Differentiability of Implicit Functions Beyond the Implicit Function Theorem

Transcription

1 Preprint Differentiability of Implicit Functions Beyond the Implicit Function Theorem Gerd 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 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 e y (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 Chemnitz, Germany, gerd.wachsmuth@mathematik.tu-chemnitz.de, part_dgl/wachsmuth

2 1 Intro We deal with the differentiability of a mapping S : u 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 , [Kantorovich and Akilov, 1964, Sec. XVII.4.2 or [Zeidler, 1995, Sec 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 (y 0, u 0 ) Y U with e(y 0, u 0 ) = 0 be given. Suppose that e : Y U Z is continuously Fréchet differentiable in a neighborhood of (y 0, u 0 ) and that the partial derivative e y (y 0, u 0 ) L(Y, Z) is bijective (i.e. continuously invertible). Then there exists a mapping S : U Y, and r y, r u > 0, such that for all (y, u) B ry (y 0 ) B ru (u 0 ) the statements y = S(u) and e(y, u) = 0 are equivalent. Moreover, the mapping S : U Y is Fréchet differentiable in B ru (u 0 ) and its derivative is given by S (u) = [ e y (S(u), u) 1 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, see [Tröltzsch, 2010, Thm. 4.24, and to the quasilinear PDE y = 0 on, n div [ a(y) y + d(y) = u in, y = 0 on, 2

3 see [Casas and Tröltzsch, 2009, Thm 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 : R n R n be a differentiable function which is not affine. We will consider the quasilinear PDE div [ g( y) = u in, y = 0 on, with u L 2 () and y W 1,p 0 () for some p (1, ). Under a growth condition for g, see Section 3 for details, we have g( y) L p (; R n ) if (and only if) y W 1,p 0 (). The weak formulation reads e(y, u), v g( y) v u v dx = 0 for all v W 1,q 0 (), W 1,q (),W 1,q 0 () = where q is the exponent conjugate to q (1, ). Note that we require 1/p+1/q 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 = W 1,p 0 (), U = L 2 (), Z = W 1,q () = W 1,q 0 (). 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 : L p (; R n ) L q (; R n ) is differentiable if and only if q < p. Therefore, in order to obtain the Fréchet differentiability of e(, u 0 ) : W 1,p 0 () W 1,q (), we have to assume q < p. The next step is to prove that e y (y 0, u 0 ) L(Y, Z) is bijective. This amounts to the solvability of the (weak formulation of the) linearized equation div [ g ( y 0 ) δy = δz in, δy = 0 on w.r.t. δy Y = W 1,p 0 () for all δz Z = W 1,q (). 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 e y (y 0, u 0 ) 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 = W 1, 0 () U = L 2 () Z = W 1, () = W 1,1 0 (). However, this requires that the quasilinear PDE and its linearization possess solutions in the space W 1, 0 () which may be difficult to achieve. 3

4 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 e y (y 0, u 0 ). Finally, we apply this theory to a class of quasilinear PDEs in Section 3, see Theorems 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 W 1,p 0 () and L p (), respectively. By p we denote the exponent conjugate to p. Moreover, we use the usual notation H0 1() = W 1,2 0 (), W 1,p () = W 1,p 0 (), and H 1 () = H0 1(). For n 1, we denote by R n, R n n and R n 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 R n and its associated operator norms are denoted by. The partial derivatives of a function e : (y, u) e(y, u) w.r.t. y and u are denoted by e y and e u, 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 e yy, e yu, e uy, and e uu. For the results below, it is sufficient that e is defined in a neighborhood of (y 0, u 0 ). 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 e y (y 0, u 0 ). 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 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

5 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 (y 0, u 0 ) Y + U such that e(y 0, u 0 ) = 0. We assume that e : Y + U Z 0 is Fréchet differentiable at (y 0, u 0 ), the partial derivative e y (y 0, u 0 ) L(Y +, Z 0 ) can be extended to an element of L(Y 0, Z 0 ), and e y (y 0, u 0 ) 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 u 0, such that e(s(u), u) = 0 for all u U and that S is Lipschitz continuous at u 0, i.e., S(u) S(u 0 ) Y + L u u 0 U for all u U. Then S : U Y 0 is Fréchet differentiable at u 0. The derivative S (u 0 ) is given in (2.1). Proof. We define S (u 0 ) h = e y ( ) 1 e u ( ) h (2.1) for h U. Here and in what follows, we abbreviate the argument (S(u 0 ), u 0 ) = (y 0, u 0 ) by ( ). By the assumptions of the theorem, we have S (u 0 ) L(U, Y 0 ). It remains to prove the estimate S(u 0 + h) S(u 0 ) S (u 0 ) h Y 0 = o( h U ) as h U 0. We have e y ( ) [ S(u 0 + h) S(u 0 ) S (u 0 ) h = e(s(u 0 + h), u 0 + h) e( ) e y ( ) [ S(u 0 + h) S(u 0 ) e u ( ) h (2.2) for all h U with h U small, i.e., u 0 + h U. Note that we have used e(s(u 0 + h), u 0 + h) = e( ) = 0. By the Fréchet differentiability of e, the right-hand side satisfies e(s(u0 + h), u 0 + h) e( ) e y ( ) [ S(u 0 + h) S(u 0 ) e u ( ) h Z 0 = o ( S(u 0 + h) S(u 0 ) Y + + h U ) as S(u 0 + h) S(u 0 ) in Y + and h 0 in U. Since S : U Y + is Lipschitz continuous at u 0, we can replace the right-hand side by o ( h U ) as h 0 in U. Together with (2.2) and e y ( ) 1 L(Z 0, Y 0 ), we have S(u 0 + h) S(u 0 ) S (u 0 ) h Y 0 = o( h U ) as h 0 in U. Hence, S (u 0 ) is the Fréchet derivative of S : U Y 0 at u 0. We remark that if the assumptions of Theorem 2.1 are satisfied, we have S (u 0 ) L(U,Y 0 ) L C, where L is the local Lipschitz constant of S : U Y + and C is the embedding constant satisfying y Y 0 C y Y + for all y Y +. Next, we address the question of continuity of S. If the assumptions of Theorem 2.1 are satisfied in a neighborhood of (y 0, u 0 ), then S is Fréchet differentiable in a neighborhood 5

6 of u 0. Moreover, the derivative S 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 e y : 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 e y. 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 (y 0, u 0 ) Y + U such that e(y 0, u 0 ) = 0. We assume that there are neighborhoods Y, U of y 0, u 0 in Y +, U, respectively, such that e : Y + U Z 0 is Fréchet differentiable in Y U, the partial derivative e y (y, u) L(Y +, Z 0 ) can be extended to an element of L(Y 0, Z 0 ) and e y (y, u) L(Y 0, Z 0 ) is bijective for every (y, u) Y U, the partial derivative e y (y, u) L(Y +, Z 0 ) can be extended to an element of L(Y /2, Z /2 ) and e y (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., S(v) S(u) Y + L(u) v u U for all u, v U. Then S : U Y 0 is Fréchet differentiable and the function S : 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 : U L(U, Y /2 ). By assumption we have Using the identities e y (S(v), v) 1 L(Z /2,Y /2 ) C for all v U. (2.3) e y (S(u), u) S (u) h + e u (S(u), u) h = 0 e y (S(v), v) S (v) h + e u (S(v), v) h = 0 and we infer 0 = e y (S(v), v) [ S (u) S (v) h + [ e u (S(u), u) e u (S(v), v) h + [ e y (S(u), u) e y (S(v), v) S (u) h 6

7 for all h U and all u, v U. By the assumptions we have e u (S(v), v) e u (S(u), u) in L(U, Z /2 ) e y (S(v), v) e y (S(u), u) in L(Y 0, Z /2 ) as v u in U (and hence, S(v) S(u) in Y + ). S (v) S (u) in L(U, Y /2 ) as v u in U. Together with (2.3) this shows 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, we obtain the existence of S 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 e y : Y + U L(Y /2, Z ) is Lipschitz continuous at (y 0, u 0 ), e : Y 0 U Z is twice Fréchet differentiable at (y 0, u 0 ), e y (y 0, u 0 ) L(Y 0, Z 0 ) can be expanded to an element of L(Y, Z ), and e y (y 0, u 0 ) L(Y, Z ) is bijective. Then S : U Y is twice Fréchet differentiable at u 0. The second derivative S (u 0 ) is given in (2.4). Proof. Let h 1, h 2 U be given. We define the second derivative S (u 0 )[h 1, h 2 by S (u 0 )[h 1, h 2 = e y ( ) 1( e yy ( ) [ S (u 0 ) h 1, S (u 0 ) h 2 + eyu ( ) [ S (u 0 ) h 1, h 2 + e uy ( ) [ h 1, S (u 0 ) h 2 + euu ( ) [ h 1, h 2 ). (2.4) Note that the bilinear form S (u 0 ) maps U U continuously (i.e. boundedly) to Y by the assumptions on e. It remains to show the estimate S (u 0 + h 1 ) S (u 0 ) S (u 0 )[h 1, L(U,Y ) = o( h 1 U ) as h 1 0 in U. By the definition of the operator norm in L(U, Y ), this is equivalent to S (u 0 + h 1 ) h 2 S (u 0 ) h 2 S (u 0 )[h 1, h 2 Y = h 2 U o( h 1 U ) for all h 2 U and as h 1 0 in U. 7

8 Similarly to the proof of Theorem 2.1, we start with e y ( ) [ S (u 0 + h 1 ) h 2 S (u 0 ) h 2 S (u 0 )[h 1, h 2 = e y (S(u 0 + h 1 ), u 0 + h 1 ) S (u 0 + h 1 ) h 2 e y ( ) S (u 0 + h 1 ) h 2 e y (S(u 0 + h 1 ), u 0 + h 1 ) S (u 0 + h 1 ) h 2 + e y ( ) S (u 0 ) h 2 e yy ( ) [ S (u 0 ) h 1, S (u 0 ) h 2 euy ( ) [ h 1, S (u 0 ) h 2 e yu ( ) [ S (u 0 ) h 1, h 2 euu ( ) [ h 1, h 2. (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 e y (S(u 0 + h 1 ), u 0 + h 1 ) S (u 0 + h 1 ) h 2 e y ( ) S (u 0 + h 1 ) h 2 e yy ( ) [ S (u 0 ) h 1, S (u 0 ) h 2 euy ( ) [ h 1, S (u 0 ) h 2. (2.6) We find that this term is equal to { e y (S(u 0 + h 1 ), u 0 + h 1 ) e y ( ) e yy ( ) [ S(u 0 + h 1 ) S(u 0 ), e uy ( ) [ h 1, } S (u 0 ) h 2 + e yy ( ) [ S(u 0 + h 1 ) S(u 0 ) S (u 0 ) h 1, S (u 0 ) h 2 + [ e y (S(u 0 + h 1 ), u 0 + h 1 ) e y ( ) [ S (u 0 + h 1 ) S (u 0 ) h 2. The L(Y 0, Z )-norm of the term in curly brackets is of order o( h 1 U ) by the assumptions on e and S. The Z -norm of the second line is also bounded by h 2 U o( h 1 U ) in virtue of the differentiability of S : U Y 0, and since e yy ( ) 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 h 1 U by the Lipschitz continuity of e y, whereas the L(U, Y /2 )-norm of the second term in square brackets goes to zero as h 1 0 by Corollary 2.2. Therefore, we have shown that the Z -norm of (2.6) is bounded by h 2 U o( h 1 U ). 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 e y (S(u 0 + h 1 ), u 0 + h 1 ) S (u 0 + h 1 ) h 2 + e y ( ) S (u 0 ) h 2 e yu ( ) [ S (u 0 ) h 1, h 2 euu ( ) [ h 1, h 2. (2.7) By the definition (2.1) of S (u), this is equal to [ e u (S(u 0 + h 1 ), u 0 + h 1 ) e u ( ) e yu ( ) [ S (u 0 ) h 1, e uu ( ) [ h 1, h 2. The Z -norm of this term is bounded by h 2 U o( h 1 U ) due to the differentiability of e u and the Lipschitz continuity of S. Altogether, we have proven the following estimate for (2.5): e y ( ) [ S (u 0 + h 1 ) h 2 S (u 0 ) h 2 S (u 0 )[h 1, h 2 Z = h 2 U o( h 1 U ). 8

9 By the invertibility of e y ( ) L(Y, Z ), this yields S (u 0 + h 1 ) S (u 0 ) S (u 0 )[h 1, L(U,Y ) = o( h 1 U ). Hence, S : U Y is twice Fréchet differentiable at u 0. 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 [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. [ 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 H0 1() of an elastic membrane Rn which is clamped at the boundary. 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 1 2 y 2 dx u y dx w.r.t. y H 1 0 (), where u L2 () is an external force, under the constraint (3.1a) y 1 a.e. in, (3.1b) where denotes the (Euclidean) norm in R n. Such problems occur for elastic-plastic beams with cross section 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 H0 1 () onto the set 9

10 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 Minimize 1 2 y 2 dx u y dx + γ 2 max ( y 1, 0 ) 2 dx (3.2) where γ > 0 is a penalty parameter (finally, one would drive γ ). The first order necessary conditions of (3.2) are given by ( ) y v u v dx + γ max ( y 1, 0 ) y v dx for all v H0 1 (). (3.3) y Since the objective in (3.2) is (strictly) convex, (3.3) is even sufficient for the optimality of y H0 1 (). 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 max ε ( ) C 2 (R) of max(0, ) satisfying max ε (x) = max(x, 0) where ε > 0. For instance, the function for all x R \ ( ε, +ε), 0 max ε(x) 1 for all x R, 0 max ε(x) ε 1 for all x R, max ε(x) = 1 ε max ( 0, min(ε x, ε + x) ) x t, max ε (x) = max ε(s) ds dt satisfies these assumptions. Now, we consider the regularized problem ( ) ( y v u v dx + γ max ε 1 1 ) y v dx for all v H0 1 (). (3.4) y Note that (3.4) is not necessarily the optimality condition for a regularized version of (3.2). By defining g : R n R n, g(z) = ( [1 + γ max ε 1 1 ) z, (3.5) z we find that (3.4) is the weak formulation of the quasilinear equation (3.6) which is dealt with in the next section. 10

11 3.2 Quasilinear Elliptic PDEs Let g C 2 (R n, R n ) and a domain R n, 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 L 2 () u y of the quasilinear PDE div g( y) = u in, (3.6a) y = 0 on. (3.6b) A function y H0 1 () is called a weak solution of (3.6) if and only if g( y) v dx = u v dx holds for all v H 1 0 (). In order to apply the integrability results of Gröger [1989 and those of Section 2, we assume that the domain 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, such that for all z, y R n the conditions (g(z) g(y)) (z y) m z y 2 R n, g(z) g(y) R n M z y R n, (3.7a) (3.7b) g (z) g (y) R n n M z y R n (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 L 2 () by assuming u W 1,p (). Lemma 3.1. For every u H 1 (), there exists a unique weak solution y(u) of (3.6). Moreover, there exists p 0 > 2, depending only on, m and M, such that for all p [2, p 0 and u W 1,p () we have y(u) W 1,p 0 () and y(u 2 ) y(u 1 ) W 1,p 0 () L u 2 u 1 W 1,p () for all u 1, u 2 W 1,p (). The Lipschitz constant L > 0 depends only on, m, M and p 0. 11

12 Proof. Due to (3.7) and Friedrichs inequality, the operator y div(g( y)) : H 1 0 () H 1 () = H 1 0 () is strongly monotone. Using the Browder-Minty theorem we infer the unique solvability of (3.6) and the a-priori estimate y(u) H 1 0 () L 1 u H 1 (), where L 1 depends only on m and the constant of Friedrichs inequality. Obviously, y(u i ), i = 1, 2, is also the weak solution of div g( y) + m y = u i + m y(u i ) in, y = 0 on. Now we can apply [Gröger, 1989, Thm. 1 and obtain the existence of p 0 > 2 and L 2 > 0, such that for all p [2, p 0 and u i W 1,p () we have y(u i ) W 1,p 0 () and y(u 2 ) y(u 1 ) W 1,p 0 () L ( ) 2 y(u2 ) y(u 1 ) W 1,p () + u 2 u 1 W 1,p (), where L 2 depends only on, m and M. Hence, by the embedding H 1 0 () W 1,p () we have y(u 2 ) y(u 1 ) W 1,p 0 () L u 2 u 1 W 1,p (). for all p [2, p 0 and L depends only on, m, M and p 0. In what follows, we fix p 0 6 such that Lemma 3.1 is satisfied. This, in particular, implies that L 2 () W 1,p 0 () (since n 3) and, therefore, we have a unique weak solution y W 1,p 0 () of (3.6) for all u L 2 (). 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, the Caratheodory condition (see [Goldberg et al., 1992, Sec. 2.1) reduces to the continuity w.r.t. z R n. Hence, g and its derivatives g and g satisfy the Caratheodory condition. We define the Nemytskii operator G which maps f L p (; R n ) 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 L p (; R n ) continuously into itself, for arbitrary p [1,, see [Goldberg et al., 1992, Sec Similarly, we define the Nemytskii operators H : L p (; R n ) L q (; R n n ) and I : L p (; R n ) L q (; R n n n ) associated with g and g. Due to (3.7b) and (3.7c), g and g satisfy the uniform boundedness condition [Goldberg et al., 1992, (UB3). Hence, H 12

13 and I are continuous from L p (; R n ) to L q (; R n n ) and L q (; R n n n ), respectively, for all p [1, and q [1, ), see [Goldberg et al., 1992, Sec 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 is continuously Fréchet differentiable and G : L p (; R n ) L q (; R n ) (3.8) G : L r (; R n ) L s (; R n ) (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 Their derivatives are given by (G (f) h)(x) = (H(f)(x)) h(x), (G (f)[h 1, h 2 )(x) = (I(f)(x)) [h 1 (x), h 2 (x), G (f) h L q (; R n ), G (f)[h 1, h 2 L s (; R n ), for almost all x, respectively. Here, f, h L p (; R n ) and f, h 1, h 2 L r (; R n ), respectively. Lipschitz continuity of the Nemytskii operator G. For the application of Theorem 2.3, we need to address the Lipschitz continuity of the operator G : L p (; R n ) L(L q (; R n ), L r (; R n )) for certain exponents p, q, r. We have G (f 1 ) h G (f 2 ) h ( L r (;R n ) = g (f 1 (x)) g (f 2 (x)) ) r 1/r h(x) r dx g (f 1 ) g (f 2 ) L rq/(q r) (;R n n ) h L q (;R n ) for q r, with the convention that rq/(q r) = if q = r. Hence, we obtain by (3.7c) G (f 1 ) G (f 2 ) L(L q (;R n ),L r (;R n )) g (f 1 ) g (f 2 ) L rq/(q r) (;R n n ) if p rq/(q r), i.e., 1/p + 1/q 1/r. M f 1 f 2 L rq/(q r) (;R n ) C M f 1 f 2 L p (;R n ) 13

14 Next, we study the solvability of the lin- Solvability of the linearized equation. earized equation div(g ( y) δy) = h in, (3.10a) δy = 0 on. (3.10b) By (3.7) we obtain g ( y) M and the uniform coercivity g ( y) z z m z 2 R n. 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 [p 0, 2, where p 0 is the exponent conjugate to p 0, i.e. 1/p 0 + 1/p 0 = 1. Lemma 3.2. Let p 0 > 2 be as above and y W 1,p 0 0 () be given. For every p [p 0, p 0 and h W 1,p (), there exists a unique weak solution δy W 1,p 0 () of (3.10). Moreover, the a-priori estimate δy W 1,p 0 () L h W 1,p () holds for L > 0 given as in Lemma 3.1. Here we use that the exponent p 0 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 p 0 be given and choose p (2, p 0 ). We define the spaces Y + = W 1,p 0 0 (), U = L 2 (), Y 0 = W 1,p 0 (), Z 0 = W 1,p (). The function e : Y 0 U Z 0 is given by e(y, u), z = W 1,p (),W 1,p 0 () = g( y(x)) z(x) u(x) z(x) dx G( y)(x) z(x) u(x) z(x) dx, where z W 1,p 0 () is an arbitrary test function. Since p 0 > p, the Nemytskii operator G : L p 0 (; R n ) L p (; R n ) is Fréchet differentiable. Therefore, e : Y + U Z 0 is 14

15 Fréchet differentiable and its derivative is given by e (y,u) (y, u)(δy, δu), z W 1,p (),W 1,p 0 () [ = G ( y) δy (x) z(x) δu(x) z(x) dx [( ) = H( y)(x) δy(x) z(x) δu(x) z(x) dx [ = g ( y(x)) δy(x) z(x) δu(x) z(x) dx. Since g ( ) 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, e y (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 R n, 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 : L 2 () W 1,p () 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 e y (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 e u is constant, it is sufficient to study the continuity of e y. Let q [2, p) be given. We define We have e y (y, u) δy, z Z /2,(Z /2 ) = Y /2 = W 1,q 0 (), Z /2 = W 1,q (). (g ( y) δy) z dx = H( y) δy, z Z /2,(Z /2 ). Since H maps L p 0 (; R n ) continuously into L pq/(p q) (; R n ) (recall q < p), e y : Y + U L(Y 0, Z /2 ) is continuous. By Lemma 3.2, e y (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 : 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) e y : Y + U L(Y /2, Z ) is Lipschitz continuous at (y 0, u 0 ), 15

16 (ii) e : Y 0 U Z is twice Fréchet differentiable at (y 0, u 0 ), (iii) e y (y 0, u 0 ) L(Y 0, Z 0 ) can be extended to an element of L(Y, Z ), and e y (y 0, u 0 ) L(Y, Z ) is bijective. In order to satisfy the third requirement, we choose r > 1 and set Y = W 1,r 0 (), Z = W 1,r (). Depending on r, (iii) may follow from Lemma 3.2. For the other two requirements (i) and (ii), we simply need 1/p 0 + 1/q 1/r and 2 2 r < p, i.e., 2/p < 1/r. We distinguish the cases p 0 > 3 and p 0 (2, 3. In the case that p 0 > 3, we could choose p (3, p 0 ) and q < 3 such that 1/p 0 +1/q = 2/3. Finally, we choose r [p 0, 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 R n, n {2, 3} be a Lipschitz domain. Suppose that g satisfies (3.7) and that p 0, which given by Lemma 3.1, satisfies p 0 > 3. Then the solution operator S : L 2 () W 1,3/2 () of (3.6) is twice Fréchet differentiable. Finally, we state a theorem applicable to the case p 0 (2, 3. In this case, one can choose p (2, p 0 ), q (2, p) and r > 1, such that 1/r 1/p 0 + 1/q and 1/r > 2/p. Therefore, one has to prove that the linearized equation (3.10) (with y = y 0 ) has a solution δy W 1,r 0 () for all h W 1,r (). This may be possible by proving the Hölder continuity of y 0 and applying [Troianiello, 1987, Thm (iv) to the adjoint equation. Theorem 3.5. Let R n, n {2, 3} be a Lipschitz domain. Suppose that g satisfies (3.7) and that p 0, which given by Lemma 3.1, satisfies p 0 (2, 3. Let r (1, p 0 /2) be given and assume that the linearized PDE (3.10) (with y = y 0 ) has a solution δy W 1,r 0 () for all h W 1,r (). Then the solution operator S : L 2 () W 1,r () of (3.6) is twice Fréchet differentiable. Application to optimal control problems Let us highlight one potential application of these differentiability results. Since Y = W 1,r 0 () L2 () for r 6/5, the solution map of the quasilinear PDE (3.6) is twice differentiable from L 2 () to L 2 (). Let f : L 2 () L 2 () 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

17 The second order sufficient condition yields that if j (ū) = 0 and j (ū)[h, h κ h 2 L 2 () for some κ > 0, then ū is a strict local minimizer and there exist δ, ε > 0, such that j(u) j(ū) + δ u ū 2 L 2 () for all u L 2 () satisfying u ū L 2 () ε, see Maurer and Zowe [1979. Note that due to the assumed differentiability of f : L 2 () L 2 (), no two-norms discrepancy occurs here. References A. Bermúdez. A mixed method for the elastoplastic torsion problem. IMA Journal of Numerical Analysis, 2: , H. Cartan. Calcul différentiel. Hermann, Paris, E. Casas and L. A. Fernández. Distributed control of systems governed by a general class of quasilinear elliptic equations. Journal of Differential Equations, 104(1):20 47, ISSN doi: /jdeq E. Casas and F. Tröltzsch. First- and second order optimality conditions for a class of optimal control problems with quasilinear ellitpic equations. SIAM Journal on Control and Optimization, 48(2): , doi: / J.C. de los Reyes. Optimal control of a class of variational inequalities of the second kind. SIAM Journal on Control and Optimization, 49(4): , doi: / L. C. Evans. A second order elliptic equation with gradient constraint. Communications in Partial Differential Equations, 4(5): , doi: / H. Goldberg, W. Kampowsky, and F. Tröltzsch. On Nemytskij operators in L p -spaces of abstract functions. Mathematische Nachrichten, 155: , ISSN X. doi: /mana K. Gröger. A W 1,p -estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Mathematische Annalen, 283: , doi: /BF R. Haller-Dintelmann, C. Meyer, J. Rehberg, and A. Schiela. Hölder continuity and optimal control for nonsmooth elliptic problems. Applied Mathematics and Optimization, 60: , doi: /s x. R. Herzog, C. Meyer, and G.. Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions. Journal of Mathematical Analysis and Applications, 382(2): , doi: /j.jmaa

18 R. Herzog, C. Meyer, and G.. C-stationarity for optimal control of static plasticity with linear kinematic hardening. SIAM Journal on Control and Optimization, 50(5): , doi: / T. H. Hildebrandt and Lawrence M. Graves. Implicit functions and their differentials in general analysis. Transactions of the American Mathematical Society, 29(1): , ISSN doi: / G. Idone, A. Maugeri, and C. Vitanza. Variational inequalities and the elastic-plastic torsion problem. Journal of Optimization Theory and Applications, 117(3): , ISSN doi: /A: L.V. Kantorovich and G.P. Akilov. Funktionalanalysis in normierten Räumen. Akademie- Verlag, H. Maurer and J. Zowe. First and second order necessary and sufficient optimality conditionsfor infinite-dimensional programming problems. Mathematical Programming, 16(1):98 110, doi: /BF G. Troianiello. Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York, F. Tröltzsch. Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels. G.. Optimal control of quasistatic plasticity with linear kinematic hardening, part II: Regularization and differentiability. submitted, E. Zeidler. Applied Functional Analysis: Main Principles and their Applications. Springer, New York,