
(1.1)
Here, Y, U, Z are Banach spaces and the function e maps (an open subset of) Y × U into
Z. This issue is usually dealt with the implicit function theorem in infinite dimensions
which goes back to Hildebrandt and Graves [1927], see, e.g., [Cartan, 1967, Thm. 4.7.1],
[Kantorovich and Akilov, 1964, Sec. XVII.4.2] or [Zeidler, 1995, Sec. 4.8]. However, the
implicit function theorem is not applicable to quasilinear partial differential equations
(PDEs) whose principal part depends nonlinearly on the gradient of the state, see the
discussion below.
The main contribution of this paper is to provide a theorem which applies to this class
of nonlinear PDEs and which yields the differentiability of the solution mapping S, see
Theorem 2.1. We also deal with second-order derivatives, see Theorem 2.3.
In the remainder of the introduction, we recall the classical implicit function theorem
(as it can be found in [Zeidler, 1995, Sec. 4.8]) and show that it cannot be applied to a
certain class of nonlinear PDEs as mentioned above.
Theorem 1.1 (Implicit Function Theorem). Let (y0 , u0 ) ∈ Y × U with e(y0 , u0 ) = 0
be given. Suppose that e : Y × U → Z is continuously Fréchet differentiable in a
neighborhood of (y0 , u0 ) and that the partial derivative ey (y0 , u0 ) ∈ L(Y, Z) is bijective
(i.e. continuously invertible).
Then there exists a mapping S : U → Y , and ry , ru > 0, such that for all (y, u) ∈
Bry (y0 ) × Bru (u0 ) the statements y = S(u) and e(y, u) = 0 are equivalent. Moreover,
the mapping S : U → Y is Fréchet differentiable in Bru (u0 ) and its derivative is given by
−1
S 0 (u) = − ey (S(u), u) eu (S(u), u).
We remark that, if e is n times continuously Fréchet differentiable, then S is also n times
Fréchet differentiable.
The implicit function theorem is very powerful and can be applied, e.g., to (the weak
formulation of) nonlinear PDEs. Let us mention two situations in which this approach
yields the differentiability of the solution map. Under suitable assumptions on the nonlinearities d and a, the implicit function theorem can be applied to the semilinear PDE
− ∆y + d(y) = u in Ω,
∂
y = 0 on ∂Ω,
∂n
see [Tröltzsch, 2010, Thm. 4.24], and to the quasilinear PDE
− div a(y) ∇y + d(y) = u in Ω,
y = 0 on ∂Ω,
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Differentiability of Implicit Functions
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see [Casas and Tröltzsch, 2009, Thm. 2.10]. Note that in the latter case the nonlinearity
a in the principal part does not depend on the gradient of the state ∇y.
However, the implicit function theorem is not applicable to the solution map of a quasilinear PDE whose nonlinearity of the principal part does depend on ∇y. We briefly
point to the difficulties appearing in this situation. To this end, let g : Rn → Rn be a
differentiable function which is not affine. We will consider the quasilinear PDE
− div g(∇y) = u in Ω,
y = 0 on ∂Ω,
with u ∈ L2 (Ω) and y ∈ W01,p (Ω) for some p ∈ (1, ∞). Under a growth condition for g,
see Section 3 for details, we have g(∇y) ∈ Lp (Ω; Rn ) if (and only if) y ∈ W01,p (Ω). The
weak formulation reads
Z
0
g(∇y) · ∇v − u v dx = 0 for all v ∈ W01,q (Ω),
e(y, u), v W −1,q (Ω),W 1,q0 (Ω) =
0
Ω
where q 0 is the exponent conjugate to q ∈ (1, ∞). Note that we require 1/p+1/q 0 ≤ 1, i.e.
q ≤ p, in order that this weak formulation is well defined. In the notation of Theorem 1.1
the spaces under consideration are
Y = W01,p (Ω),
0
Z = W −1,q (Ω) = W01,q (Ω)? .
U = L2 (Ω),
In order to ensure the Fréchet differentiability of e with respect to y, we need to differentiate the Nemytskii operator associated with the nonlinear function g. It is well known,
see, e.g. Goldberg et al. [1992], that a norm gap is required whenever g is not affine.
That is, g : Lp (Ω; Rn ) → Lq (Ω; Rn ) is differentiable if and only if q < p. Therefore, in
order to obtain the Fréchet differentiability of e(·, u0 ) : W01,p (Ω) → W −1,q (Ω), we have
to assume q < p.
The next step is to prove that ey (y0 , u0 ) ∈ L(Y, Z) is bijective. This amounts to the
solvability of the (weak formulation of the) linearized equation
− div g 0 (∇y0 ) ∇δy = δz in Ω,
δy = 0 on ∂Ω
w.r.t. δy ∈ Y = W01,p (Ω) for all δz ∈ Z = W −1,q (Ω). This solvability requires p ≤ q.
Hence, the differentiability of e requires a norm gap and this norm gap indeed renders the
invertibility of the linearized PDE ey (y0 , u0 ) impossible. We conclude that the implicit
function theorem is not applicable to quasilinear PDEs whose principal part depends
nonlinearly on the gradient of the state.
We briefly remark that the implicit function theorem may be applicable in the setting
Y = W01,∞ (Ω)
Z = W −1,∞ (Ω) = W01,1 (Ω)? .
U = L2 (Ω)
However, this requires that the quasilinear PDE and its linearization possess solutions in
the space W01,∞ (Ω) which may be difficult to achieve.
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Differentiability of Implicit Functions
Wachsmuth
The outline of the paper is as follows. Theorems ensuring the (twice) Fréchet differentiability of the implicit map S are presented in Section 2. These theorems take into account
that we have to use different spaces for the differentiation of e and the invertibility of
ey (y0 , u0 ). Finally, we apply this theory to a class of quasilinear PDEs in Section 3, see
Theorems 3.3–3.5.
Notation
For a Banach space X we denote by X ? its topological dual space. The continuous
embedding of a Banach space X into another Banach space Y is denoted by X ,→ Y .
The space of bounded linear operators from X into Y is denoted by L(X, Y ).
We use the usual Sobolev spaces (including homogeneous Dirichlet boundary conditions)
and Lebesgue spaces, which are denoted by W01,p (Ω) and Lp (Ω), respectively. By p0
we denote the exponent conjugate to p. Moreover, we use the usual notation H01 (Ω) =
0
W01,2 (Ω), W −1,p (Ω) = W01,p (Ω)? , and H −1 (Ω) = H01 (Ω)? .
For n ≥ 1, we denote by Rn , Rn×n and Rn×n×n the space of real vectors (of length n),
the space of matrices and the space of third-order tensors, respectively. The Euclidean
norm in Rn and its associated operator norms are denoted by |·|.
The partial derivatives of a function e : (y, u) 7→ e(y, u) w.r.t. y and u are denoted by ey
and eu , respectively. The total Fréchet derivative of e (w.r.t. (y, u)) is denoted by e(y,u) .
The second order partial derivatives are denoted by eyy , eyu , euy , and euu .
For the results below, it is sufficient that e is defined in a neighborhood of (y0 , u0 ).
However, in interest of a clear presentation, we assume that e is defined on the whole
product space Y × U .
2 Main Theorems
The introductory example suggests to work with different norms for the differentiation
of e and for the invertibility of ey (y0 , u0 ). Note that the so-called two-norms discrepancy
in infinite dimensional optimization also requires to work with different norms for the
differentiation of a functional and for the coercivity of its second derivative, see [Tröltzsch,
2010, Sect. 4.10.2].
2.1 First order derivative
First, we address the differentiability of the implicitly defined mapping S. Note that,
in contrast to the implicit function theorem, we have to assume the existence of the
implicit map S. However, in context of nonlinear PDEs this is not a problem, since a
well-developed solution theory exists in many cases, which usually provides the existence
4
Differentiability of Implicit Functions
Wachsmuth
of a solution map. In these cases, our theory can be used to prove the differentiability of
this mapping.
Theorem 2.1. Let Y 0 , Y + , U, Z 0 be Banach spaces such that Y + ,→ Y 0 . Let e :
Y 0 × U → Z 0 be given and (y0 , u0 ) ∈ Y + × U such that e(y0 , u0 ) = 0. We assume
that
• e : Y + × U → Z 0 is Fréchet differentiable at (y0 , u0 ),
• the partial derivative ey (y0 , u0 ) ∈ L(Y + , Z 0 ) can be extended to an element of
L(Y 0 , Z 0 ), and ey (y0 , u0 ) ∈ L(Y 0 , Z 0 ) is bijective (i.e. continuously invertible).
Moreover, we assume that there exists a solution map S : U → Y + , where U is a
neighborhood of u0 , such that e(S(u), u) = 0 for all u ∈ U and that S is Lipschitz
continuous at u0 , i.e., kS(u) − S(u0 )kY + ≤ L ku − u0 kU for all u ∈ U.
Then S : U → Y 0 is Fréchet differentiable at u0 . The derivative S 0 (u0 ) is given in (2.1).
Proof. We define
S 0 (u0 ) h = −ey (·)−1 eu (·) h
(2.1)
for h ∈ U . Here and in what follows, we abbreviate the argument (S(u0 ), u0 ) = (y0 , u0 )
by (·). By the assumptions of the theorem, we have S 0 (u0 ) ∈ L(U, Y 0 ). It remains to
prove the estimate kS(u0 + h) − S(u0 ) − S 0 (u0 ) hkY 0 = o(khkU ) as khkU → 0.
We have
−ey (·) S(u0 + h) − S(u0 ) − S 0 (u0 ) h = e(S(u0 + h), u0 + h) − e(·)
(2.2)
− ey (·) S(u0 + h) − S(u0 ) − eu (·) h
for all h ∈ U with khkU small, i.e., u0 + h ∈ U. Note that we have used e(S(u0 + h), u0 +
h) = e(·) = 0. By the Fréchet differentiability of e, the right-hand side satisfies
e(S(u0 + h), u0 + h) − e(·) − ey (·) S(u0 + h) − S(u0 ) − eu (·) h 0
Z
= o kS(u0 + h) − S(u0 )kY + + khkU
as S(u0 + h) → S(u0 ) in Y + and h → 0 in U . SinceS : U → Y + is Lipschitz continuous
at u0 , we can replace the right-hand side by o khkU as h → 0 in U . Together with (2.2)
and ey (·)−1 ∈ L(Z 0 , Y 0 ), we have kS(u0 + h) − S(u0 ) − S 0 (u0 ) hkY 0 = o(khkU ) as h → 0
in U . Hence, S 0 (u0 ) is the Fréchet derivative of S : U → Y 0 at u0 .
We remark that if the assumptions of Theorem 2.1 are satisfied, we have kS 0 (u0 )kL(U,Y 0 ) ≤
L C, where L is the local Lipschitz constant of S : U → Y + and C is the embedding
constant satisfying kykY 0 ≤ C kykY + for all y ∈ Y + .
Next, we address the question of continuity of S 0 . If the assumptions of Theorem 2.1 are
satisfied in a neighborhood of (y0 , u0 ), then S is Fréchet differentiable in a neighborhood
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Differentiability of Implicit Functions
Wachsmuth
of u0 . Moreover, the derivative S 0 is continuous if we assume the partial derivatives of
e to be continuous, see (2.1). However, for the case of quasilinear PDEs mentioned in
the introduction the derivative ey : Y + → L(Y 0 , Z 0 ) is not continuous. This is shown
in Section 3.2. Hence, we have to introduce additional weaker spaces Y −/2 and Z −/2 for
the continuity of ey . This is made precise in the following corollary.
Corollary 2.2. Let Y 0 , Y + , Y −/2 , U, Z 0 , Z −/2 be Banach spaces such that Y + ,→ Y 0 ,→
Y −/2 and Z 0 ,→ Z −/2 . Let e : Y 0 × U → Z 0 be given and (y0 , u0 ) ∈ Y + × U such
that e(y0 , u0 ) = 0. We assume that there are neighborhoods Y, U of y0 , u0 in Y + , U,
respectively, such that
• e : Y + × U → Z 0 is Fréchet differentiable in Y × U,
• the partial derivative ey (y, u) ∈ L(Y + , Z 0 ) can be extended to an element of
L(Y 0 , Z 0 ) and ey (y, u) ∈ L(Y 0 , Z 0 ) is bijective for every (y, u) ∈ Y × U,
• the partial derivative ey (y, u) ∈ L(Y + , Z 0 ) can be extended to an element of
L(Y −/2 , Z −/2 ) and ey (y, u) ∈ L(Y −/2 , Z −/2 ) is bijective for every (y, u) ∈ Y × U
with uniformly bounded inverses,
• the Fréchet derivative e(y,u) : Y + × U → L(Y 0 × U, Z −/2 ) is continuous in Y × U.
Moreover, we assume that there exists a solution map S : U → Y + , such that e(S(u), u) =
0 for all u ∈ U and S is Lipschitz continuous in U, i.e., kS(v) − S(u)kY + ≤ L(u) kv − ukU
for all u, v ∈ U.
Then S : U → Y 0 is Fréchet differentiable and the function S 0 : U → L(U, Y −/2 ) is
continuous.
Proof. The assumptions of Theorem 2.1 are satisfied for all elements of U. Therefore,
S : U → Y 0 is Fréchet differentiable. It remains to show the continuity of S 0 : U →
L(U, Y −/2 ). By assumption we have
key (S(v), v)−1 kL(Z −/2 ,Y −/2 ) ≤ C
for all v ∈ U.
Using the identities
ey (S(u), u) S 0 (u) h + eu (S(u), u) h = 0 and
ey (S(v), v) S 0 (v) h + eu (S(v), v) h = 0
we infer
0 = ey (S(v), v) S 0 (u) − S 0 (v) h
+ eu (S(u), u) − eu (S(v), v) h
+ ey (S(u), u) − ey (S(v), v) S 0 (u) h
6
(2.3)
Differentiability of Implicit Functions
Wachsmuth
for all h ∈ U and all u, v ∈ U. By the assumptions we have
eu (S(v), v) → eu (S(u), u) in L(U, Z −/2 )
ey (S(v), v) → ey (S(u), u) in L(Y 0 , Z −/2 )
as v → u in U (and hence, S(v) → S(u) in Y + ). Together with (2.3) this shows
S 0 (v) → S 0 (u) in L(U, Y −/2 ) as v → u in U .
2.2 Second order derivative
In this section we address the existence of the second Fréchet derivative of S. Just as for
the continuity for S 0 , we obtain the existence of S 00 only in an even weaker space Y − .
Theorem 2.3. Let the assumptions of Corollary 2.2 be satisfied. Let Y − , Z − be Banach
spaces, such that Y −/2 ,→ Y − and Z −/2 ,→ Z − . We assume that
• ey : Y + × U → L(Y −/2 , Z − ) is Lipschitz continuous at (y0 , u0 ),
• e : Y 0 × U → Z − is twice Fréchet differentiable at (y0 , u0 ),
• ey (y0 , u0 ) ∈ L(Y 0 , Z 0 ) can be expanded to an element of L(Y − , Z − ), and
ey (y0 , u0 ) ∈ L(Y − , Z − ) is bijective.
Then S : U → Y − is twice Fréchet differentiable at u0 . The second derivative S 00 (u0 ) is
given in (2.4).
Proof. Let h1 , h2 ∈ U be given. We define the second derivative S 00 (u0 )[h1 , h2 ] by
S 00 (u0 )[h1 , h2 ] = −ey (·)−1 eyy (·) S 0 (u0 ) h1 , S 0 (u0 ) h2 + eyu (·) S 0 (u0 ) h1 , h2
(2.4)
+ euy (·) h1 , S 0 (u0 ) h2 + euu (·) h1 , h2 .
Note that the bilinear form S 00 (u0 ) maps U × U continuously (i.e. boundedly) to Y − by
the assumptions on e.
It remains to show the estimate
0
S (u0 + h1 ) − S 0 (u0 ) − S 00 (u0 )[h1 , ·]
= o(kh1 kU ) as h1 → 0 in U.
L(U,Y − )
By the definition of the operator norm in L(U, Y − ), this is equivalent to
0
S (u0 + h1 ) h2 − S 0 (u0 ) h2 − S 00 (u0 )[h1 , h2 ] − = kh2 kU o(kh1 kU )
Y
for all h2 ∈ U and as h1 → 0 in U.
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Differentiability of Implicit Functions
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Similarly to the proof of Theorem 2.1, we start with
− ey (·) S 0 (u0 + h1 ) h2 − S 0 (u0 ) h2 − S 00 (u0 )[h1 , h2 ] =
ey (S(u0 + h1 ), u0 + h1 ) S 0 (u0 + h1 ) h2 − ey (·) S 0 (u0 + h1 ) h2
− ey (S(u0 + h1 ), u0 + h1 ) S 0 (u0 + h1 ) h2 + ey (·) S 0 (u0 ) h2
− eyy (·) S 0 (u0 ) h1 , S 0 (u0 ) h2 − euy (·) h1 , S 0 (u0 ) h2
− eyu (·) S 0 (u0 ) h1 , h2 − euu (·) h1 , h2 .
(2.5)
Now we proceed by estimating the Z − -norm of the sum of the first and the third lines
and the sum of the second and the fourth lines on the right-hand side of (2.5).
We have to estimate
ey (S(u0 + h1 ), u0 + h1 ) S 0 (u0 + h1 ) h2 − ey (·) S 0 (u0 + h1 ) h2
− eyy (·) S 0 (u0 ) h1 , S 0 (u0 ) h2 − euy (·) h1 , S 0 (u0 ) h2 .
(2.6)
We find that this term is equal to
n
o
ey (S(u0 + h1 ), u0 + h1 ) − ey (·) − eyy (·) S(u0 + h1 ) − S(u0 ), · − euy (·) h1 , · S 0 (u0 ) h2
+ eyy (·) S(u0 + h1 ) − S(u0 ) − S 0 (u0 ) h1 , S 0 (u0 ) h2
+ ey (S(u0 + h1 ), u0 + h1 ) − ey (·) S 0 (u0 + h1 ) − S 0 (u0 ) h2 .
The L(Y 0 , Z − )-norm of the term in curly brackets is of order o(kh1 kU ) by the assumptions
on e and S. The Z − -norm of the second line is also bounded by kh2 kU o(kh1 kU ) in virtue
of the differentiability of S : U → Y 0 , and since eyy (·) is bounded from Y 0 × Y 0 into Z − .
The L(Y −/2 , Z − )-norm of the term inside the first pair of square brackets on the third
line is bounded by kh1 kU by the Lipschitz continuity of ey , whereas the L(U, Y −/2 )-norm
of the second term in square brackets goes to zero as h1 → 0 by Corollary 2.2. Therefore,
we have shown that the Z − -norm of (2.6) is bounded by kh2 kU o(kh1 kU ).
Now we prove an estimate for the sum of the second and the fourth line of the right-hand
side of (2.5), that is of the expression
− ey (S(u0 + h1 ), u0 + h1 ) S 0 (u0 + h1 ) h2 + ey (·) S 0 (u0 ) h2
− eyu (·) S 0 (u0 ) h1 , h2 − euu (·) h1 , h2 .
(2.7)
By the definition (2.1) of S 0 (u), this is equal to
h
i
eu (S(u0 + h1 ), u0 + h1 ) − eu (·) − eyu (·) S 0 (u0 ) h1 , · − euu (·) h1 , · h2 .
The Z − -norm of this term is bounded by kh2 kU o(kh1 kU ) due to the differentiability of
eu and the Lipschitz continuity of S.
Altogether, we have proven the following estimate for (2.5):
ey (·) S 0 (u0 + h1 ) h2 − S 0 (u0 ) h2 − S 00 (u0 )[h1 , h2 ] − = kh2 kU o(kh1 kU ).
Z
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Differentiability of Implicit Functions
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By the invertibility of ey (·) ∈ L(Y − , Z − ), this yields
0
S (u0 + h1 ) − S 0 (u0 ) − S 00 (u0 )[h1 , ·]
= o(kh1 kU ).
L(U,Y − )
Hence, S : U → Y − is twice Fréchet differentiable at u0 .
3 Applications
In this section, we are going to apply the abstract theory of Section 2 to a general class
of quasilinear PDEs. This class is not only of interest in its own right, see, e.g., Casas
and Fernández [1993], but it also appears in the context of regularization of variational
inequalities, namely as a regularization of Bingham flows, see, e.g. [de los Reyes, 2011,
Sec. 6.1], and of gradient obstacle problems, see Section 3.1.
We remark that similar results can be obtained for systems of quasilinear PDEs, see also
[Gröger, 1989, Rem. 14], and for nonlinear, small-strain elasticity, where the material law
is given by a nonlinear relation σ = F (ε). Such systems appear e.g. in each time step of
semi-discretized problems plasticity with kinematic hardening, see Wachsmuth [2012], or
as regularizations of static plasticity problems, see Herzog et al. [2012]. Note that in the
case of nonlinear elasticity problems, one has to replace the regularity result of Gröger
[1989] by the one of Herzog et al. [2011].
3.1 Regularized Gradient Obstacle Problem
In this section we give a concrete example which is an instance of the class of general
quasilinear elliptic equations discussed in the next section.
We consider the deflection y ∈ H01 (Ω) of an elastic membrane Ω ⊂ Rn which is clamped
at the boundary ∂Ω. Moreover, the membrane is restricted by a gradient obstacle, i.e.,
the (Euclidean) norm of ∇y should stay below 1. We arrive at the problem of minimizing
the membrane energy
Z
Z
1
2
|∇y| dx −
u y dx
(3.1a)
2 Ω
Ω
w.r.t. y ∈ H01 (Ω), where u ∈ L2 (Ω) is an external force, under the constraint
|∇y| ≤ 1 a.e. in Ω,
(3.1b)
where |·| denotes the (Euclidean) norm in Rn . Such problems occur for elastic-plastic
beams with cross section Ω under torsion, see, e.g., Evans [1979], Bermúdez [1982], Idone
et al. [2003].
It is straightforward to see that the unique solution of (3.1) coincides with the projection
w.r.t. the energy norm of the unconstrained minimizer of (3.1a) in H01 (Ω) onto the set
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Differentiability of Implicit Functions
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of functions satisfying (3.1b). Since projections are in general not differentiable, the
solution mapping of (3.1) is not differentiable.
It is often desirable to approximate (3.1) with an equation whose solution mapping is
differentiable. To this end, we replace (3.1) by an unconstrained minimization problem
in which the violation of the constraint (3.1b) enters the objective as an inexact penalty
term. That is, we consider
Z
Z
Z
2
1
γ
2
Minimize
u y dx +
|∇y| dx −
max |∇y| − 1, 0 dx
(3.2)
2 Ω
2 Ω
Ω
where γ > 0 is a penalty parameter (finally, one would drive γ → ∞). The first order
necessary conditions of (3.2) are given by
Z
Z
∇y · ∇v
max |∇y| − 1, 0
∇y · ∇v − u v dx + γ
dx for all v ∈ H01 (Ω). (3.3)
|∇y|
Ω
Ω
Since the objective in (3.2) is (strictly) convex, (3.3) is even sufficient for the optimality
of y ∈ H01 (Ω). Due to the max operator involved in (3.3), the solution mapping of (3.3) is
still not differentiable. Hence, we use a smooth replacement maxε (·) ∈ C 2 (R) of max(0, ·)
satisfying
maxε (x) = max(x, 0) for all x ∈ R \ (−ε, +ε),
0 ≤ max0ε (x) ≤ 1 for all x ∈ R,
0 ≤ max00ε (x) ≤ ε−1
for all x ∈ R,
where ε > 0. For instance, the function
max00ε (x)
1
= max 0, min(ε − x, ε + x) ,
ε
Z
x
Z
t
maxε (x) =
−∞
−∞
max00ε (s) ds dt
satisfies these assumptions. Now, we consider the regularized problem
Z
Z
1 ∇y · ∇v dx for all v ∈ H01 (Ω). (3.4)
∇y · ∇v − u v dx + γ
maxε 1 −
|∇y|
Ω
Ω
Note that (3.4) is not necessarily the optimality condition for a regularized version of
(3.2). By defining
g : Rn → Rn ,
h
1 i
g(z) = 1 + γ maxε 1 −
z,
|z|
(3.5)
we find that (3.4) is the weak formulation of the quasilinear equation (3.6) which is dealt
with in the next section.
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Differentiability of Implicit Functions
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3.2 Quasilinear Elliptic PDEs
Let g ∈ C 2 (Rn , Rn ) and a domain Ω ∈ Rn , n ≤ 3 be given. Note that the following
analysis is not restricted to n ≤ 3, but rather this bound is chosen for simplicity of the
presentation. We shall study the solution map L2 (Ω) 3 u 7→ y of the quasilinear PDE
− div g(∇y) = u
y=0
in Ω,
(3.6a)
on ∂Ω.
(3.6b)
A function y ∈ H01 (Ω) is called a weak solution of (3.6) if and only if
Z
Z
u v dx
g(∇y) · ∇v dx =
Ω
Ω
holds for all v ∈ H01 (Ω).
In order to apply the integrability results of Gröger [1989] and those of Section 2, we assume that the domain Ω is a Lipschitz domain and, hence, regular in the sense of [Gröger,
1989, Def. 2], see also [Haller-Dintelmann et al., 2009, Sec. 5]. Moreover, we assume that
there are constants m, M, M 0 > 0, such that for all z, y ∈ Rn the conditions
(g(z) − g(y)) · (z − y) ≥ m |z − y|2Rn ,
(3.7a)
|g(z) − g(y)|Rn ≤ M |z − y|Rn ,
(3.7b)
0
0
0
|g (z) − g (y)|Rn×n ≤ M |z − y|Rn
(3.7c)
are satisfied and g(0) = 0 holds. Note that (3.7) is fulfilled for the choice (3.5) in
the previous section. The last assumption (3.7c) is only needed for the second order
derivative.
We remark that the analysis is not restricted to the case of Dirichlet boundary conditions,
but also mixed boundary conditions as considered in Gröger [1989] could be handled.
Solvability of the nonlinear equation. First, we address the solvability of (3.6).
For convenience of the presentation, we relax u ∈ L2 (Ω) by assuming u ∈ W −1,p (Ω).
Lemma 3.1. For every u ∈ H −1 (Ω), there exists a unique weak solution y(u) of (3.6).
Moreover, there exists p0 > 2, depending only on Ω, m and M, such that for all p ∈ [2, p0 ]
and u ∈ W −1,p (Ω) we have y(u) ∈ W01,p (Ω) and
ky(u2 ) − y(u1 )kW 1,p (Ω) ≤ L ku2 − u1 kW −1,p (Ω)
0
for all u1 , u2 ∈ W −1,p (Ω). The Lipschitz constant L > 0 depends only on Ω, m, M and
p0 .
11
Differentiability of Implicit Functions
Wachsmuth
Proof. Due to (3.7) and Friedrichs’ inequality, the operator y 7→ − div(g(∇y)) : H01 (Ω) →
H −1 (Ω) = H01 (Ω)? is strongly monotone. Using the Browder-Minty theorem we infer the
unique solvability of (3.6) and the a-priori estimate
ky(u)kH01 (Ω) ≤ L1 kukH −1 (Ω) ,
where L1 depends only on m and the constant of Friedrichs’ inequality.
Obviously, y(ui ), i = 1, 2, is also the weak solution of
− div g(∇y) + m y = ui + m y(ui )
y=0
in Ω,
on ∂Ω.
Now we can apply [Gröger, 1989, Thm. 1] and obtain the existence of p0 > 2 and L2 > 0,
such that for all p ∈ [2, p0 ] and ui ∈ W −1,p (Ω) we have y(ui ) ∈ W01,p (Ω) and
ky(u2 ) − y(u1 )kW 1,p (Ω) ≤ L2 ky(u2 ) − y(u1 )kW −1,p (Ω) + ku2 − u1 kW −1,p (Ω) ,
0
where L2 depends only on Ω, m and M . Hence, by the embedding H01 (Ω) ,→ W −1,p (Ω)
we have
ky(u2 ) − y(u1 )kW 1,p (Ω) ≤ L ku2 − u1 kW −1,p (Ω) .
0
for all p ∈ [2, p0 ] and L depends only on Ω, m, M and p0 .
In what follows, we fix p0 ≤ 6 such that Lemma 3.1 is satisfied. This, in particular,
implies that L2 (Ω) ,→ W −1,p0 (Ω) (since n ≤ 3) and, therefore, we have a unique weak
solution y ∈ W 1,p0 (Ω) of (3.6) for all u ∈ L2 (Ω).
Nemytskii operators associated with the nonlinearity g. In the sequel, we need
several results on the mapping properties, continuity and differentiability of Nemytskii
operators. We refer to Goldberg et al. [1992] for these results.
Since g does not depend on the spatial variable x ∈ Ω, the Caratheodory condition (see
[Goldberg et al., 1992, Sec. 2.1]) reduces to the continuity w.r.t. z ∈ Rn . Hence, g and
its derivatives g 0 and g 00 satisfy the Caratheodory condition.
We define the Nemytskii operator G which maps f ∈ Lp (Ω; Rn ) to the function G(f )(x) =
g(f (x)). Note that (3.7b) implies that g satisfies a growth condition (see [Goldberg
et al., 1992, Sec. 2.1]). Hence, G maps Lp (Ω; Rn ) continuously into itself, for arbitrary
p ∈ [1, ∞], see [Goldberg et al., 1992, Sec. 2.4].
Similarly, we define the Nemytskii operators H : Lp (Ω; Rn ) → Lq (Ω; Rn×n ) and I :
Lp (Ω; Rn ) → Lq (Ω; Rn×n×n ) associated with g 0 and g 00 . Due to (3.7b) and (3.7c), g 0 and
g 00 satisfy the uniform boundedness condition [Goldberg et al., 1992, (UB3)]. Hence, H
12
Differentiability of Implicit Functions
Wachsmuth
and I are continuous from Lp (Ω; Rn ) to Lq (Ω; Rn×n ) and Lq (Ω; Rn×n×n ), respectively,
for all p ∈ [1, ∞] and q ∈ [1, ∞), see [Goldberg et al., 1992, Sec. 2.4].
Differentiability of the Nemytskii operator G. Now, let exponents p, q, r, s be
given, such that 1 ≤ q < p and 2 ≤ 2 s < r are satisfied. Using the mapping properties
of G, H and I, we can apply [Goldberg et al., 1992, Thms. 7 and 9] to infer that
G : Lp (Ω; Rn ) → Lq (Ω; Rn )
(3.8)
is continuously Fréchet differentiable and
G : Lr (Ω; Rn ) → Ls (Ω; Rn )
(3.9)
is twice continuously Fréchet differentiable. Note that the nonlinear Nemytskii operator
G will not be (twice) differentiable if p ≤ q (or r ≤ 2 s), see [Goldberg et al., 1992,
Sec. 3.1]. Their derivatives are given by
(G0 (f ) h)(x) = (H(f )(x)) h(x),
(G00 (f )[h1 , h2 ])(x) = (I(f )(x)) [h1 (x), h2 (x)],
G0 (f ) h ∈ Lq (Ω; Rn ),
G00 (f )[h1 , h2 ] ∈ Ls (Ω; Rn ),
for almost all x ∈ Ω, respectively. Here, f , h ∈ Lp (Ω; Rn ) and f , h1 , h2 ∈ Lr (Ω; Rn ),
respectively.
Lipschitz continuity of the Nemytskii operator G0 . For the application of Theorem 2.3, we need to address the Lipschitz continuity of the operator G0 : Lp (Ω; Rn ) →
L(Lq (Ω; Rn ), Lr (Ω; Rn )) for certain exponents p, q, r. We have
1/r
Z 0
g 0 (f 1 (x)) − g 0 (f 2 (x))r |h(x)|r dx
G (f 1 ) h − G0 (f 2 ) h r
=
n
L (Ω;R )
Ω
≤ kg 0 (f 1 ) − g 0 (f 2 )kLrq/(q−r) (Ω;Rn×n ) khkLq (Ω;Rn )
for q ≥ r, with the convention that rq/(q −r) = ∞ if q = r. Hence, we obtain by (3.7c)
0
G (f 1 ) − G0 (f 2 ) q
≤ kg 0 (f 1 ) − g 0 (f 2 )kLrq/(q−r) (Ω;Rn×n )
L(L (Ω;Rn ),Lr (Ω;Rn ))
≤ M 0 kf 1 − f 2 kLrq/(q−r) (Ω;Rn )
≤ C M 0 kf 1 − f 2 kLp (Ω;Rn )
if p ≥ rq/(q − r), i.e., 1/p + 1/q ≤ 1/r.
13
Differentiability of Implicit Functions
Wachsmuth
Solvability of the linearized equation. Next, we study the solvability of the linearized equation
− div(g 0 (∇y) ∇δy) = h
δy = 0
in Ω,
(3.10a)
on ∂Ω.
(3.10b)
By (3.7) we obtain |g 0 (∇y)| ≤ M and the uniform coercivity g 0 (∇y) z · z ≥ m |z|2Rn .
Hence, similarly to Lemma 3.1 we obtain a result on the linearized equation. By considering the adjoint of the linearized equation, we obtain even the solvability of the
linearized equation (3.10) for p ∈ [p00 , 2], where p00 is the exponent conjugate to p0 , i.e.
1/p0 + 1/p00 = 1.
Lemma 3.2. Let p0 > 2 be as above and y ∈ W01,p0 (Ω) be given. For every p ∈ [p00 , p0 ]
and h ∈ W −1,p (Ω), there exists a unique weak solution δy ∈ W01,p (Ω) of (3.10). Moreover,
the a-priori estimate
kδykW 1,p (Ω) ≤ L khkW −1,p (Ω)
0
holds for L > 0 given as in Lemma 3.1.
Here we use that the exponent p0 in Lemma 3.1 does not depend on the differential
operator itself, but only on its Lipschitz and strong monotonicity constants (i.e. M and
m), see also [Gröger, 1989, Thm. 1].
First derivative of the solution mapping. Now, we are going to apply Theorem 2.1
to (3.6). Let p0 be given and choose p ∈ (2, p0 ). We define the spaces
Y + = W01,p0 (Ω),
U = L2 (Ω),
Y 0 = W01,p (Ω),
Z 0 = W −1,p (Ω).
The function e : Y 0 × U → Z 0 is given by
Z
he(y, u), ziW −1,p (Ω),W 1,p0 (Ω) =
g(∇y(x)) · ∇z(x) − u(x) z(x) dx
0
ZΩ
=
G(∇y)(x) · ∇z(x) − u(x) z(x) dx,
Ω
1,p0
where z ∈ W0 (Ω) is an arbitrary test function. Since p0 > p, the Nemytskii operator
G : Lp0 (Ω; Rn ) → Lp (Ω; Rn ) is Fréchet differentiable. Therefore, e : Y + × U → Z 0 is
14
Differentiability of Implicit Functions
Wachsmuth
Fréchet differentiable and its derivative is given by
he(y,u) (y, u)(δy, δu), ziW −1,p (Ω),W 1,p0 (Ω)
0
Z
0
=
G (∇y) ∇δy (x) · ∇z(x) − δu(x) z(x) dx
Ω
Z
H(∇y)(x) ∇δy(x) · ∇z(x) − δu(x) z(x) dx
=
ZΩ
0
g (∇y(x)) ∇δy(x) · ∇z(x) − δu(x) z(x) dx.
=
Ω
Since |g 0 (·)| ≤ M , the derivative e(y,u) (y, u) can be expanded to an element of L(Y 0 ×
U, Z 0 ) for all y ∈ Y + , u ∈ U . By Lemma 3.2, ey (y, u) : Y 0 → Z 0 is bijective for all
y ∈ Y + and u ∈ U . By Lemma 3.1, we infer the existence of the solution mapping
S : U → Y + . Therefore, the application of Theorem 2.1 yields
Theorem 3.3. Let Ω ⊂ Rn , n ∈ {2, 3} be a Lipschitz domain. Suppose that g satisfies
(3.7a) and (3.7b). Then there exists p > 2 such that the solution operator S : L2 (Ω) →
W 1,p (Ω) of (3.6) is Fréchet differentiable.
Continuity of the first derivative of the solution mapping. In order to apply
Corollary 2.2 we have to choose spaces Y −/2 , Z −/2 , such that additionally the derivatives
ey (y, u) can be expanded to elements of L(Y −/2 , Z −/2 ) with uniformly bounded inverses
and that e(y,u) : Y + × U → L(Y 0 × U, Z −/2 ) is continuous. Since eu is constant, it is
sufficient to study the continuity of ey .
Let q ∈ [2, p) be given. We define
Y −/2 = W01,q (Ω),
Z −/2 = W −1,q (Ω).
We have
Z
hey (y, u) δy, ziZ −/2 ,(Z −/2 )? =
Ω
(g 0 (∇y) ∇δy) ∇z dx = hH(∇y) ∇δy, ∇ziZ −/2 ,(Z −/2 )? .
Since H maps Lp0 (Ω; Rn ) continuously into Lpq/(p−q) (Ω; Rn ) (recall q < p), ey : Y + ×
U → L(Y 0 , Z −/2 ) is continuous. By Lemma 3.2, ey (y, u) ∈ L(Y −/2 , Z −/2 ) is bijective
and its inverse is bounded by L.
Hence, we can apply Corollary 2.2 and obtain the continuity of S 0 : U → Y −/2 .
Second derivative of the solution mapping. In order to apply Theorem 2.3, we
have to choose spaces Y − , Z − , such that additionally
(i) ey : Y + × U → L(Y −/2 , Z − ) is Lipschitz continuous at (y0 , u0 ),
15
Differentiability of Implicit Functions
Wachsmuth
(ii) e : Y 0 × U → Z − is twice Fréchet differentiable at (y0 , u0 ),
(iii) ey (y0 , u0 ) ∈ L(Y 0 , Z 0 ) can be extended to an element of L(Y − , Z − ), and ey (y0 , u0 ) ∈
L(Y − , Z − ) is bijective.
In order to satisfy the third requirement, we choose r > 1 and set
Y − = W01,r (Ω),
Z − = W −1,r (Ω).
Depending on r, (iii) may follow from Lemma 3.2. For the other two requirements (i)
and (ii), we simply need 1/p0 + 1/q ≤ 1/r and 2 ≤ 2 r < p, i.e., 2/p < 1/r.
We distinguish the cases p0 > 3 and p0 ∈ (2, 3].
In the case that p0 > 3, we could choose p ∈ (3, p0 ) and q < 3 such that 1/p0 +1/q = 2/3.
Finally, we choose r ∈ [p00 , 3/2]. In this case, Lemma 3.2 ensures the solvability of the
linearized equation w.r.t. the spaces Z − and Y − .
Applying Theorem 2.3 yields that
Theorem 3.4. Let Ω ⊂ Rn , n ∈ {2, 3} be a Lipschitz domain. Suppose that g satisfies
(3.7) and that p0 , which given by Lemma 3.1, satisfies p0 > 3.
Then the solution operator S : L2 (Ω) → W 1,3/2 (Ω) of (3.6) is twice Fréchet differentiable.
Finally, we state a theorem applicable to the case p0 ∈ (2, 3]. In this case, one can
choose p ∈ (2, p0 ), q ∈ (2, p) and r > 1, such that 1/r ≥ 1/p0 + 1/q and 1/r > 2/p.
Therefore, one has to prove that the linearized equation (3.10) (with y = y0 ) has a
solution δy ∈ W01,r (Ω) for all h ∈ W −1,r (Ω). This may be possible by proving the
Hölder continuity of ∇y0 and applying [Troianiello, 1987, Thm. 3.16 (iv)] to the adjoint
equation.
Theorem 3.5. Let Ω ⊂ Rn , n ∈ {2, 3} be a Lipschitz domain. Suppose that g satisfies
(3.7) and that p0 , which given by Lemma 3.1, satisfies p0 ∈ (2, 3]. Let r ∈ (1, p0 /2)
be given and assume that the linearized PDE (3.10) (with y = y0 ) has a solution δy ∈
W01,r (Ω) for all h ∈ W −1,r (Ω).
Then the solution operator S : L2 (Ω) → W 1,r (Ω) of (3.6) is twice Fréchet differentiable.
Application to optimal control problems Let us highlight one potential application
of these differentiability results. Since Y − = W01,r (Ω) ,→ L2 (Ω) for r ≥ 6/5, the solution
map of the quasilinear PDE (3.6) is twice differentiable from L2 (Ω) to L2 (Ω). Let
f : L2 (Ω) × L2 (Ω) → R be twice differentiable. Then the twice differentiability of S
implies that one can apply second order (necessary and sufficient) optimality conditions
to the problem
Minimize j(u) = f (S(u), u).
16
Differentiability of Implicit Functions
Wachsmuth
The second order sufficient condition yields that if j 0 (ū) = 0 and j 00 (ū)[h, h] ≥ κ khk2L2 (Ω)
for some κ > 0, then ū is a strict local minimizer and there exist δ, ε > 0, such that
j(u) ≥ j(ū) + δ ku − ūk2L2 (Ω)
for all u ∈ L2 (Ω) satisfying ku − ūkL2 (Ω) ≤ ε, see Maurer and Zowe [1979]. Note that due
to the assumed differentiability of f : L2 (Ω) × L2 (Ω), no two-norms discrepancy occurs
here.
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