Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 349–362

HIGHER ORDER VARIATIONAL PROBLEMS ON TWO-DIMENSIONAL DOMAINS

Michael Bildhauer and Martin Fuchs

Saarland University, Department of Mathematics, P.O. Box 15 11 50 DE-66041 Saarbrücken, Germany; bibi@math.uni-sb.de, fuchs@math.uni-sb.de

Abstract. Let u: R^{2} ⊃ Ω → R^{M} denote a local minimizer of J[w] = ∫_{Ω} f(∇^{k}w) dx, where k ≥ 2 and ∇^{k}w is the tensor of all k^{th} order (weak) partial derivatives. Assuming rather general growth and ellipticity conditions for f, we prove that u actually belongs to the class C^{k,α}(Ω; R^{M}) by the way extending the result of [BF2] to the higher order case by using different methods. A major tool is a lemma on the higher integrability of functions established in [BFZ].

1. Introduction

Let Ω denote a bounded domain in R^{2} and consider a function u: Ω → R^{M} which locally minimizes the variational integral

J[w, Ω] = ∫_{Ω} f(∇^{k}w) dx,

where ∇^{k}w represents the tensor of all k^{th} order (weak) partial derivatives. Our main concern is the investigation of the smoothness properties of such local minimizers under suitable assumptions on the energy density f. For the first order case (i.e. k = 1) we have rather general results which can be found for example in the textbooks of Morrey [Mo], Ladyzhenskaya and Ural’tseva [LU], Gilbarg and Trudinger [GT] or Giaquinta [Gi], for an update of the history including recent contributions we refer to [Bi]. In order to keep our exposition simple (and only for this reason) we consider the scalar case (i.e. M = 1) and restrict ourselves to variational problems involving the second (generalized) derivative. Then our variational problem is related to the theory of plates: one may think of u: Ω → R as the displacement in vertical direction from the flat state of an elastic plate. The classical case of a potential f with quadratic growth is discussed in the monographs of Ciarlet and Rabier [CR], Necǎs and Hlávácek [NH], Chudinovich and Constanda [CC] or Friedman [Fr], further references are contained in Zeidler’s book [Ze]. We also like to remark that plates with other hardening laws (logarithmic and power growth case) together with an additional obstacle have been studied in the papers [BF1] and [FLM] but not with optimal regularity results. The purpose of this note is to present a rather satisfying regularity theory for a quite large class of potentials allowing even anisotropic growth.

2000 Mathematics Subject Classification: Primary 49N60, 74K20.

To be precise let M denote the space of all (2×2)-matrices and suppose that we are given a function f: M → [0, ∞) of class C^{2} which satisfies with exponents 1 < p ≤ q < ∞ the anisotropic ellipticity estimate

(1.1)  λ(1 + |ξ|^{2})^{(p−2)/2}|σ|^{2} ≤ D^{2}f(ξ)(σ, σ) ≤ Λ(1 + |ξ|^{2})^{(q−2)/2}|σ|^{2}

for all ξ, σ ∈ M with positive constants λ, Λ. Note that (1.1) implies the growth condition

(1.2)  a|ξ|^{p} − b ≤ f(ξ) ≤ A|ξ|^{q} + B

with suitable constants a, A > 0, b, B ≥ 0. Let

J[w, Ω] = ∫_{Ω} f(∇^{2}w) dx, ∇^{2}w = (∂_{α}∂_{β}w)_{1≤α,β≤2}.

We say that a function u ∈ W_{p,loc}^{2}(Ω) is a local J -minimizer if and only if J[u, Ω′] < ∞ for any subdomain Ω′ ⋐ Ω and

J[u, Ω′] ≤ J[v, Ω′]

for all v ∈ W_{p,loc}^{2}(Ω) such that u − v ∈ W̊_{p}^{2}(Ω′) (here W_{p,loc}^{k}(Ω) etc. denote the standard Sobolev spaces, see [Ad]). Note that (1.1) implies the strict convexity of f. Therefore, given a function u_{0} ∈ W_{q}^{2}(Ω), the direct method ensures the existence of a unique J -minimizer u in the class

{v ∈ W_{p}^{2}(Ω) : J[v, Ω] < ∞, v − u_{0} ∈ W̊_{p}^{2}(Ω)}

which motivates the discussion of local J -minimizers. Our main result reads as follows:

Theorem 1.1. Let u denote a local J -minimizer under condition (1.1). Assume further that

(1.3)  q < min(2p, p + 2)

holds. Then u is of class C^{2,α}(Ω) for any 0 < α < 1.

Remark 1.1. (i) Clearly the result of Theorem 1.1 extends to local minimizers of the variational integral

I[w, Ω] = ∫_{Ω} f(∇^{2}w) dx + ∫_{Ω} g(∇w) dx,

where f is as before and where g denotes a density of class C^{2} satisfying

0 ≤ D^{2}g(ξ)(η, η) ≤ c(1 + |ξ|^{2})^{(s−2)/2}|η|^{2}

for some suitable exponent s. In case p ≥ 2 any finite number is admissible for s, in case p < 2 we require the bound s ≤ 2p/(2 − p). The details are left to the reader.

(ii) Without loss of generality we may assume that q ≥ 2: if (1.1) holds with some exponent q < 2, then of course (1.1) is true with q replaced by q̄:= 2 and (1.3) continues to hold for the new exponent.

(iii) If we consider the higher order variational integral ∫_{Ω} f(∇^{k}w) dx with k ≥ 2 and f satisfying (1.1), then (1.3) implies that local minimizers u ∈ W_{p,loc}^{k}(Ω) actually belong to the space C^{k,α}(Ω).

(iv) The degree of smoothness of u can be improved by standard arguments provided f is sufficiently regular.

(v) A typical example of an energy J satisfying the assumptions of Theorem 1.1 is given by

J[w, Ω] = ∫_{Ω} |∇^{2}w|^{2} dx + ∫_{Ω}(1 + |∂_{1}∂_{2}w|^{2})^{q/2} dx

with some exponent q ∈ (2, 4).

(vi) Our arguments can easily be adjusted to prove C^{k,α} -regularity of local minimizers u ∈ W_{p(x),loc}^{k}(Ω) of the energy ∫_{Ω}(1 + |∇^{k}w|^{2})^{p(x)/2} dx provided that 1 < p_{∗} ≤ p(x) ≤ p^{∗} < ∞ for some numbers p_{∗}, p^{∗} and if p(x) is sufficiently smooth. Another possible extension concerns the logarithmic case, i.e. we now consider the variational integral ∫_{Ω} |∇^{k}w| ln(1+|∇^{k}w|) dx and its local minimizers which have to be taken from the corresponding higher order Orlicz–Sobolev space.

The proof of Theorem 1.1 is organized as follows: we first introduce some suitable regularization and then prove the existence of higher order weak derivatives for this approximating sequence in Step 2. Here we also derive a Caccioppoli-type inequality using difference quotient methods. In a third step we deduce uniform higher integrability of the second generalized derivatives for any finite exponent. From this together with a lemma established in [BFZ] we finally obtain our regularity result in the last two steps.

2. Proof of Theorem 1.1

Step 1. Approximation. Let us fix some open domains Ω_{1} ⋐ Ω_{2} ⋐ Ω and denote by ū_{m} the mollification of u with radius 1/m, in particular

∥ū_{m} − u∥_{W_{p}^{2}(Ω_{2})} ⟶^{m→∞} 0.

Jensen’s inequality implies

J[ū_{m}, Ω_{2}] ≤ J[u, Ω_{2}] + τ_{m},

where τ_{m} → 0 as m → ∞. This, together with the lower semicontinuity of the functional J, shows that

(2.1)  J[ū_{m}, Ω_{2}] ⟶^{m→∞} J[u, Ω_{2}].

Next let

ϱ_{m} := ∥ū_{m} − u∥_{W_{p}^{2}(Ω_{2})}[∫_{Ω_{2}}(1 + |∇^{2}ū_{m}|^{2})^{q/2} dx]^{−1},

which obviously tends to 0 as m → ∞. With these preliminaries we introduce the regularized functional

J_{m}[w, Ω_{2}] := ϱ_{m} ∫_{Ω_{2}}(1 + |∇^{2}w|^{2})^{q/2} dx + J[w, Ω_{2}]

and the corresponding regularizing sequence {u_{m}} as the sequence of the unique solutions to the problems

(2.2)  J_{m}[ · , Ω_{2}] → min in ū_{m} + W̊_{q}^{2}(Ω_{2}).

By (2.1) and (2.2) we have

\begin{aligned}J_{m}[u_{m}, Ω_{2}] & ≤ J_{m}[ū_{m}, Ω_{2}] \\ & = ∥ū_{m} − u∥_{W_{p}^{2}(Ω_{2})} + J[ū_{m}, Ω_{2}] ⟶^{m→∞} J[u, Ω_{2}], \end{aligned}

hence one gets

(2.3)  lim sup_{m→∞} J_{m}[u_{m}, Ω_{2}] ≤ J[u, Ω_{2}].

On account of (2.3) and the growth of f we may assume

u_{m} ⇁^{m→∞} : û in W_{p}^{2}(Ω_{2}).

Moreover, lower semicontinuity gives

J[û, Ω_{2}] ≤ lim inf_{m→∞} J[u_{m}, Ω_{2}],

which together with (2.3) and the strict convexity of f implies û = u (here we also note that û − u ∈ W̊_{p}^{2}(Ω_{2})). Summarizing the results it is shown up to now that (as m → ∞)

(2.4)  \begin{matrix}u_{m} & ⇁ u in W_{p}^{2}(Ω_{2}), \\ J_{m}[u_{m}, Ω_{2}] & → J[u, Ω_{2}]. \end{matrix}

Step 2. Existence of higher order weak derivatives. In this second step we will prove that (f_{m}(ξ) := ϱ_{m}(1 + |ξ|^{2})^{q/2} + f(ξ))

(2.5) \begin{matrix}∫_{Ω_{2}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx \\ ≤ c(∥∇η∥_{∞}^{2} + ∥∇^{2}η∥_{∞}^{2}) ∫_{spt∇η} |D^{2}f_{m}(∇^{2}u_{m})|[|∇^{2}u_{m}|^{2} + |∇u_{m}|^{2}] dx, \end{matrix}

where η ∈ C_{0}^{∞}(Ω_{2}), 0 ≤ η ≤ 1, η ≡ 1 on Ω_{1} and where we take the sum over repeated indices. To this purpose let us recall the Euler equation

(2.6)  ∫_{Ω_{2}} Df_{m}(∇^{2}u_{m}) : ∇^{2}φ = 0 for all φ ∈ W̊_{q}^{2}(Ω_{2}).

If Δ_{h} denotes the difference quotient in the coordinate direction e_{α}, α = 1, 2, then the test function Δ_{−h}(η^{6}Δ_{h}u_{m}) is admissible in (2.6) with the result

(2.7)  ∫_{Ω_{2}} Δ_{h}{Df_{m}(∇^{2}u_{m})} : ∇^{2}(η^{6}Δ_{h}u_{m}) dx = 0.

Now denote by ℬ_{x} the bilinear form

ℬ_{x} = ∫_{0}^{1} D^{2}f_{m}(∇^{2}u_{m}(x) + th∇^{2}(Δ_{h}u_{m})(x)) dt,

and observe that

Δ_{h}\{Df_{m}(∇^{2}u_{m})\}(x) = \frac{1}{h} ∫_{0}^{1} \frac{d}{dt}Df_{m}(∇^{2}u_{m}(x)
+ t[∇^{2}u_{m}(x + he_{α}) − ∇^{2}u_{m}(x)]) dt

= \frac{1}{h} ∫_{0}^{1} \frac{d}{dt}Df_{m}(∇^{2}u_{m}(x) + ht∇^{2}(Δ_{h}u_{m})(x)) dt

= ℬ_{x}(∇^{2}(Δ_{h}u_{m})(x), · ), hence (2.7) can be written as

∫_{Ω_{2}} ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}(η^{6}Δ_{h}u_{m})) dx = 0,

which means that we have

(2.8)

∫_{Ω_{2}} η^{6}ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}(Δ_{h}u_{m})) dx
= − ∫_{Ω_{2}} ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}η^{6}Δ_{h}u_{m}) dx
− 2 ∫_{Ω_{2}} ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇η^{6} ⊙ ∇(Δ_{h}u_{m})) dx
=: −T_{1} − 2T_{2}.

To handle T_{1} we just observe ∂_{α}∂_{β}η^{6} = 30∂_{α}η∂_{β}ηη^{4} + 6∂_{α}∂_{β}ηη^{5} , for T_{2} we use ∇η^{6} = 6η^{5}∇η. The Cauchy–Schwarz inequality for the bilinear form ℬ_{x} implies

|T_{2}| = 6|∫_{Ω_{2}} ℬ_{x}(η^{3}∇^{2}(Δ_{h}u_{m}), η^{2}∇η ⊙ ∇(Δ_{h}u_{m})) dx|

≤ 6[∫_{Ω_{2}} ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}(Δ_{h}u_{m}))η^{6} dx]^{1/2}

× [∫_{Ω_{2}} ℬ_{x}(∇η ⊙ ∇(Δ_{h}u_{m}), ∇η ⊙ ∇(Δ_{h}u_{m}))η^{4} dx]^{1/2},

an analogous estimate being valid for T_{1} . Absorbing terms, (2.8) turns into

(2.9)

∫_{Ω_{2}} η^{6}ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}(Δ_{h}u_{m})) dx
≤ c(∥∇η∥_{∞}^{2} + ∥∇^{2}η∥_{∞}^{2}) ∫_{spt∇η} |ℬ_{x}|(|∇(Δ_{h}u_{m})|^{2} + |Δ_{h}u_{m}|^{2}) dx.

Next we estimate (note that in the following calculations we always assume, without loss of generality, q ≥ 2, compare Remark 1.1(ii)) for h sufficiently small

∫_{spt∇η} |ℬ_{x}| |∇(Δ_{h}u_{m})|^{2} dx
≤ ∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2} + h^{2}|∇^{2}(Δ_{h}u_{m})|^{2})^{(q−2)/2}|∇(Δ_{h}u_{m})|^{2} dx
≤ c[∫_{spt∇η} |∇(Δ_{h}u_{m})|^{q/2} dx
+ ∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2} + h^{2}|∇^{2}(Δ_{h}u_{m})|^{2})^{q/2} dx]
≤ c ∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2})^{q/2} dx.

In a similar way we estimate ∫_{spt∇η} |ℬ_{x}| |Δ_{h}u_{m}|^{2} dx and end up with

(2.10)

lim sup_{h→0} ∫_{Ω_{2}} η^{6}ℬ_{x}(∇^{2}(Δ_{h}u_{m}), ∇^{2}(Δ_{h}u_{m})) dx

≤ c(∥∇η∥_{∞}^{2} + ∥∇^{2}η∥_{∞}^{2}) ∫_{spt∇η}(1 + |∇u_{m}|^{2} + |∇^{2}u_{m}|^{2})^{q/2} dx.

Since q ≥ 2 is assumed, (2.10) implies that ∇^{2}u_{m} ∈ W_{2,loc}^{1}(Ω_{2}) and

Δ_{h}(∇^{2}u_{m}) ⟶^{h→0} ∂_{α}(∇^{2}u_{m}) in L_{loc}^{2}(Ω_{2}) and a.e.

Remark 2.1. With (2.10) we have

|Δ_{h}{Df_{m}(∇^{2}u_{m})}|^{q/(q−1)} ∈ L_{loc}^{1}(Ω_{2}) uniformly with regard to h, and, as a consequence,

Df_{m}(∇^{2}u_{m}) ∈ W_{q/(q−1),loc}^{1}(Ω_{2}).

This follows exactly as outlined in the calculations after (3.12) of [BF3].

With the above convergences and Fatou’s lemma we find the lower bound ∫_{Ω_{2}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx

for the left-hand side of (2.10) which gives using (1.1)

\begin{matrix}∫_{Ω_{2}} η^{6}(1 + |∇^{2}u_{m}|^{2})^{(p−2)/2}|∇^{3}u_{m}|^{2} dx \\ ≤ c(∥∇η∥_{∞}^{2} + ∥∇^{2}η∥_{∞}^{2}) ∫_{spt∇η}(1 + |∇u_{m}|^{2} + |∇^{2}u_{m}|^{2})^{q/2} dx < ∞, \end{matrix}

in particular

(2.11)  h_{m} := (1 + |∇^{2}u_{m}|^{2})^{p/4} ∈ W_{2,loc}^{1}(Ω_{2}).

But (2.11) implies h_{m} ∈ L_{loc}^{r}(Ω_{2}) for any r < ∞, i.e.

(2.12)  ∇^{2}u_{m} ∈ L_{loc}^{t}(Ω_{2}) for any t < ∞.

Using Fatou’s lemma again we obtain from (2.8)

(2.13)

∫_{Ω_{2}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx

≤ lim inf_{h→0} ∫_{Ω_{2}} η^{6}Δ_{h}{Df_{m}(∇^{2}u_{m})} : ∇^{2}(Δ_{h}u_{m}) dx

= lim inf_{h→0} − ∫_{Ω_{2}} Δ_{h}{Df_{m}(∇^{2}u_{m})} : [∇^{2}η^{6}Δ_{h}u_{m} + 2∇η^{6} ⊙ ∇(Δ_{h}u_{m})] dx.

On account of (2.12), Remark 2.1 and Vitali’s convergence theorem we may pass to the limit h → 0 on the right-hand side of (2.13) and obtain

\begin{matrix}∫_{Ω_{2}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx \\ ≤ − ∫_{Ω_{2}} D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∇^{2}η^{6}∂_{α}u_{m} + 2∇η^{6} ⊙ ∇∂_{α}u_{m}) dx. \end{matrix}

This immediately gives (2.5) by repeating the calculations leading from (2.8) to (2.9).

Step 3. Uniform higher integrability of ∇^{2}u_{m}. Let χ denote any real number satisfying χ > p/(2p−q), moreover we set α = χp/2. For all discs B_{r} ⋐ B_{R} ⋐ Ω_{2} any η ∈ C_{0}^{∞}(B_{R}), η ≡ 1 on B_{r}, |∇^{k}η| ≤ c/(R − r)^{k}, k = 1, 2, we have by Sobolev’s inequality

∫_{B_{r}}(1 + |∇^{2}u_{m}|^{2})^{α} dx ≤ ∫_{B_{R}}(η^{3}h_{m})^{2χ} dx ≤ c[∫_{B_{R}} |∇(η^{3}h_{m})|^{t} dx]^{2χ/t},

where t ∈ (1, 2) satisfies 2χ = 2t/(2 − t). Hölder’s inequality implies

\begin{aligned}∫_{B_{r}}(1 + |∇^{2}u_{m}|^{2})^{α} dx & ≤ c(r, R)[∫_{B_{R}} |∇(η^{3}h_{m})|^{2} dx]^{χ} \\ & ≤ c(r, R)[∫_{B_{R}} η^{6}|∇h_{m}|^{2} dx + ∫_{spt∇η} |∇η^{3}|^{2}h_{m}^{2} dx]^{χ}. \end{aligned}

Observing that obviously

∫_{spt∇η} |∇η^{3}|^{2}h_{m}^{2} dx ≤ c(r, R) ∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2})^{p/2} dx

and that by (2.5)

∫_{B_{R}} η^{6}|∇h_{m}|^{2} dx ≤ c(r, R) ∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2})^{(q−2)/2}[|∇^{2}u_{m}|^{2} + |∇u_{m}|^{2}] dx
≤ c(r, R)[∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2})^{q/2} dx + ∫_{spt∇η} |∇u_{m}|^{q} dx],

we deduce

(2.14)

∫_{B_{r}}(1 + |∇^{2}u_{m}|^{2})^{α} dx ≤ c(r, R)[∫_{spt∇η}(1 + |∇^{2}u_{m}|^{2})^{q/2} dx
+ ∫_{spt∇η} |∇u_{m}|^{q} dx]^{χ},

where c(r, R) = c(R − r)^{−β} for some suitable β > 0. For discussing (2.14) we first note that the term ∫_{spt∇η} |∇u_{m}|^{q} dx causes no problems. In fact, since ∥u_{m}∥_{W_{p}^{2}(Ω_{2})} ≤ c < ∞ we know that ∇u_{m} ∈ L_{loc}^{t}(Ω_{2}) for any t < ∞ in case p ≥ 2. If p < 2, then we have local L^{t} -integrability of ∇u_{m} provided that t < 2p/(2 − p), but q < 2p < 2p/(2 − p) on account of (1.3). As a consequence, we may argue exactly as in [ELM] or [Bi, p. 60], to derive from (2.14) by interpolation and hole-filling (here q < 2p enters in an essential way)

(2.15) ∇^{2}u_{m} ∈ L_{loc}^{t}(Ω_{2}) for any t < ∞ and uniformly with regard to m.

Note that (2.15) implies with Step 2 the uniform bound

(2.16)  ∫_{Ω_{2}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx ≤ c(η) < ∞,

in particular (2.16) shows

(2.17)  h_{m} ∈ W_{2,loc}^{1}(Ω_{2}) uniformly with regard to m.

Remark 2.2. (i) If u is a local J -minimizer subject to an additional constraint of the form u ≥ ψ a.e. on Ω for a sufficiently regular function ψ: Ω → R, then it is an easy exercise to adjust the technique used in [BF1] to the present situation which means that we still have (2.15) so that (recall (2.4)) u ∈ W_{t,loc}^{2}(Ω) for any t < ∞, hence u ∈ C^{1,α}(Ω) for all 0 < α < 1. In [Fr, Theorem 10.6, p. 98], it is shown for the special case f(w) = |Δw|^{2} that actually u ∈ C^{2}(Ω) is true, and it would be interesting to see if this result also holds for the energy densities discussed here.

(ii) We remark that the proof of (2.15) just needs the inequality q < 2p, whereas the additional assumption q < p + 2 enters in the next step.

Step 4. C^{2} -regularity. Now we consider an arbitrary disc B_{2R} ⋐ Ω_{1} and η ∈ C_{0}^{∞}(B_{2R}) satisfying η ≡ 1 on B_{R} and |∇η| ≤ c/R, |∇^{2}η| ≤ c/R^{2} . Moreover we denote by T_{2R} the annulus T_{2R} := B_{2R}−B_{R} and by P_{m} a polynomial function 358 Michael Bildhauer and Martin Fuchs of degree less than or equal to 2. Exactly as in Step 2 (replacing u_{m} by u_{m} −P_{m}) we obtain

∫_{B_{2R}} η^{6}D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m}) dx
≤ − ∫_{T_{2R}} D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∇^{2}η^{6}∂_{α}[u_{m} − P_{m}]
+ 2∇η^{6} ⊙ ∇∂_{α}(u_{m} − P_{m})) dx.

With the notation

H_{m} := [D^{2}f_{m}(∇^{2}u_{m})(∂_{α}∇^{2}u_{m}, ∂_{α}∇^{2}u_{m})]^{1/2}, σ_{m} := Df_{m}(∇^{2}u_{m}) we therefore have ∫_{B_{2R}} η^{6}H_{m}^{2} dx ≤ c ∫_{T_{2R}} |∇σ_{m}|[|∇^{2}η^{6}| |∇u_{m} − ∇P_{m}| + |∇η^{6}| |∇^{2}u_{m} − ∇^{2}P_{m}|] dx. Moreover, by the Cauchy–Schwarz inequality and (1.1)

|∇σ_{m}|^{2} ≤ H_{m}[D^{2}f_{m}(∇^{2}u_{m})(∂_{α}σ_{m}, ∂_{α}σ_{m})]^{1/2} ≤ H_{m}|∇σ_{m}|Γ_{m}^{(q−2)/4}, where Γ_{m} := 1 + |∇^{2}u_{m}|^{2} . Finally we let

h̃_{m} := max[Γ_{m}^{(q−2)/4}, Γ_{m}^{(2−p)/4}]

and obtain

|∇σ_{m}| ≤ cH_{m}Γ_{m}^{(q−2)/4} ≤ cH_{m}h̃_{m},

hence

(2.18)  \begin{aligned}∫_{B_{2R}} η^{6}H_{m}^{2} dx & ≤ c ∫_{T_{2R}} H_{m}h̃_{m}[|∇^{2}η^{6}| |∇u_{m} − ∇P_{m}| \\ & + |∇η^{6}| |∇^{2}u_{m} − ∇^{2}P_{m}|] dx. \end{aligned}

Letting γ = 4/3 we discuss the right-hand side of (2.18):

\begin{matrix}∫_{T_{2R}} H_{m}h̃_{m}|∇η^{6}| |∇^{2}u_{m} − ∇^{2}P_{m}| dx \\ ≤ \frac{c}{R}[∫_{B_{2R}}(H_{m}h̃_{m})^{γ} dx]^{1/γ}[∫_{B_{2R}} |∇^{2}u_{m} − ∇^{2}P_{m}|^{4} dx]^{1/4}. \end{matrix}

Next the choice of P_{m} is made more precise by the requirement

(2.19)  ∇^{2}P_{m} = ∫−_{B_{2R}} ∇^{2}u_{m} dx.

Then Sobolev–Poincaré’s inequality together with the definition of h̃_{m} gives

\begin{aligned}[∫_{B_{2R}} |∇^{2}u_{m} − ∇^{2}P_{m}|^{4} dx]^{1/4} & ≤ c[∫_{B_{2R}} |∇^{3}u_{m}|^{γ} dx]^{1/γ} \\ & ≤ c[∫_{B_{2R}}(H_{m}h̃_{m})^{γ} dx]^{1/γ}, \end{aligned}

hence

(2.20)  ∫_{T_{2R}} H_{m}h̃_{m}|∇η^{6}| |∇^{2}u_{m} − ∇^{2}P_{m}| dx ≤ \frac{c}{R}[∫_{B_{2R}}(H_{m}h̃_{m})^{γ} dx]^{2/γ}.

To handle the remaining term on the right-hand side of (2.18) we need in addition to (2.19)

∫−_{B_{2R}}(∇u_{m} − ∇P_{m}) dx = 0,

which can be achieved by adjusting the linear part of P_{m}. Then we have by Poincaré’s inequality

∫_{B_{2R}} H_{m}h̃_{m}|∇^{2}η^{6}| |∇u_{m} − ∇P_{m}| dx

≤ \frac{c}{R^{2}} [∫_{B_{2R}}(H_{m}h̃_{m})^{γ} dx]^{1/γ}[∫_{B_{2R}} |∇u_{m} − ∇P_{m}|^{4} dx]^{1/4}

≤ \frac{c}{R}[∫_{B_{2R}}(H_{m}h̃_{m})^{γ} dx]^{1/γ}[∫_{B_{2R}} |∇^{2}u_{m} − ∇^{2}P_{m}|^{4} dx]^{1/4},

and the right-hand side is bounded by the right-hand side of (2.20). Hence, recalling (2.18) and (2.20), we have established the inequality

(2.21)  [∫−_{B_{R}} H_{m}^{2} dx]^{γ/2} ≤ c ∫−_{B_{2R}}(H_{m}h̃_{m})^{γ} dx.

Given this starting inequality we like to apply the following lemma which is proved in [BFZ].

Lemma 2.1. Let d > 1, β > 0 be two constants. With a slight abuse of notation let f, g, h now denote any non-negative functions on Ω ⊂ R^{n} satisfying

f ∈ L_{loc}^{d}(Ω), exp(βg^{d}) ∈ L_{loc}^{1}(Ω), h ∈ L_{loc}^{d}(Ω).

Suppose that there is a constant C > 0 such that

[∫−_{B} f^{d} dx]^{1/d} ≤ C ∫−_{2B} fg dx + C[∫−_{2B} h^{d} dx]^{1/d}

holds for all balls B = B_{r}(x) with 2B = B_{2r}(x) ⋐ Ω. Then there is a real number c_{0} = c_{0}(n, d, C) such that if h^{d} log^{c_{0}β}(e+h) ∈ L_{loc}^{1}(Ω), then the same is true for f. Moreover, for all balls B as above we have

∫−_{B} f^{d} log^{c_{0}β}[e + \frac{f}{∥f∥_{d,2B}} ] dx ≤ c[∫−_{2B} exp(βg^{d}) dx][∫−_{2B} f^{d} dx]

+ c ∫−_{2B} h^{d} log^{c_{0}β}[e + \frac{h}{∥f∥_{d,2B}} ] dx,

where c = c(n, d, β, C) > 0 and ∥f∥_{d,2B} = (∫ −_{2B} f^{d} dx)^{1/d}.

The appropriate choices in the setting at hand are d = 2/γ = 3/2, f = H_{m}^{γ}, g = h̃_{m}^{γ}, h ≡ 0. We claim that

∫−_{B_{2R}} exp(h̃_{m}^{2}β) dx ≤ c and ∫−_{B_{2R}} H_{m}^{2} dx ≤ c

for a constant being uniform in m. The uniform bound of the second integral follows from (2.16); thus let us discuss the first one. By (2.17) and Trudinger’s inequality (see e.g. Theorem 7.15 of [GT]) we know that for any disc B_{ϱ} ⋐ Ω_{1}

∫_{B_{ϱ}} exp(β_{0}h_{m}^{2}) dx ≤ c(ϱ) < ∞,

where β_{0} just depends on the uniformly bounded quantities ∥h_{m}∥_{W_{2}^{1}(Ω_{1})} . This implies for any β > 0 and κ ∈ (0, 1)

∫_{B_{ϱ}} exp(βh_{m}^{2−κ}) dx ≤ c(ϱ, β, κ) < ∞.

Moreover, on account of q < p + 2 we have

Γ_{m}^{(q−2)/2} ≤ h_{m}^{2−κ} and clearly Γ_{m}^{(2−p)/2} ≤ h_{m}^{2−κ}

for κ sufficiently small, which gives our claim and we may indeed apply the lemma with the result

∫−_{B_{ϱ}} H_{m}^{2} log^{c_{0}β}(e + H_{m}) dx ≤ c(β, ϱ) < ∞

for all discs B_{ϱ} ⊂ Ω_{1} and all β > 0. Thus we have established the counterparts of (2.7) and (2.10) in [BFZ], and exactly the same arguments as given there lead to (2.11) from [BFZ]. Thus we deduce the uniform continuity of the sequence {σ_{m}} (see again [BFZ], end of Section 2), hence we have uniform convergence σ_{m} →: σ for some continuous tensor σ. In order to identify σ with Df(∇^{2}u), we recall the weak convergence stated in (2.4) and also observe that ∇^{2}u_{m} → ∇^{2}u a.e. which can be deduced along the same lines as in Lemma 4.5c) of [BF3], we also refer to Proposition 3.29 iii) of [Bi]. Therefore Df(∇^{2}u) is a continuous function, i.e. ∇^{2}u is of class C^{0} , and finally u ∈ C^{2}(Ω) follows.

Step 5. C^{2,α} -regularity of u. To finish the proof of Theorem 1.1 we observe that with Step 4 we get from (2.5) the estimate

∫_{Ω_{1}} |∇^{3}u_{m}|^{2} dx ≤ c(Ω_{1}) < ∞,

in particular one has for α = 1, 2

U := ∂_{α}u ∈ W_{2,loc}^{2}(Ω).

Moreover we have

∫_{Ω} D^{2}f_{m}(∇^{2}u_{m})(∇^{2}∂_{α}u_{m}, ∇^{2}φ) dx = 0 for any φ ∈ C_{0}^{∞}(Ω).

Together with the convergences (as m → ∞)

D^{2}f_{m}(∇^{2}u_{m}) → D^{2}f(∇^{2}u) in L_{loc}^{∞}(Ω),
∇^{2}∂_{α}u_{m} ⇁ ∇^{2}U in L_{loc}^{2}(Ω)

we therefore arrive at the limit equation

∫_{Ω} D^{2}f(∇^{2}u)(∇^{2}U, ∇^{2}φ) dx = 0.

Hence U is a weak solution of an equation with continous coefficients and u ∈ C^{2,α}(Ω) for any 0 < α < 1 follows from [GM, Theorem 4.1].

362 Michael Bildhauer and Martin Fuchs

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Received 1 April 2005