DUALITY BASED A POSTERIORI ERROR ESTIMATES FOR HIGHER ORDER VARIATIONAL INEQUALITIES

Annales Academiæ Scientiarum Fennicæ Mathematica

Volumen 33, 2008, 475–490

DUALITY BASED A POSTERIORI ERROR ESTIMATES FOR HIGHER ORDER VARIATIONAL INEQUALITIES

WITH POWER GROWTH FUNCTIONALS

Michael Bildhauer, Martin Fuchs and Sergey Repin

Universität des Saarlandes, Fachbereich 6.1 Mathematik

Postfach 15 11 50, D-66041 Saarbrücken, Germany; [email protected] Universität des Saarlandes, Fachbereich 6.1 Mathematik

Postfach 15 11 50, D-66041 Saarbrücken, Germany; [email protected] V. A. Steklov Mathematical Institute, St. Petersburg Branch Fontanka 27, 191011 St. Petersburg, Russia; [email protected]

Abstract. We consider variational inequalities of higher order with p-growth potentials over a domain in the plane by the way including the obstacle problem for a plate with power hardening law. Using duality methods we prove a posteriori error estimates of functional type for the difference of the exact solution and any admissible comparision function.

1. Introduction

On a bounded Lipschitz domain Ω⊂R² we consider the minimization problem

(P) J[u] :=

Z

Ω

π_p(∇²u)dx→min inK,

where the classKconsists of all functions v from the spaceW^◦2_p(Ω) s.t.v(x)≥Ψ(x) onΩ, the potentialπ_p is given by the formulaπ_p(E) := ¹_p|E|^p for symmetric(2×2)- matrices E and ∇²u represents the matrix of the second generalized derivatives.

It is assumed that Ψ∈W_p²(Ω) is a given function s.t. Ψ|_∂Ω <0 and Ψ(x₀) >0 for some point x₀ ∈ Ω and the exponent p is arbitrarily chosen in the interval 1 < p < ∞. For a definition of the Sobolev spaces W^{◦ 2}_p(Ω), W_p²(Ω) and related classes we refer to [Ad].

We recall that by Sobolev’s embedding theorem the functionsΨandv ∈W_p²(Ω) have a representative inC⁰(Ω) and that this observation can be used to show that the class K is not empty (compare [FLM]), which means that (P) has a unique solutionu∈K.

The second order obstacle problem(P)is of some physical relevance: consider a plate which is clamped at the boundary and whose undeformed state is represented by the regionΩ. If some outer forces are applied acting in vertical direction, then the

2000 Mathematics Subject Classification: Primary 65N15, 65K10, 74K20, 49J40, 49M29.

Key words: A posteriori error estimates, higher order variational inequalities, duality meth- ods, power growth, elastic plates with obstacles.

equilibrium configuration can be found as a minimizer of an energy with principal

part Z

Ω

g(∇²w)dx,

where the mechanical properties of the plate are characterized by the given convex functiong. In the case of linear elastic plates, we haveg =π₂and since our exponent pis arbitrary we can include any power-hardening law. In particular, for p close to 1we have an approximation of perfectly plastic plates. The new feature of problem (P) however is that the plate has to respect the given side condition.

In various papers mainly the regularity properties of minimizers (with or without obstacle) for different functionsg have been investigated, we refer to [Se] for the case of plastic behaviour, whereas the case of nearly linear growth is studied in [FLM]

and [BF1]. For the “classical case” p= 2 we refer, e.g., to [Fr].

In the present note we concentrate on a posteriori error estimates of functional type for the solution of problem (P) by combining the methods developed by the third author for first order obstacle problems with quadratic potentials (see [Re]

and also [NR1], Chapter 8) with the techniques introduced in [BR] and [BFR] for unconstrained variational problems with a power growth potential. To be more precise, letu∈K denote the solution of problem (P) and consider any functionv from the class of comparison functions. Then our goal is to prove the estimate (1.1) k∇²u− ∇²vkL^p ≤M(v,D),

whereD stands for the set of known data and whereM is a non-negative functional depending on v, on the data of the problem such as p, Ω, Ψ and on “parameters”

which are under our disposal. M should satisfy the following requirements:

i.) the value of M is easy to calculate for any choice of an admissible function v;

ii.) the estimate is consistent in the sense that

M(v,D) = 0 if and only if v =u, moreover M(vk,D)→0 if k∇²vk− ∇²ukL^p →0;

iii.) M provides a realistic upper bound for the quantity k∇²u− ∇²vk_L^p. Of course iii.) means that for obtaining the bound (1.1) one carefully tries to avoid

“over-estimation” so that (1.1) can be used for a reliable verification of approximative solutions obtained by various numerical methods. As already outlined in [BR] and [BFR] the casesp≥2and1< p <2require different techniques: in Section 3 we will prove an estimate like (1.1) ifp≥2, which—without a priori estimates for the exact solution—we could not verify for the subquadratic case. Therefore, in Section 4, we pass to the dual variational problem (P)^∗ and discuss a variant of (1.1) involving the dual solutionσ^∗. We like to remark that just for the case of technical simplicity we did not include terms like R

Ωuf dx or R

Ω∇u·F dx with functions f: Ω → R, F: Ω→R² into our variational integralJ. These quantities can be added without substantial changes providedf and F satisfy suitable integrability assumptions. In

the same spirit we could include the double obstacle problemΨ≤u≤Φ combined with different boundary conditions being compatible with the obstacle(s).

It is worth noting that if we start from any functionΨ∈W_p²(Ω)and if we require that K is not empty, then all our calculations remain valid if Ω⊂R^d, d ≥2, with constants partially depending on d.

2. Preliminaries

Basic facts in duality theory. We recall some facts from duality theory (see, e.g. [ET]) valid for all 1 < p < ∞: if we let q := _p−1^p , Y^∗ := L^q(Ω;R^2×2_sym), Y :=L^p(Ω;R^2×2_sym) and

`(v, τ^∗) :=

Z

Ω

£τ^∗ :∇²v−π_p^∗(τ^∗)¤ dx,

whereπ_p^∗ is the conjugate function of πp, i.e. π_p^∗(E) = ¹_q|E|^q, then J[v] = sup

τ^∗∈Y^∗

`(v, τ^∗) and

J[u] =J^∗[σ^∗],

whereJ^∗[τ^∗] := inf_v∈K`(v, τ^∗),τ^∗ ∈Y^∗, is the dual functional,σ^∗ denotes its unique maximizer, and u∈Kis the unique solution of the problem (P).

Clarkson’s inequality. If p ≥ 2, then we will use a version of Clarkson’s inequality [Cl] presented in [MM] for tensor-valued functionsτ1, τ2 ∈Y:

(2.1)

Z

Ω

h¯¯τ₁+τ₂ 2

¯¯^p+¯

¯τ₁−τ₂ 2

¯¯^pi

dx ≤ 1

2kτ₁k^p_Lp+ 1

2kτ₂k^p_Lp. If p <2, then we apply this inequality to the dual variable.

Basic deviation estimate. In the superquadratic case, i.e. if p ≥ 2, (2.1) implies for allv ∈K

°°∇²(u−v)°

°^p

L^p ≤p2^p h1

2J[v] +1

2J[u]−J

hu+v 2

ii ,

and since ^u+v₂ ∈K we deduce from the J-minimality of u that

(2.2) °

°∇²(u−v)°

°^p

L^p ≤p2^p−1 h

J[v]−J[u]

i .

If p < 2, then the basic deviation estimate (2.2) takes the form (2.3) and is derived as follows: withv ∈K chosen arbitrarily we have

1

q2^1−qkτ^∗−σ^∗k^q_Lq ≤ 1 q

Z

Ω

|τ^∗|^qdx+1 q

Z

Ω

|σ^∗|^qdx− 2 q

Z

Ω

¯¯

¯τ^∗+σ^∗ 2

¯¯

¯^qdx

= Z

Ω

h1

q |τ^∗|^q−τ^∗ :∇²v i

dx+ Z

Ω

h1

q |σ^∗|^q−σ^∗ :∇²v i

dx

−2 Z

Ω

h1 q

¯¯

¯τ^∗+σ^∗ 2

¯¯

¯^q− σ^∗+τ^∗ 2 :∇²v

i dx

=−`(v, τ<sup>∗</sup>)−`(v, σ^∗) + 2`

³

v,τ^∗+σ^∗ 2

´

≤ sup

v1∈K

³

−`(v₁, τ^∗)

´

+ sup

v2∈K

³

−`(v₂, σ^∗)

´ + 2`

³

v,τ^∗+σ^∗ 2

´

=− inf

v1∈K`(v₁, τ^∗)− inf

v2∈K`(v<sub>2</sub>, σ<sup>∗</sup>) + 2`

³

v,τ^∗+σ^∗ 2

´

=−J^∗[τ^∗]−J^∗[σ^∗] + 2`

³

v,τ^∗+σ^∗ 2

´ ,

and sincev ∈K is under our disposal, we may pass to the inf w.r.t. v ∈K on the r.h.s. with the result

1

q 2^1−qkσ^∗−τ^∗k^q_Lq ≤ −J^∗[τ^∗]−J^∗[σ^∗] + 2J^∗

hσ^∗+τ^∗ 2

i , and theJ^∗-maximality of σ^∗ implies

(2.3) 1

q2^1−qkσ^∗−τ^∗k^q_Lq ≤J^∗[σ^∗]−J^∗[τ^∗].

A modified functional. Following [Re] we introduce a relaxtion of (P): for λ∈Λ :={ρ∈L^q(Ω) :ρ≥0 a.e.}

we let

(P_λ) J_λ[w] := J[w]− Z

Ω

λ(w−Ψ)dx→min in W^◦2p(Ω).

Clearly (P_λ)is well-posed with unique solution u_λ. Also we note that sup

λ∈Λ

J_λ[w] =J[w]− inf

λ∈Λ

Z

Ω

λ(w−Ψ)dx=

(J[w], if w∈K +∞, if w /∈K.

Letting

L(w, τ^∗, λ) :=

Z

Ω

£∇²w:τ^∗ −π_p^∗(τ^∗)−λ(w−Ψ)¤

dx, w ∈W^◦2_p(Ω), τ^∗ ∈Y^∗, λ∈Λ, we define the dual functional

J_λ^∗[τ^∗] := inf

w∈W^◦2p(Ω)

L(w, τ^∗, λ) with unique maximizer σ_λ^∗ and get

(2.4) J_λ[u_λ] = J_λ^∗[σ^∗_λ].

The reader should observe that J_λ^∗[τ^∗]> −∞ for τ^∗ ∈ Y^∗ implies that τ^∗ is in the class

Q^∗_λ :=

n

η∈Y^∗ : Z

Ω

£η^∗ :∇²w−λw¤

dx= 0 for all w∈W^◦2_p(Ω) o

, which means that in the distributional sense τ^∗ = (τ_αβ^∗ ) satisfies

div(divτ^∗) :=∂_α(∂_βτ_αβ^∗ ) = λ.

In this case we have

(2.5) J_λ^∗[τ^∗] =

Z

Ω

£−π^∗_p(τ^∗) +ψλ¤ dx.

We further note that

◦inf

W²p(Ω)

Jλ ≤inf

K Jλ = inf

v∈K

h Z

Ω

πp(∇²v)dx− Z

Ω

λ(v−Ψ)dx i

≤ inf

v∈K

Z

Ω

πp(∇²v)dx= inf

K J.

(2.6)

3. Estimates for the superquadratic case Let us now state our first result:

Theorem 3.1. Letp≥2. With the notation introduced above we have for any v ∈K, for any η^∗ ∈Y^∗, for all λ∈Λ and for any choice of β >0the estimate

°°∇²(u−v)°°^p

L^p ≤p2^p−1 n

D_p£

∇²v, η^∗¤ +£

2^2−q(3−q) + 1 qβ^−q¤

d(η^∗)^q +1

pβ^p°°|η^∗|^q−2η^∗− ∇²v°°^p

L^p+ Z

Ω

λ(v−Ψ)dx o

, (3.1)

where

Dp[ρ,κ^∗] :=

Z

Ω

£πp(ρ) +π^∗_p(κ^∗)−ρ:κ^∗¤ dx

forρ∈Y, κ^∗ ∈Y^∗ and

d(κ^∗) := inf

τ^∗∈Q^∗_λkκ^∗−τ^∗k_L^q.

If in addition div(divη^∗)∈L^q(Ω), then we have the inequality

(3.2) d(η^∗)≤C_p(Ω)°

°λ−div(divη^∗)°

°L^q. The constantC_p(Ω) is defined in formula (3.8).

Remark 3.1. Note that all the quantities on the r.h.s. of (3.1) are non-negative, and the r.h.s. of (3.1) vanishes if and only if

∇²v =|η^∗|^q−2η^∗, div(divη^∗) =λ, λ(v−Ψ) = 0.

Letw∈K. Then the validity of the above equations gives Z

Ω

|∇²v|^p−2∇²v :∇²(w−v)dx= Z

Ω

η^∗ :∇²(w−v)dx= Z

Ω

λ(w−v)dx

= Z

Ω

λ(w−Ψ)dx+ Z

Ω

λ(Ψ−v)dx = Z

Ω

λ(w−Ψ)dx≥0, which means that v is the unique solution of Problem(P).

If d is estimated via (3.2) then all the functions on the r.h.s. of (3.1) are either known or in our disposal. Thus (3.1) gives a practical way to measure the accuracy.

Having proved Theorem 3.1 we will give variants of (3.1) by optimizing the functionλ.

Proof of Theorem 3.1. We recall (2.6), i.e.

J[u]≥ inf

W◦²p(Ω)

J_λ,

so that according to (2.4)

J[u]≥J_λ^∗[σ^∗_λ]≥J_λ^∗[τ^∗]

for all choices of λ∈Λ and τ^∗ ∈Q^∗_λ. This gives in combination with (2.2)

(3.3) °

°∇²(u−v)°

°^p

L^p ≤p2^p−1£

J[v]−J_λ^∗[τ^∗]¤ .

By (2.5) we find that J[v]−J_λ^∗[τ^∗] =

Z

Ω

h

πp(∇²v) +π_p^∗(τ^∗)−Ψλ i

dx

= Z

Ω

h

π_p(∇²v) +π_p^∗(τ^∗)−τ^∗ :∇²v i

dx+ Z

Ω

λ(v−Ψ)dx

=D_p h

∇²v, τ^∗ i

+ Z

Ω

λ(v−Ψ)dx,

and according to (3.3) we have shown that (3.4) °°∇²(u−v)°°^p

L^p ≤p2^p−1 n

D_p[∇²v, τ^∗] + Z

Ω

λ(v−Ψ)dx o

valid for allv ∈K, τ^∗ ∈Q^∗_λ and λ ∈Λ.

Consider any tensor η^∗ ∈Y^∗. Then (3.4) and the convexity of π_p^∗ imply

°°∇²(u−v)°

°^p

L^p ≤p2^p−1n

D_p[∇²v, η^∗] + Z

Ω

£π^∗_p(τ^∗)−π^∗_p(η^∗)−(τ^∗−η^∗) :∇²v¤ dx +

Z

Ω

λ(v−Ψ)dx o

≤p2^p−1 n

D_p[∇²v, η^∗] + Z

Ω

h

|τ^∗|^q−2τ^∗− ∇²v i

: (τ^∗−η^∗)dx (3.5)

+ Z

Ω

λ(v−Ψ)dx o

=p2^p−1 n

D_p[∇²v, η^∗] + Z

Ω

£|τ^∗|^q−2τ^∗− |η^∗|^q−2η^∗¤

: (τ^∗−η^∗)dx +

Z

Ω

£|η^∗|^q−2η^∗− ∇²v¤

: (τ^∗−η^∗)dx+ Z

Ω

λ(v−Ψ)dx o

. As demonstrated in [BR] we have

Z

Ω

h

|τ^∗|^q−2τ^∗− |η^∗|^q−2η^∗ i

: (τ^∗−η^∗)dx≤2^2−q(3−q)kτ^∗ −η^∗k^q_Lq, moreover, from Hölder’s inequality it follows that

Z

Ω

£|η^∗|^q−2η^∗− ∇²v¤

: (τ^∗−η^∗)dx≤°

°|η^∗|^q−2η^∗− ∇²vk_L^pkτ^∗−η^∗k_L^q

≤ 1

pβ^p°°|η^∗|^q−2η^∗− ∇²v°°^p

L^p+ 1

qβ^−qkτ^∗−η^∗k^q_Lq, where in the last line we used Young’s inequality with some β >0. Inserting these estimates into (3.5) and taking the inf w.r.t. τ^∗ ∈Q^∗_λ, inequality (3.1) is proved.

To establish the second part of the theorem, we consider η^∗ ∈ Y^∗ with the property div(divη^∗)∈ L^q(Ω). Then inf_τ^∗_∈Q^∗_λ°°η^∗ −τ^∗°°

L^q is attained for some τ^∗ ∈ Q^∗_λ, and °

°η^∗ −τ^∗°

°L^q is a measure for the distance from η^∗ to Q^∗_λ. Letting λ :=

λ−∂_α∂_βη_αβ^∗ we find (with an obvious meaning of Q^∗_λ)

ρ^∗inf∈Q^∗_λ

1

q kρ^∗−η^∗k^q_Lq =− sup

κ^∗∈Q^∗_λ

h

− 1

qkκ^∗k^q_Lq

i , (3.6)

sup

κ^∗∈Q^∗_λ

h

−1

qkκ^∗k^q_Lq

i

= inf

w∈W^◦2p(Ω)

Z

Ω

h1 p

¯¯∇²w¯

¯^p−λw i

dx.

(3.7)

Forw∈W^◦2p(Ω) we have by Poincaré’s inequality

(3.8) kwk_L^p ≤C_p(Ω)k∇²wk_L^p

with a positive constantC_p(Ω) depending on p and Ω. Using (3.8) on the r.h.s. of (3.7), we see that the r.h.s. of (3.7) is bounded from below by

inft≥0

h1

pt^p −C_p(Ω)tkλk_L^qi , and this inf is attained at

t₀ :=

h

kλk_L^qC_p(Ω) i ¹

p−1.

From (3.6) we therefore get

¤

(3.9) inf

ρ^∗∈Q^∗_λkρ^∗−η^∗k_L^q =:d(η^∗)≤C_p(Ω)kλ−div(divη^∗)k_L^q.

Now we discuss two variants of how to choose λ ∈Λ in a suitable way.

Variant 1. Given v ∈ K and η^∗ ∈Y^∗ s.t. div(divη^∗)∈ L^q(Ω) we let (following [Re])

(λ = 0 on[v >Ψ], λ =£

div(divη^∗)¤

⊕ on[v = Ψ],

wheref⊕ := max(0, f),fª := min(0, f), hencef =f⊕+fªfor real-valued functions f. With this choice of λ we get

Z

Ω

λ(v−Ψ)dx= 0, moreover

Z

Ω

|λ−div(divη^∗)|^qdx= Z

[v>Ψ]

|div(divη^∗)|^qdx+ Z

[v=Ψ]

¯¯£

div(divη^∗)¤

ª

¯¯^qdx,

and we arrive at

Corollary 3.1. Let p ≥ 2. For any v ∈ K, for all η^∗ ∈ Y^∗ s.t. div(divη^∗) ∈ L^q(Ω) and for all β >0 we have with

K_p(Ω, β) := C_p^q(Ω) h

2^2−q(3−q) + 1 q β^−q

i

the estimate

°°∇²(v−u)°°^p

L^p ≤p2^p−1 (

D_p[∇²v, η^∗] +K_p(Ω, β)

" Z

[v>Ψ]

¯¯div(divη^∗)¯¯^qdx

+ Z

[v=Ψ]

¯¯¡

div(divη^∗)¢

ª

¯¯^qdx

# +1

pβ^p°

°|η^∗|^q−2η^∗ − ∇²v°

°^p

L^p

) . ¤ (3.10)

Assume that the r.h.s. of (3.10) is zero for some triple (v, η^∗, β). Then ∇²v =

|η^∗|^q−2η^∗ together with

div(divη^∗) = 0 on[v >Ψ], h

div(divη^∗) i

ª = 0 on[v = Ψ].

This implies for anyw∈K Z

Ω

|∇²v|^p−2∇²v :∇²(w−v)dx= Z

Ω

η^∗ :∇²(w−v)dx

= Z

Ω

div(divη^∗)(w−v)dx

= Z

[v=Ψ]

£div(divη^∗)¤

⊕(w−Ψ)dx≥0, and thereforev coincides with the solution u of (P).

Variant 2. As an alternative to (3.10) we again follow ideas of [Re] and estimate d(η^∗)^q on the r.h.s. of (3.1) by C_p^q(Ω)°

°λ−div(divη^∗)°

°^q

L^q (recall (3.2)) and then try to findλ∈Λ s.t.

Z

Ω

λ(v−Ψ)dx+K_p(Ω, β) Z

Ω

¯¯λ−div(divη^∗)¯¯^qdx

becomes minimal for a fixed triple (v, η^∗, β). Of course this can be achieved by pointwise minimization of the function

f(t) :=t¡

v(x)−Ψ(x)¢ +K¯

¯t−δ(x)¯

¯^q

on[0,∞). Here K :=K_p(Ω, β), δ := div(divη^∗), and in the following we will omit the fixed argumentx ∈Ω. Note that f is strictly convex on R and f(t)→+∞ as t→ ±∞, thus there is a unique numbert₀ ∈Rs.t. f(t₀) = inf_Rf. Fromf⁰(t₀) = 0 it follows that

0 = v−Ψ +Kq|t0−δ|^q−2(t0−δ), and since v ≥Ψ, we must have t0 ≤δ, hence

(δ−t₀)^q−1 = 1

Kq(v−Ψ), i.e. t₀ =δ− h 1

qK (v−Ψ) i ¹

q−1.

Case 1. t0 <0. Sincef is strictly increasing on [t0,∞), we get mint≥0 f(t) =f(0) =K|δ|^q.

Case 2. t0 ≥0. Then we have

mint≥0 f(t) = f(t₀)

For v ∈K,η^∗ ∈Y^∗ s.t. div(divη^∗)∈L^q(Ω) and β >0 we define the sets Ω⁺ :=

n

x∈Ω : div(divη^∗)(x)≥

h 1

qKp(Ω, β)

¡v(x)−Ψ(x)¢iq−1¹ o , Ω⁻ := Ω−Ω⁺

and setλ= 0onΩ⁻andλ=£ ₁

qKp(Ω,β)(v(x)−Ψ(x))¤ ¹

q−1 onΩ⁺. We further introduce the quantity

ε(v, η^∗, β) :=

Z

Ω⁻

K_p(Ω, β)|div(divη^∗)|^qdx +

Z

Ω⁺

h

(v−Ψ)

³

div(divη^∗)−

n v−Ψ qKp(Ω, β)

o ¹

q−1´

+K_p(Ω, β)

n v−Ψ qKp(Ω, β)

o_pi dx.

With the above choice ofλ it is immediate that

µ∈Λinf h Z

Ω

µ(v−Ψ)dx+K_p(Ω, β) Z

Ω

¯¯µ−div(divη^∗)¯

¯^qdx i

≤ε(v, η^∗, β), and forp= 2this estimate reduces to the one given in Remark 2 of [Re]. Summing up we arrive at

Corollary 3.2. With the notation introduced above we have in case p≥2 (3.11) °

°∇²(u−v)°

°^p

L^p ≤p2^p−1 h

Dp[∇²v, η^∗] +1 pβ^p°

°|η^∗|^q−2η^∗−∇²v°

°^p

L^p+ε(v, η^∗, β) i

valid for allv ∈K, for anyη^∗ ∈Y^∗ such thatdiv(divη^∗)∈L^q(Ω)and for any choice of β >0.

Note that the r.h.s. of (3.11) vanishes only on the exact solution. Estimate (3.11) gives an upper bound on the error related to the approximations of variational inequalities. It should be emphasized that it does not require a priori knowledge of the form of the unknown free boundaries.

Elimination of the quantity div(divη^∗). Let us finally reconsider the quan- tity

d(η^∗) := inf

τ^∗∈Q^∗_λkη^∗ −τ^∗k_L^q for a tensor η^∗ ∈Y^∗ and a function λ∈Λ. We have

(3.12) inf

τ^∗∈Q^∗_λ

1

qkη^∗−τ^∗k^q_Lq = inf

ρ^∗∈Y^∗ sup

w∈W^◦2p(Ω)

L(w, ρ^∗),

where

L(w, ρ^∗) :=

Z

Ω

£π_p^∗(η^∗−ρ^∗) +ρ^∗ :∇²w−λw¤ dx.

In fact, it holds sup

w∈W^◦2p

L(w, ρ^∗) =





+∞, if ρ^∗ ∈/ Q^∗_λ, Z

Ω

π_p^∗(η^∗−ρ^∗)dx, if ρ^∗ ∈Q^∗_λ,

and this implies (3.12). Now, using standard results from duality theory, we have

(3.13) inf

ρ^∗∈Y^∗ sup

w∈W^◦2p

L(w, ρ^∗) = sup

w∈W^◦2p

ρ^∗inf∈Y^∗L(w, ρ^∗).

Since

ρ^∗inf∈Y^∗L(w, ρ^∗) = inf

κ^∗∈Y^∗L(w, η^∗−κ^∗), we get from (3.12) and (3.13)

(3.14) inf

τ^∗∈Q^∗_λ

1

qkη^∗−τ^∗k_L^q = sup

w∈W^◦2p

κ^∗inf∈Y^∗

Z

Ω

£π^∗_p(κ^∗) + (η^∗−κ^∗) :∇²w−λw¤ dx.

Note that (3.14) corresponds to formula (4.1) established in [NR2] for linear equa- tions related to the biharmonic operator. Proceeding as in this reference, we write

κ^∗inf∈Y^∗

Z

Ω

£π_p^∗(κ^∗) + (η^∗−κ^∗) :∇²w−λw¤ dx

=− sup

κ^∗∈Y^∗

Z

Ω

£κ^∗ :∇²w−π^∗_p(κ^∗)¤ dx+

Z

Ω

£η^∗ :∇²w−λw¤ dx

=− Z

Ω

£π_p(∇²w)−η^∗ :∇²w+λw¤ dx.

Inserting this into (3.14) we have shown that

(3.15) inf

τ^∗∈Q^∗_λ

1

qkη^∗ −τ^∗k^q_Lq =− inf

w∈W^◦2p

Z

Ω

£π_p(∇²w)−η^∗ :∇²w+λw¤ dx.

Ifdiv(divη^∗)∈L^q(Ω), then (3.15) reduces to (3.7), and we arrive at (3.9). Without further information concerning η^∗ (3.15) just states that the quantity d(η^∗) can be obtained by “solving” an auxiliary variational problem, which means to compute a lower bound for the functional

w7→

Z

Ω

£π_p(∇²w)−η^∗ :∇²w+λw¤ dx

defined on the space W^{◦ 2}_p(Ω). Here of course no side condition enters but for each choice of η^∗ and λ a new problem has to be considered.

A rather natural assumption concerning η^∗ ∈Y^∗ is the requirement that divη^∗ =¡

∂αη^∗_αβ¢

1≤β≤2

is in the space L^q(Ω). Then Z

Ω

£π_p(∇²w)−η^∗ :∇²w+λw¤ dx

= Z

Ω

£π_p(∇²w) + divη^∗· ∇w+λw¤ dx

= Z

Ω

£π_p(∇²w) + (divη^∗−y^∗)· ∇w+ (λ−divy^∗)w¤ dx,

wherey^∗ is a vector-function from L^q(Ω) such that divy^∗ ∈L^q(Ω). From (3.15) we get by Hölder’s inequality

1

qd(η^∗)^q ≤ − inf

w∈W^◦2p(Ω)

h1

pk∇²wk^p_Lp− kdivη^∗−y^∗k_L^qk∇wk_L^p

− kλ−divy^∗k_L^qkwk_L^p i

. (3.16)

From the Poincaré inequality we have for all v ∈W^◦1_p(Ω)

(3.17) kvk_L^p ≤K_p(Ω)k∇vk_L^p

and applying (3.17) tow ∈W^◦2_p(Ω) as well as to the vectorial function∇w ∈W^◦1_p(Ω) we see that the r. h. s. of (3.16) is bounded from above by

− inf

w∈W^◦2p(Ω)

"

1

pk∇²wk^p_Lp−Kp(Ω)kdivη^∗−y^∗kL^qk∇²wkL^p

−K_p(Ω)²kλ−divy^∗k_L^qk∇²wk_L^p

#

≤ −inf

t≥0

h1

pt^p−K_p(Ω)£

kdivη^∗−y^∗k_L^q+K_p(Ω)kλ−divy^∗k_L^q¤ t

i

= 1

qK_p(Ω)^q£

kdivη^∗−y^∗k_L^q +K_p(Ω)kλ−divy^∗k_L^q¤_q . Let us summarize our results:

Theorem 3.2. Suppose that we are given η^∗ ∈ L^q(Ω) and λ ∈ Λ. Then we have

τ^∗inf∈Q^∗_λ

1

qkη^∗−τ^∗k^q_Lq =− inf

w∈W^◦2p(Ω)

Z

Ω

£π_p(∇²w)−η^∗ :∇²w+λw¤ dx.

If in addition divη^∗ ∈L^q(Ω), then

τ^∗inf∈Q^∗_λ

1

qkη^∗−τ^∗kL^q ≤ 1

qKp(Ω)^q h

kdivη^∗−y^∗kL^q +Kp(Ω)kλ−divy^∗kL^q

i_q , where y^∗ is any vector-function in L^q(Ω) s.t. divy^∗ ∈ L^q(Ω) and where K_p(Ω) is defined according to (3.17).

From the proof it is immediate that the statements of Theorem 3.2 are also valid in the case1< p <2.

4. Estimates for the subquadratic case Obviously we have for τ^∗ ∈Q^∗_λ

J_λ^∗[τ^∗] = inf

w∈W^◦2p(Ω)

Z

Ω

£l(w, τ^∗)−λ(w−Ψ)¤ dx

≤ inf

w∈K

Z

Ω

£l(w, τ^∗)−λ(w−Ψ)¤

dx≤ inf

w∈K

Z

Ω

l(w, τ^∗)dx=J^∗[τ^∗], moreoverJ^∗[σ^∗] = J[u]≤J[w] for any w∈K, hence we get from (2.3)

(4.1) °

°τ^∗−σ^∗k^q_Lq ≤q2^q−1 h

J[w]−J_λ^∗[τ^∗] i

.

Observe that (4.1) exactly corresponds to (3.3), and as outlined in Section 3 in- equality (4.1) can be rewritten as

(4.2) °

°τ^∗−σ^∗k^q_Lq ≤q2^q−1 n

D_p[∇²w, τ^∗] + Z

Ω

λ(w−Ψ)dx o

valid for all τ^∗ ∈ Q^∗_λ, w ∈ K and λ ∈ Λ, D_p[∇²w, τ^∗] having the same meaning as before. If nowη^∗ ∈Y^∗ and τ^∗ ∈Q^∗_λ, then (using (4.2))

°°η^∗−σ^∗°

°^q

L^q ≤³°

°σ^∗−τ^∗°

°L^q +°

°η^∗−τ^∗°

°L^q

´_q

≤2^q−1³°

°σ^∗−τ^∗°

°^q

L^q +°

°η^∗−τ^∗°

°^q

L^q

´

≤q4^q−1 h

D_p[∇²w, τ^∗] + Z

Ω

λ(w−Ψ)dx i

+ 2^q−1°

°η^∗−τ^∗°

°^q

L^q. (4.3)

The quantity D_p[∇²w, τ^∗] can be estimated as before:

D_p[∇²w, τ^∗]≤D_p[∇²w, η^∗] + Z

Ω

£|τ^∗|^q−2τ^∗− ∇²w¤ :¡

τ^∗−η^∗¢ dx

=D_p[∇²w, η^∗] + Z

Ω

£|τ^∗|^q−2τ^∗− |η^∗|^q−2η^∗¤

: (τ^∗−η^∗)dx +

Z

Ω

£|η^∗|^q−2η^∗ − ∇²w¤

: (τ^∗−η^∗)dx

=:D_p[∇²w, η^∗] +I₁+I₂. (4.4)

According to the calculations after (3.5) in [BFR] we have I₁ ≤(q−1)°°τ^∗−η^∗°°²

L^q

h°°τ^∗−η^∗°°

L^q + 2°°η^∗°°

L^q

i_q−2 , and for I₂ we get with Hölder’s and Young’s inequality

I₂ ≤ β 2

°°|η^∗|^q−2η^∗− ∇²w°

°²

L^p+ 1 2β

°°τ^∗−η^∗°

°²

L^q,