Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 549–558

SECOND ORDER OBSTACLE PROBLEMS FOR VECTORIAL FUNCTIONS AND

INTEGRANDS WITH SUBQUADRATIC GROWTH

M. Fuchs, Li Gongbao, and O. Martio

Universität des Saarlandes, Fachbereich 9 Mathematik D-66213 Saarbrücken, Germany Wuhan Institute of Physics and Mathematics, Young Scientist Lab. of Math. Physics P.O. Box 71010, Wuhan 430071, P.R. China University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland; olli.martio@helsinki.fi

Abstract. It is shown that the obstacle problem associated with the second order variational integral

∫_{Ω}(1 + |∇^{2}u|^{2})^{p/2} dx

has a unique solution u in the class C^{1,α} for any α < 1 (n = 2) and for α = 1 − 1/p (n = 3).

1. Introduction and main results

Let Ω denote an open bounded set in R^{n}, n = 2 or n = 3, having Lipschitz boundary ∂Ω. For k ∈ N and 1 ≤ q ≤ ∞ we define the Sobolev space W_{q}^{k}(Ω, R^{M}) of vector-valued functions u: Ω → R^{M}, M ≥ 1, in the usual way (see, e.g. [A]). The subspace W̊_{q}^{k}(Ω, R^{M}) is introduced as the closure of C_{0}^{∞}(Ω, R^{M}) in W_{q}^{k}(Ω, R^{M}), and we say that a measurable function u: Ω → R^{M} belongs to the local space W_{q,loc}^{k}(Ω, R^{M}) if u ∈ W_{q}^{k}(Ω′, R^{M}) for any subregion Ω′ with compact closure in Ω. Suppose now that p ∈ (1, 2) is a fixed number satisfying p > \frac{3}{2} in case n = 3 and that a function Φ ∈ W_{2}^{3}(Ω, R^{M}) satisfies

Φ |_{∂Ω}< 0, i.e. Φ^{i} < 0 on ∂Ω for i = 1, …, M.

We consider the variational integral

J(u) = ∫_{Ω}(1 + |∇^{2}u|^{2})^{p/2} dx

1991 Mathematics Subject Classification: Primary 49J10, 49N60, 73C50.

Part of this paper was written during the first author’s stay at the Department of Mathematics of the University of Helsinki in August and December 1996. He greatly appreciates the friendly atmosphere provided by the members of the department. The second author was partially supported by the Academy of Finland and NSFC.

which makes sense for u in the space W_{p}^{2}(Ω, R^{M}). Note that W_{p}^{2}(Ω, R^{M}) ⊂ C^{o}(\overline{Ω}, R^{M}) for any p > 1, if n = 2, and for p > \frac{3}{2} , if n = 3. Here ∇^{2}u denotes the matrix (∂_{α}∂_{β}u^{i})_{1≤α,β≤n, 1≤i≤M} of all second generalized derivatives. We then look for solutions of the obstacle problem

(V)  \{ \begin{matrix}to find u ∈ K := \{w ∈ W̊_{p}^{2}(Ω, R^{M}) : w^{i} ≥ Φ^{i} a.e., i = 1, …, M\} \\ such that J(u) = inf_{K} J. \end{matrix}

The scalar case M = 1 is of some physical interest: consider a plate whose undeformed state is represented by a region Ω ⊂ R^{2} . If some outer forces are applied acting in vertical direction, then the equilibrium configuration can be found as a minimizer of the energy

I(u) = ∫_{Ω} g(∇^{2}u) dx + potential terms

subject to appropriate boundary conditions. The physical properties of the plate are characterized in terms of the given convex function g: R^{2×2} → [0, ∞). In the case of elastic plates (compare [F]) we have g(E) = |E|^{2} (up to physical constants), for perfectly plastic plates (see [S]) g is of linear growth near infinity, hence the choice

g(E) = (1 + |E|^{2})^{p/2}

provides some interpolation between these two cases which is a suitable model for plastic plates with power hardening. So, for n = 2 and M = 1, our variational problem (V) reduces to the obstacle problem for pseudo-plastic plates, i.e. plates with power hardening, where now the plate is forced to lie above some function Φ and, in addition, it is clamped at the boundary.

Let us now state the main result of this paper.

Theorem 1.1. Problem (V) admits a unique solution u ∈ K. We have u ∈ C^{1,α}(Ω, R^{M}) for any α < 1, if n = 2, and u ∈ C^{1,1−1/p}(Ω, R^{M}), if n = 3.

The proof of Theorem 1.1 is organized in several steps: in Section 2 we first show that the unique minimizer can be obtained with the help of a suitable approximation. This means that we replace our integrand g(E) = (1 + |E|^{2})^{p/2} by the sequence

g_{δ}(E) = \frac{1}{2}δ|E|^{2} + g(E), δ > 0,

and study the more regular obstacle problem

(V)_{δ}  \{ \begin{matrix}to find u_{δ} ∈ K′ := \{w ∈ W̊_{2}^{2}(Ω, R^{M}) : w ≥ Φ\} \\ such that J_{δ}(u_{δ}) := ∫_{Ω} g_{δ}(∇^{2}u_{δ}) dx = inf_{w∈K′} J_{δ}(w) \end{matrix}

Second order obstacle problems for vectorial functions and integrands 551

whose solutions converge weakly to the solution of (V).

In Section 3 we use this result to show that the scalar function (1+|∇^{2}u|^{2})^{p/4} is in the space W_{2,loc}^{1}(Ω) which will be deduced in this space from uniform bounds for (1 + |∇^{2}u_{δ}|^{2})^{p/4} . The regularity properties of u are then a consequence of Sobolev’s embedding theorem. It should be noted that similar arguments in a different setting have been used in [FR] and [FS].

2. Approximation

First we want to show that our variational problem (V) admits a unique solution which will be immediate as soon as we can show that the class K is non-empty. This follows from

Lemma 2.1. Suppose that Φ ∈ W_{2}^{3}(Ω, R^{M}) satisfies Φ|_{∂Ω} < 0. Then there exists Φ_{0} ∈ W̊_{2}^{2}(Ω, R^{M}) with the property Φ_{0} ≥ Φ in Ω, in particular Φ_{0} ∈ K.

Proof. We may assume that M = 1. By Sobolev’s embedding theorem (recall n = 2 or 3) we have Φ ∈ C^{o}(\overline{Ω}), and since Φ|_{∂Ω} < 0, there is a subdomain Ω′ ⊂⊂ Ω such that Φ < 0 on \overline{Ω} − Ω′. Let c := max\{0, max\overline{Ω} Φ\} and Φ_{0} := cη, where η ∈ C_{0}^{∞}(Ω) satisfies 0 ≤ η ≤ 1 on Ω as well as η = 1 on Ω′. Clearly Φ_{0} ∈ K′ ⊂ K.

We recall the following characterization of W̊_{q}^{m}(G); in this lemma the function f ∈ W_{q}^{m}(R^{n}) is supposed to be (m, q)-quasicontinuous. In particular, the lemma applies to continuous functions in W_{q}^{m}(R^{n}).

Lemma 2.2 (see [AH, Theorem 9.1.3]). Let m be a positive integer, 1 < q < ∞ and f ∈ W_{q}^{m}(R^{n}). Let G denote an open subset of R^{n}, K = R^{n} − G. Then the following statements are equivalent:

(a) D^{β}f|_{K} = 0 for all multi-indices β, |β| ≤ m − 1,

(b) f ∈ W̊_{q}^{m}(G),

(c) for any ε > 0 and any compact set F ⊂ G there is a function η ∈ C_{0}^{∞}(G) such that η = 1 on F, 0 ≤ η ≤ 1 and ∥f − ηf∥_{W_{q}^{m}(G)} < ε.

Lemma 2.3. Suppose that Φ, K and K′ are defined as in Section 1. Then K′ is dense in K with respect to the norm ∥ · ∥_{W_{p}^{2}(Ω)} .

Proof. Again we may assume that M = 1.

Step I: Given any ε > 0 and an arbitrary function v ∈ K, we first show that there exists a function w ∈ K with compact support such that ∥v − w∥_{W_{p}^{2}(Ω)} < ε.

To this end we notice that there is a subdomain Ω′ ⊂⊂ Ω such that Φ < 0 on \overline{Ω} − Ω′ (recall that Φ is continuous on \overline{Ω}). By letting v = 0 on R^{n} − Ω we may assume that v ∈ W_{p}^{2}(R^{n}). We then use Lemma 2.2 with f = v, F = \overline{Ω′}, G = Ω, m = 2, q = p to see that there is η ∈ C_{0}^{∞}(Ω), 0 ≤ η ≤ 1, η = 1 on \overline{Ω′} such that

(2.1)  ∥v − ηv∥_{W_{p}^{2}(Ω)} < ε.

We claim that ηv ≥ Φ: in fact, ηv = v ≥ Φ on \overline{Ω′}. For any x_{0} ∈ Ω − \overline{Ω′} we have

η(x_{0})v(x_{0}) ≥ 0 > Φ(x_{0}) if v(x_{0}) ≥ 0

and (observe 0 ≤ η ≤ 1)

η(x_{0})v(x_{0}) ≥ v(x_{0}) ≥ Φ(x_{0}) if v(x_{0}) < 0.

Thus w = ηv belongs to K with spt w ⊂⊂ Ω and ∥v − w∥_{W_{p}^{2}(Ω)} < ε according to (2.1).

Step II: We show that for any ε > 0 and any w ∈ K with compact support there exists a v_{ε} ∈ K′ such that

(2.2)  ∥w − v_{ε}∥_{W_{p}^{2}(Ω)} < ε.

This together with the first step will complete the proof of Lemma 2.3. For Step II choose η ∈ C_{0}^{∞}(Ω) satisfying 0 ≤ η ≤ 1 and η = 1 on \overline{Ω′′} for some set Ω′′ ⊂⊂ Ω containing spt (w) and with the property that

Φ < 0 on Ω − \overline{Ω′′}.

Let w_{ρ} denote the mollification of w with radius ρ > 0 (see, e.g. [A]). Using the fact that spt w is compact in Ω, we find ρ_{0} = ρ_{0}(ε) such that for all 0 < ρ < ρ_{0} we have

(2.3)  \{ \begin{matrix}w_{ρ} ∈ C_{0}^{∞}(Ω), spt w_{ρ} ⊂ Ω′′, \\ ∥w − w_{ρ}∥_{L^{∞}(Ω)} < \frac{1}{2}ε∥∇^{2}η∥_{p} =: μ, \\ ∥w − w_{ρ}∥_{W_{p}^{2}(Ω)} < \frac{1}{2}ε. \end{matrix}

Let us fix ρ ∈ (0, ρ_{0}) and let v_{ε} = η(w_{ρ} + μ). Then v_{ε} ∈ C_{0}^{∞}(Ω) and v_{ε} ≥ Φ in Ω which follows from (2.3). It is also immediate that (2.3) implies (2.2).

Remark 2.1. From the proof of Lemma 2.3 we actually get that {w ∈ C_{0}^{∞}(Ω, R^{M}) : w ≥ Φ in Ω} is dense in K with respect to ∥ · ∥_{W_{p}^{2}(Ω)} .

By Lemma 2.1 we know that K′ ≠ ∅, hence there is a unique solution u_{δ} of problem (V)_{δ}. Let u ∈ K (≠ ∅) denote the unique solution of (V). Then we have

Lemma 2.4. The solution u_{δ} ⇁ u weakly in W_{p}^{2}(Ω), J_{δ}(u_{δ}) → J(u) and \frac{1}{2}δ ∫_{Ω} |∇^{2}u_{δ}|^{2} dx → 0 as δ ↓ 0.

Proof. Let Φ_{0} ∈ K′ denote the function defined in Lemma 2.1. Then for 0 < δ ≤ 1 we have

J_{δ}(u_{δ}) ≤ J_{δ}(Φ_{0}) ≤ J_{1}(Φ_{0}) =: c_{1},

hence

∫_{Ω}(1 + |∇^{2}u_{δ}|^{2})^{p/2} dx ≤ c_{1},

which shows that sup_{0<δ≤1} ∥u_{δ}∥_{W_{p}^{2}(Ω)} < ∞. After passing to a subsequence we may assume that

u_{δ} ⇁_{δ↓0} ũ weakly in W_{p}^{2}(Ω, R^{M})

for some function ũ. Since we may also assume that u_{δ} → ũ a.e. (after passing to another subsequence if necessary), we clearly get ũ ∈ K.

For any w ∈ K′ we have

J_{δ}(u_{δ}) ≤ J_{δ}(w) ⟶_{δ↓0} J(w)

and by the lower semicontinuity

J(ũ) ≤ lim inf_{δ↓0} J(u_{δ}) ≤ lim inf_{δ↓0} J_{δ}(u_{δ}) =: α ≤ lim sup_{δ↓0} J_{δ}(u_{δ}) =: β

so that

(2.4)  J(ũ) ≤ α ≤ β ≤ J(w) for all w ∈ K′.

By Lemma 2.3, K′ is dense in K with respect to ∥ · ∥_{W_{p}^{2}(Ω)} . So (2.4) in fact holds for w ∈ K and hence ũ coincides with the unique solution u of (V). Therefore u_{δ} ⇁ u in W_{p}^{2}(Ω, R^{M}) for the whole sequence. Choosing w = u in (2.4) we get α = β = J(u), i.e. J(u) = lim_{δ↓0} J_{δ}(u_{δ}). Using

J(u) ≤ J(u_{δ}) ≤ J_{δ}(u_{δ})

we see that also

J(u_{δ}) ⟶_{δ↓0} J(u),

in conclusion

\frac{1}{2}δ ∫_{Ω} |∇^{2}u_{δ}|^{2} dx ⟶_{δ↓0} 0,

and Lemma 2.4 is established.

3. Existence of higher order weak derivatives

Lemma 3.1. Let u ∈ K denote the unique solution of problem (V). Then we have (1 + |∇^{2}u|^{2})^{p/4} ∈ W_{2,loc}^{1}(Ω).

Accepting Lemma 3.1 for the moment we see by the embedding theorem that

|∇^{2}u| ∈ L_{loc}^{t}(Ω)

for any finite t, if n = 2, and for t ≤ 3p, if n = 3, hence ∇u ∈ C^{0,α}(Ω, R^{nM}) for any α < 1, if n = 2, and for α = 1−1/p in case n = 3. This proves Theorem 1.1.

Proof of Lemma 3.1. We fix a coordinate direction e_{γ} ∈ R^{n}, γ = 1, …, n, and define for h ≠ 0 and a function f

Δ_{h}f(x) = \frac{1}{h}(f(x + he_{γ}) − f(x)).

Let {u_{δ}} denote the sequence introduced in Section 2. With δ fixed we consider ε > 0 satisfying εh^{−2} < \frac{1}{2} and define

v_{ε} = u_{δ} + εΔ_{−h}(η^{6}Δ_{h}[u_{δ} − Φ])

with η ∈ C_{0}^{2}(Ω) such that 0 ≤ η ≤ 1. Observing

\begin{aligned}v_{ε}(x) & = Φ(x) + [1 − \frac{ε}{h^{2}} η^{6}(x − he_{γ}) − \frac{ε}{h^{2}} η^{6}(x)](u_{δ} − Φ)(x) \\ & + \frac{ε}{h^{2}} η^{6}(x − he_{γ})(u_{δ} − Φ)(x − he_{γ}) + \frac{ε}{h^{2}} η^{6}(x)(u_{δ} − Φ)(x + he_{γ}) \end{aligned}

we see that v_{ε} ≥ Φ, i.e. v_{ε}^{i} ≥ Φ^{i} for each component, hence v_{ε} ∈ K′, which gives J_{δ}(u_{δ}) ≤ J_{δ}(v_{ε}), and we deduce

∫_{Ω} \frac{1}{ε} \{g_{δ}(∇^{2}u_{δ} + ε∇^{2}[Δ_{−h}(η^{6}Δ_{h}(u_{δ} − Φ))]) − g_{δ}(∇^{2}u_{δ})\} dx ≥ 0.

Passing to the limit ε → 0 we infer

∫_{Ω} Dg_{δ}(∇^{2}u_{δ}) : ∇^{2}(Δ_{−h}[η^{6}Δ_{h}(u_{δ} − Φ)]) dx ≥ 0,

Dg_{δ} denoting the gradient of g_{δ}. Using “integration by parts” for Δ_{−h} we end up with the result

(3.1)  ∫_{Ω} Δ_{h}{Dg_{δ}(∇^{2}u_{δ})} : ∇^{2}(η^{6}Δ_{h}[u_{δ} − Φ]) dx ≤ 0.

Second order obstacle problems for vectorial functions and integrands 555 Introducing ξ_{t} := ∇^{2}u_{δ} + thΔ_{h}(∇^{2}u_{δ}) we may write

\begin{matrix}Δ_{h}\{Dg_{δ}(∇^{2}u_{δ})\} : & ∇^{2}(η^{6}Δ_{h}[u_{δ} − Φ]) \\ = ∫_{0}^{1} D^{2}g_{δ}(ξ_{t})(Δ_{h}∇^{2}u_{δ}, ∇^{2}(η^{6}Δ_{h}[u_{δ} − Φ])) dt. \end{matrix}

Let us further define the bilinear form

B_{x}(X, Y) = ∫_{0}^{1} D^{2}g_{δ}(ξ_{t}(x))(X, Y) dt

for x ∈ Ω and matrices X, Y . Then (3.1) takes the form

(3.2)  ∫_{Ω} B_{x}(Δ_{h}∇^{2}u_{δ}, ∇^{2}(η^{6}Δ_{h}[u_{δ} − Φ])) dx ≤ 0.

We have

\begin{aligned}∇^{2}(η^{6}Δ_{h}u_{δ}) & = η^{6}∇^{2}Δ_{h}u_{δ} + (∂_{α}∂_{β}ηΔ_{h}u_{δ} + ∂_{α}η^{6}∂_{β}Δ_{h}u_{δ} + ∂_{β}η∂_{α}Δ_{h}u_{δ})_{1≤α,β≤n} \\ & =: η^{6}∇^{2}Δ_{h}u_{δ} + T_{h}, \end{aligned}

and (3.2) implies

∫_{Ω} η^{6}B_{x}(Δ_{h}∇^{2}u_{δ}, Δ_{h}∇^{2}u_{δ}) dx ≤ ∫_{Ω} B_{x}(Δ_{h}∇^{2}u_{δ}, ∇^{2}(η^{6}Δ_{h}Φ) − T_{h}) dx. On the right-hand side we may apply Cauchy–Schwarz’s inequality in the form

B_{x}(X, Y) ≤ B_{x}(X, X)^{1/2}B_{x}(Y, Y)^{1/2}.

This together with Young’s inequality leads to ∫_{Ω}η^{6}B_{x}(Δ_{h}∇^{2}u_{δ}, Δ_{h}∇^{2}u_{δ}) dx (3.3)

≤ c_{1}(η) ∫_{spt η}∥B_{x}∥(|Δ_{h}Φ|^{2} + |∇Δ_{h}Φ|^{2} + |∇^{2}Δ_{h}Φ|^{2} + |Δ_{h}u_{δ}|^{2} + |∇Δ_{h}u_{δ}|^{2}) dx

for some constant c_{1} depending on η.

It is easy to check that the following bounds hold for the parameter dependent bilinear form D^{2}g_{δ}(Z)(X, Y):

(3.4)  ∥D^{2}g_{δ}(Z)∥ = sup_{|X|=1} D^{2}g_{δ}(Z)(X, X)
≤ δ + c_{2}(p)(1 + |Z|^{2})^{(p/2)−1} ≤ δ + c_{2}(p),
(3.5)  D^{2}g_{δ}(Z)(X, X) = δX : X + p(1 + |Z|^{2})^{(p/2)−1}|X|^{2}
+ p(p − 2)(1 + |Z|^{2})^{(p/2)−2}(Z : X)^{2} ≥ δ|X|^{2}.

Inserting this into (3.3), we find that

∫_{Ω} η^{6}|Δ_{h}∇^{2}u_{δ}|^{2} dx ≤ c_{3}(δ, η, p){∥u_{δ}∥_{W_{2}^{2}(Ω)}^{2} + ∥ϕ∥_{W_{2}^{3}(Ω)}^{2}},

therefore u_{δ} ∈ W_{2,loc}^{3}(Ω, R^{M}). For this reason we can replace Δ_{h} in (3.1) by the derivative ∂_{γ}. Then, following the calculations after (3.1), we see that (3.3) has to be replaced by (summation over γ)

∫_{Ω}η^{6}D^{2}g_{δ}(∇^{2}u_{δ})(∂_{γ}∇^{2}u_{δ}, ∂_{γ}∇^{2}u_{δ}) dx (3.6)

≤ c_{4}(η) ∫_{spt η}∥D^{2}g_{δ}(∇^{2}u_{δ})∥(|∇u_{δ}|^{2} + |∇^{2}u_{δ}|^{2} + |∇ϕ|^{2} + |∇^{2}ϕ|^{2} + |∇^{3}ϕ|^{2}) dx,

c_{4} being independent of δ. Recall from Section 2 that

δ ∫_{Ω} |∇^{2}u_{δ}|^{2} dx → 0 as δ ↓ 0.

Our assumption concerning p together with sup_{δ>0} ∥u_{δ}∥_{W_{p}^{2}(Ω)} < ∞ implies that {∇u_{δ}} is uniformly bounded in L^{2}(Ω, R^{nM}). In view of (3.4) it remains to discuss

∫_{Ω} ∥D^{2}g_{δ}(∇^{2}u_{δ})∥|∇^{2}u_{δ}|^{2} dx

≤ δ ∫_{Ω} |∇^{2}u_{δ}|^{2} dx + c_{2}(p) ∫_{Ω}(1 + |∇^{2}u_{δ}|^{2})^{(p/2)−1}|∇^{2}u_{δ}|^{2} dx

≤ δ ∫_{Ω} |∇^{2}u_{δ}|^{2} dx + c_{2}(p) ∫_{Ω}(1 + |∇^{2}u_{δ}|^{2})^{p/2} dx.

We therefore get from (3.6)

(3.7)  sup_{δ>0} ∫_{Ω′} D^{2}g_{δ}(∇^{2}u_{δ})(∂_{γ}∇^{2}u_{δ}, ∂_{γ}∇^{2}u_{δ}) dx < ∞

for each subdomain Ω′ of Ω with compact closure.

Let us introduce the auxiliary function h_{δ} := (1 + |∇^{2}u_{δ}|^{2})^{p/4} . It is easy to check that (using the equation in (3.5))

\begin{aligned}|∇h_{δ}|^{2} & = ∂_{γ}h_{δ}∂_{γ}h_{δ} ≤ \frac{1}{4}p^{2}(1 + |∇^{2}u_{δ}|^{2})^{(p/2)−2}|∇^{2}u_{δ}|^{2}∂_{γ}∇^{2}u_{δ} : ∂_{γ}∇^{2}u_{δ} \\ & ≤ c_{5}(p)D^{2}g_{δ}(∇^{2}u_{δ})(∂_{γ}∇^{2}u_{δ}, ∂_{γ}∇^{2}u_{δ}), \end{aligned}

hence {h_{δ}}_{δ>0} is bounded in W_{2,loc}^{1}(Ω) which is a consequence of (3.7). Let h ∈ W_{2,loc}^{1}(Ω) denote a weak limit as δ ↓ 0 of some subsequence. We claim

(3.8)  h = (1 + |∇^{2}u|^{2})^{p/4}.

For proving (3.8) let us write

J_{δ}(u_{δ}) − J(u) = \frac{δ}{2} ∫_{Ω} |∇^{2}u_{δ}|^{2} dx + J(u_{δ}) − J(u)

= \frac{δ}{2} ∫_{Ω} |∇^{2}u_{δ}|^{2} dx + ∫_{Ω} Dg(∇^{2}u) : (∇^{2}u_{δ} − ∇^{2}u) dx

+ ∫_{Ω} ∫_{0}^{1} D^{2}g((1 − t)∇^{2}u + t∇^{2}u_{δ})

× (∇^{2}u_{δ} − ∇^{2}u, ∇^{2}u_{δ} − ∇^{2}u)(1 − t) dt dx.

From J_{δ}(u_{δ}) → J(u), δ ↓ 0, we infer

(3.9) lim_{δ↓0}{∫_{Ω} Dg(∇^{2}u) : (∇^{2}u_{δ}−∇^{2}u) dx+∫_{Ω} ∫_{0}^{1} D^{2}g(⋯)(⋯)(1−t) dt dx} = 0.

On the other hand, minimality of u implies

(3.10)  ∫_{Ω} Dg(∇^{2}u) : (∇^{2}u_{δ} − ∇^{2}u) dx ≥ 0.

Next we observe the estimate

\begin{aligned}∫_{Ω} ∫_{0}^{1} D^{2}g & (⋯)(⋯, ⋯)(1 − t) dt dx ≥ (3.5) \\ & c_{6}(p) ∫_{Ω}(1 + |∇^{2}u_{δ}|^{2} + |∇^{2}u|^{2})^{(p/2)−1}|∇^{2}u_{δ} − ∇^{2}u|^{2} dx =: α_{δ}. \end{aligned}

By (3.9) and (3.10) we must have α_{δ} → 0. Hence we can choose a subsequence such that

(3.11)  (1 + |∇^{2}u_{δ}|^{2} + |∇^{2}u|^{2})^{(p/2)−1}|∇^{2}u_{δ} − ∇^{2}u|^{2} ⟶_{δ↓0} 0 a.e.

Clearly we may assume that also h_{δ} → h a.e. By definition of h_{δ} this implies

|∇^{2}u_{δ}|^{2} → h^{4/p} − 1 a.e.,

and the function h^{4/p} − 1 is finite a.e. Returning to (3.11) and taking care of our observation that (1 + |∇^{2}u_{δ}|^{2} + |∇^{2}u|^{2})^{(p/2)−1} has a pointwise limit a.e. which is not zero, we finally get

∇^{2}u_{δ} → ∇^{2}u a.e. on Ω,

and in conclusion h_{δ} → (1+|∇^{2}u|^{2})^{p/4} which proves (3.8). The proof of Lemma 3.1 is complete.

558 M. Fuchs, Li Gongbao, and O. Martio

References

[A] Adams, A.: Sobolev Spaces. - Academic Press, 1975.

[AH] Adams, D.R., and L.I. Hedberg: Function Spaces and Potential Theory. - Springer-Verlag, 1996.

[F] Friedman, A.: Variational Principles and Free-Boundary Problems. - John Wiley & Sons, 1982.

[FR] Fuchs, M., and J. Reuling: A modification of the blow-up technique for variational integrals with subquadratic growth. - J. Math. Anal. Appl. (to appear).

[FS] Fuchs, M., and G. Seregin: A regularity theory for variational integrals with L ln L-growth. - Calculus of Variations (to appear).

[S] Seregin, G.: Differentiability properties of weak solutions of certain variational problems in the theory of perfect elastoplastic plates. - Appl. Math. Optim. 28, 1993, 307–335.

Received 20 March 1997