LIPSCHITZ REGULARITY FOR SOME ASYMPTOTICALLY SUBQUADRATIC PROBLEMS

Chiara Leone* - Antonia Passarelli di Napoli** - Anna Verde***

Dipartimento di Matematica “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy *e-mail:chileone@unina.it **e-mail:antonia.passarelli@unina.it ***e-mail:anverde@unina.it

ABSTRACT. We establish a local Lipschitz regularity result for local minimizers of variational integrals under the assumption that the integrand becomes appropriately elliptic at infinity. The exponent that measures the ellipticity of the integrand is assumed to be less than two.

AMS Classifications. 35B65, 35J70.

Key words. Local minimizers, decay estimates, asymptotic behaviour.

1 Introduction

In this note we deal with the local Lipschitz regularity for local minimizers of variational integrals of the type

I(u) = ∫_{Ω} f(Du)dx,  (1)

where Ω is a bounded open subset of IR^{n}, u : Ω → IR^{N} is a vector valued function and Du stands for the total derivative of u.

A function u ∈ W^{1,p}(Ω, IR^{N}) is a local minimizer of I(u) if I(u) ≤ I(u + φ), for every test function φ ∈ W_{0}^{1,p}(Ω, IR^{N}) with compact support in Ω.

The regularity of local minimizers of I has been widely investigated in case the integrand f ∈ C^{2} is assumed to behave like |ξ|^{p} and to be convex or quasiconvex. In 1977, K. Uhlenbeck (see [14]), proved everywhere C^{1,α} regularity for local minimizers of functional of this type, with p ≥ 2. This result was generalized first allowing dependence of the integrand on (x, u) (see [11], [12]), and next considering the case 1 < p < 2 (see [1] , [5]).

But, in view of applications, it is interesting also the study of the regularity of minimizers of functionals with nonconvex or, more generally, not stronlgy elliptic integrands. Simple examples show that for such integrands minimizers need not to be C^{1}.

On the other hand, C^{0,1} regularity results are avalaible when f ∈ C^{2} is asymptotically, in a C^{2}-sense, quadratic or superquadratic at infinity (see [4] for the case p = 2 and [12] and [13] for the case p > 2).

The heuristic idea of this kind of results is that the gradients of minimizers can not be very large, otherwise the problem becomes “regular” and so good estimates are available. A motivation of this fact has been suggested in [12].

As far as we know, no regularity results are available in case the integrand f ∈ C^{2} is asymptotically, in a C^{2}-sense, subquadratic at infinity.

In this paper we fiil this gap extending the results found in [4], [12], and [13] to the case 1 < p < 2. More precisely we have the following theorem.

Theorem 1.1. Let p ∈ (1, 2), let f : IR^{nN} → IR satisfy the following assumptions for a suitable μ > 0

f ∈ C^{2}(IR^{nN})  (2)

|D^{2}f(ξ)| ≤ L(μ^{2} + |ξ|^{2})^{\frac{p−2}{2}} , for all ξ ∈ IR^{nN}  (3)

lim_{|ξ|→∞} |D^{2}f(ξ) − D^{2}H(ξ)|(μ^{2} + |ξ|^{2})^{\frac{2−p}{2}} = 0,  (4)

where H(ξ) := (μ^{2} + |ξ|^{2})^{\frac{p}{2}} . If u ∈ W^{1,p}(Ω, IR^{N}) is a local minimizer of

I(u) = ∫_{Ω} f(Du)dx,

then Du is locally bounded in Ω. Moreover, for almost every x_{0} ∈ Ω, we have

|Du(x_{0})| ≤ C(1 + (∫_{B_{R_{0}}(x_{0})} |Du|^{p}dx)^{\frac{1}{p}} ),  (5)

where R_{0} = dist(x_{0}, ∂Ω) and C depends on n, N, L, p, μ.

Let us finally remark that, since our assumption on f concerns only its behaviour for large deformations, not assuming, for example, that f is everywhere convex or quasiconvex, there is no evidence that a minimizer of (1) exists. In other words, here we are dealing with the question of regularity separately from questions of existence of minimizers. We have also to mention that a recent work of Dolzmann and Kristensen (see [6]) has been concerned with finding local higher integrability with large exponents of minimizers when f ∈ C^{0} approaches at infinity, in a C^{0}-sense, the p-Dirichlet integrand, for some arbitrary p > 1.

2 Technical Lemmas

In the sequel Ω will denote a bounded open set of IR^{n} and B_{R}(x_{0}) the ball {x ∈ IR^{n} : |x − x_{0}| < R}. To simplify the notation, the letter c will denote any positive constant, which may vary throughout the paper. If u ∈ L^{p}, for any B_{R}(x_{0}) we set

u_{x_{0},R} = \frac{1}{|B_{R}|} ∫_{B_{R}(x_{0})} u(x)dx = ∫_{B_{R}(x_{0})} u(x)dx.  (6)

In the following p will be a number in the interval (1, 2); for μ > 0 and for ξ ∈ IR^{k} we define the following function

V (ξ) = (μ^{2} + |ξ|^{2})^{\frac{p−2}{4}} ξ.  (7)

We begin by giving the following basic inequality (see Lemma 2.1 in [1]).

Lemma 2.1. For every γ ∈ (−\frac{1}{2}, 0) and μ ≥ 0 we have

1 ≤ \frac{∫_{0}^{1}(μ^{2} + |η + t(ξ − η)|^{2})^{γ}dt}{(μ^{2} + |η|^{2} + |ξ|^{2})^{γ}} ≤ \frac{8}{2γ + 1},  (8)

for all ξ, η not both zero if μ = 0.

Next Lemma contains some useful properties of the function V (see [7]).

Lemma 2.2. Let p ∈ (1, 2) and V be the function defined in (7), then for every ξ, η ∈ IR^{k}, t > 0

\begin{matrix}(1 + μ^{2})^{\frac{p−2}{4}} min\{|ξ|, |ξ|^{\frac{p}{2}} \} ≤ |V (ξ)| ≤ |ξ|^{\frac{p}{2}} \\ |V (tξ)| ≤ max\{t, t^{\frac{p}{2}} \}|V (ξ)| \\ |V (ξ + η)| ≤ c(p)[|V (ξ)| + |V (η)|] \\ \frac{p}{2}|ξ − η| ≤ _{\frac{|V (ξ)−V (η)|}{(μ^{2}+|ξ|^{2}+|η|^{2})^{\frac{p−2}{4}}}} ≤ c(k, p)|ξ − η| \\ |V (ξ) − V (η)| ≤ c(k, p)|V (ξ − η)|. \end{matrix}  (9)

The next regularity theorem can be found in [1] (see Proposition 2.11 ).

Proposition 2.3. Let x_{0} ∈ Ω, B_{R}(x_{0}) ⊂ Ω, and let u ∈ W^{1,p}(Ω, IR^{N}). If v is a minimizer of the functional

∫_{B_{R}(x_{0})} H(Dw)dx

such that

w − u ∈ W_{0}^{1,p}(B_{R}(x_{0}), IR^{N})

then there exist two constants c > 0 and σ > 0, both independent of μ and of R, such that

∫_{B_{ρ}(x_{0})} |V (Dv) − V (Dv)_{x_{0},ρ}|^{2}dx
≤ c(\frac{ρ}{R})^{σ}∫_{B_{R}(x_{0})} |V (Dv) − V (Dv)_{x_{0},R}|^{2}dx  (10)

for every ρ < R.

Finally we will prove the following Lemma, that is the analogue of Lemma 5.1 in [12] when 1 < p < 2.

Lemma 2.4. Let p ∈ (1, 2) and let f, g : IR^{nN} → IR be two functions such that, for some μ > 0,

f, g ∈ C^{2}(IR^{nN})  (11)

|D^{2}f(ξ)| , |D^{2}g(ξ)| ≤ L(μ^{2} + |ξ|^{2})^{\frac{p−2}{2}} , for all ξ ∈ IR^{nN}  (12)

lim_{|ξ|→∞} |D^{2}f(ξ) − D^{2}g(ξ)|(μ^{2} + |ξ|^{2})^{\frac{2−p}{2}} = 0.  (13)

Then for every ε > 0 there exist σ(ε) > 0 and ω(ε) > 0 such that the following inequality

\begin{matrix}|⟨∫_{0}^{1}(1 − t)[D^{2}f(tξ + (1 − t)ξ_{0}) − D^{2}g(tξ + (1 − t)ξ_{0})]dt(ξ − ξ_{0}), (ξ − ξ_{0})⟩| \\ ≤ c ω(ε)(|ξ − ξ_{0}|^{2} + λ^{2})(μ^{2} + |ξ|^{2} + |ξ_{0}|^{2})^{\frac{p−2}{2}} \end{matrix} (14)

holds for all ξ, ξ_{0} ∈ IR^{nN} and λ ≥ 0 satisfying |ξ_{0}|^{2} + λ^{2} > σ^{2}(ε). Here lim_{ε→0} ω(ε) = 0 and c depends only on L and p.

Proof. Assumption (13) implies that for every ε > 0 there exists Λ(ε) > 0 such that if |ξ| > Λ(ε) then

|D^{2}f(ξ) − D^{2}g(ξ)| ≤ ε(μ^{2} + |ξ|^{2})^{\frac{p−2}{2}}  (15)

Now, for ξ_{0} ∈ IR^{nN}, consider the set I = {t ∈ [0, 1] : |tξ + (1 − t)ξ_{0}| ≤ Λ(ε)} and split the integral in the left hand side of (14) into the sum of the integrals on I and (0, 1) \ I, respectively. Thus, thanks to Lemma 2.1 and inequality (15),

\begin{matrix}|⟨∫_{(0,1)\I}(1 − t)[D^{2}f(tξ + (1 − t)ξ_{0}) − D^{2}g(tξ + (1 − t)ξ_{0})]dt(ξ − ξ_{0}), (ξ − ξ_{0})⟩| \\ ≤ \frac{8}{p−1}ε|ξ − ξ_{0}|^{2}(μ^{2} + |ξ|^{2} + |ξ_{0}|^{2})^{\frac{p−2}{2}} . \end{matrix} (16)

Assumption (12), Hölder’s inequality and again Lemma 2.1 yield

\begin{matrix}|⟨∫_{I}(1 − t)[D^{2}f(tξ + (1 − t)ξ_{0}) − D^{2}g(tξ + (1 − t)ξ_{0})]dt(ξ − ξ_{0}), (ξ − ξ_{0})⟩| \\ ≤ 2L ∫_{I}(μ^{2} + |tξ + (1 − t)ξ_{0}|^{2})^{\frac{p−2}{2}} dt|ξ − ξ_{0}|^{2} \\ ≤ 2L|I|^{\frac{1}{α}} (∫_{0}^{1}(μ^{2} + |tξ + (1 − t)ξ_{0}|^{2})^{\frac{p−2}{2} α′}dt)^{\frac{1}{α′}} |ξ − ξ_{0}|^{2} \\ ≤ 2L(\frac{8}{(p−2)α′+1})^{\frac{1}{α′}} |I|^{\frac{1}{α}} |ξ − ξ_{0}|^{2}(μ^{2} + |ξ|^{2} + |ξ_{0}|^{2})^{\frac{p−2}{2}} \end{matrix} (17)

where \frac{1}{α} + \frac{1}{α′} = 1 and α′ is choosen strictly less than \frac{1}{2−p} in order to apply Lemma 2.1 with γ = \frac{p−2}{2}α′. Let us now estimate |I|. If S denotes the segment with end points ξ, ξ_{0}, we have

|I| = \frac{|S ∩ \{ξ ∈ IR^{nN} : |ξ| ≤ Λ(ε)\}|}{|ξ − ξ_{0}|} ≤ \frac{2Λ(ε)}{|ξ − ξ_{0}|}.

Let us choose

σ^{2}(ε) = (1 + \frac{4L}{ε})^{2} Λ^{2}(ε) + (\frac{4L}{ε})^{2} Λ^{2}(ε),

and note that if

|ξ − ξ_{0}| > \frac{4L}{ε}Λ(ε)

then

|I| ≤ \frac{ε}{2L}.

On the other hand if

|ξ − ξ_{0}| ≤ \frac{4L}{ε}Λ(ε),

let λ^{2} + |ξ_{0}|^{2} > σ^{2}(ε); then

λ^{2} > (\frac{4L}{ε}Λ(ε))^{2} or |ξ_{0}|^{2} > (1 + \frac{4L}{ε})^{2} Λ^{2}(ε).

In the first case we have

|I|^{\frac{1}{α}} |ξ − ξ_{0}|^{2} ≤ 2^{4− \frac{1}{α}} L^{2− \frac{1}{α}} Λ^{\frac{1}{α}} (ε) (\frac{Λ(ε)}{ε})^{2− \frac{1}{α}} ≤ (2L)^{− \frac{1}{α}} λ^{2}ε^{\frac{1}{α}} .

In the second case one can easily see that |I| = 0.

3 Proof of the main result

We are now in position to prove Theorem 1.1. Proof of Theorem 1.1. Let x_{0} ∈ Ω, R ≤ dist(x_{0}, ∂Ω), and let v be a minimizer of

∫_{B_{R}(x_{0})} H(Dw)dx

such that

w − u ∈ W_{0}^{1,p}(B_{R}(x_{0}), IR^{N}).

From Proposition 2.3, we have that

∫_{B_{ρ}(x_{0})} |V (Du) − V (Du)_{x_{0},ρ}|^{2}dx
= ∫_{B_{ρ}(x_{0})} |V (Du) − V (Dv) + V (Dv)_{x_{0},ρ} + V (Dv) − V (Dv)_{x_{0},ρ} − V (Du)_{x_{0},ρ}|^{2}dx
≤ 2∫_{B_{ρ}(x_{0})} |V (Du) − V (Dv)|^{2}dx + ∫_{B_{ρ}(x_{0})} |V (Dv) − V (Dv)_{x_{0},ρ}|^{2}dx
≤ c(\frac{ρ}{R})^{σ}∫_{B_{R}(x_{0})} |V (Dv) − V (Dv)_{x_{0},R}|^{2}dx
+ c(\frac{R}{ρ})^{n}∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx
= c(\frac{ρ}{R})^{σ}∫_{B_{R}(x_{0})} |V (Dv) − V (Du) + V (Du)_{x_{0},ρ} + V (Du) − V (Du)_{x_{0},ρ} − V (Dv)_{x_{0},R}|^{2}dx
+ c(\frac{R}{ρ})^{n}∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx
≤ c(\frac{ρ}{R})^{σ}∫_{B_{R}(x_{0})} |V (Du) − V (Du)_{x_{0},R}|^{2}dx
+ c(\frac{R}{ρ})^{n}∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx  (18)

for every ρ < R. Choose ξ_{0,R} such that V (ξ_{0,R}) = (V (Du))_{x_{0},R}. It is easy to see that v is also a local minimizer of

∫_{B_{R}(x_{0})} g(Dw)dx

such that

w − u ∈ W_{0}^{1,p}(B_{R}(x_{0}), IR^{N}),

where

g(ξ) = H(ξ) + f(ξ_{0,R}) − H(ξ_{0,R}) + [Df(ξ_{0,R}) − DH(ξ_{0,R})](ξ − ξ_{0,R}). The Euler equation for v and the ellipticity of g yield

∫_{B_{R}(x_{0})}[g(Du) − g(Dv)]dx
= ∫_{B_{R}(x_{0})}⟨∫_{0}^{1}(1 − t)D^{2}g(Dv + t(Du − Dv))dt(Du − Dv), (Du − Dv)⟩ dx
≥ c ∫_{B_{R}(x_{0})} ∫_{0}^{1}(1 − t)(μ^{2} + |Dv + t(Du − Dv)|^{2})^{\frac{p−2}{2}} dt|Du − Dv|^{2} dx
≥ c ∫_{B_{R}(x_{0})}(μ^{2} + |Du|^{2} + |Dv|^{2})^{\frac{p−2}{2}} |Du − Dv|^{2} dx
≥ c ∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2} dx,  (19)

where we used lemma 2.2 (note that c > 0 depends only on p, n, N). Now, since u is a minimizer for f, we write

\begin{matrix}∫_{B_{R}(x_{0})}[g(Du) − g(Dv)]dx = ∫_{B_{R}(x_{0})}[g(Du) − f(Du)] dx \\ +∫_{B_{R}(x_{0})}[f(Du) − f(Dv)] dx + ∫_{B_{R}(x_{0})}[f(Dv) − g(Dv)]dx \\ ≤ ∫_{B_{R}(x_{0})}[g(Du) − f(Du)] dx + ∫_{B_{R}(x_{0})}[f(Dv) − g(Dv)]dx \end{matrix}  (20)

and observe that

\begin{matrix}f(ξ) − g(ξ) = \\ ⟨∫_{0}^{1}(1 − t)[D^{2}f((1 − t)ξ_{0,R} + tξ) − D^{2}g((1 − t)ξ_{0,R} + tξ)]dt \\ ·(ξ − ξ_{0,R}), (ξ − ξ_{0,R})⟩. \end{matrix}  (21)

Setting

λ^{2} = (μ^{2} + |ξ_{0,R}|^{2})^{\frac{2−p}{2}}∫_{B_{R}(x_{0})} |V (Du) − (V (Du))_{x_{0},R}|^{2}dx,  (22)

from Lemma 2.2 it follows that , for every ε > 0, if

(μ^{2} + |ξ_{0,R}|^{2})^{\frac{2−p}{2}}∫_{B_{R}(x_{0})} |V (Du)|^{2} dx > σ^{2}(ε),  (23)

then

|ξ_{0,R}|^{2} + λ^{2} > σ^{2}(ε).

Thus, thanks to Lemma 2.4, (19), (20), and (21) imply

∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx

≤ c ω(ε)\{∫_{B_{R}(x_{0})} |Du − ξ_{0,R}|^{2}(μ^{2} + |Du|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx

+ λ^{2}∫_{B_{R}(x_{0})}(μ^{2} + |Du|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx

+ ∫_{B_{R}(x_{0})} |Dv − ξ_{0,R}|^{2}(μ^{2} + |Dv|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx

+ λ^{2}∫_{B_{R}(x_{0})}(μ^{2} + |Dv|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx\}.

Since (μ^{2} + |Du|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} , (μ^{2} + |Dv|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} ≤ (μ^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} , we have by the definition of λ^{2} at (22)

λ^{2}∫_{B_{R}(x_{0})}(μ^{2} + |Du|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx ≤ ∫_{B_{R}(x_{0})} |V (Du) − (V (Du))_{x_{0},R}|^{2}dx,

λ^{2}∫_{B_{R}(x_{0})}(μ^{2} + |Dv|^{2} + |ξ_{0,R}|^{2})^{\frac{p−2}{2}} dx ≤ ∫_{B_{R}(x_{0})} |V (Du) − (V (Du))_{x_{0},R}|^{2}dx.

Then, by (9)_{iv} we have

∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx

≤ c ω(ε){∫_{B_{R}(x_{0})} |V (Du) − V (ξ_{0,R})|^{2}dx + ∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx}

Hence, there exists ε_{0} such that for every ε < ε_{0}, we get

∫_{B_{R}(x_{0})} |V (Du) − V (Dv)|^{2}dx ≤ c ω(ε)∫_{B_{R}(x_{0})} |V (Du) − V (ξ_{0,R})|^{2}dx

which together with (18) yields

∫_{B_{ρ}(x_{0})} |V (Du) − (V (Du))_{x_{0},ρ}|^{2}dx

≤ c ((\frac{ρ}{R})^{n+σ} + ω(ε)) ∫_{B_{R}(x_{0})} |V (Du) − (V (Du))_{x_{0},R}|^{2}dx,

for every 0 < ρ < R and for a constant c = c(L, p, n, N). Choose 0 < τ < 1, ε_{1} < ε_{0} such that

2cτ^{σ} < \frac{1}{2}, ω(ε_{1}) < τ^{n+σ}

Then for every R < R_{0} =dist(x_{0}, ∂Ω) we have

(μ^{2} + |ξ_{0,R}|^{2})^{\frac{2−p}{2}}∫_{B_{R}(x_{0})} |V (Du)|^{2} dx ≤ σ^{2}(ε_{1})  (24)

or

∫_{B_{τR}(x_{0})} |V (Du) − (V (Du))_{x_{0},τR}|^{2}dx ≤ \frac{1}{2}∫_{B_{R}(x_{0})} |V (Du) − (V (Du))_{x_{0},R}|^{2}dx (25)

Fix x_{0} such that

lim_{r→0}∫_{B_{r}(x_{0})} |V (Du) − (V (Du))(x_{0})|^{2} dx = 0.

Now, we set

0 ≤ R_{1} = inf{0 < R ≤ R_{0} : (24) holds for R or R = R_{0}}.

If R_{1} = 0, then (24) holds for a sequence R_{j} → 0, from which we deduce that

|V (Du)(x_{0})| ≤ σ(ε_{1}),  (26)

where σ(ε_{1}) = c(p, L, n, N, Λ). If 0 < R_{1} ≤ R_{0} let, for k ∈ IN, k ≥ 2,

R_{k} = τ^{k−1}R_{1},

which is less than or equal to R_{1}, and note from (25) that

∫_{B_{R_{k}} (x_{0})} |V (Du) − (V (Du))_{x_{0},R_{k}}|^{2}dx

≤ \frac{1}{2^{k−1}}∫_{B_{R_{1}}(x_{0})} |V (Du) − (V (Du))_{x_{0},R_{2}}|^{2}dx

≤ \frac{1}{2^{k−1}}∫_{B_{R_{1}}(x_{0})} |V (Du)|^{2}dx.

Since

|(V (Du))_{x_{0},R_{k+1}}−(V (Du))_{x_{0},R_{k}}| ≤ (\frac{1}{τ^{n}}∫_{B_{R_{k}} (x_{0})} |V (Du) − (V (Du))_{x_{0},R_{k}}|^{2}dx)^{\frac{1}{2}} ,

we get

|V (Du)(x_{0})| ≤ ∑_{k=2}^{∞} |(V (Du))_{x_{0},R_{k+1}} − (V (Du))_{x_{0},R_{k}}| + |(V (Du))_{x_{0},R_{1}}|

≤ c (∫_{B_{R_{1}}(x_{0})} |V (Du)|^{2}dx)^{\frac{1}{2}} .

If R_{1} < R_{0} one obtains by (24)

(∫_{B_{R_{1}}(x_{0})} |V (Du)|^{2}dx)^{\frac{1}{2}} ≤ μ^{\frac{p−2}{2}} c(ε_{1}, p, L, n, N) = c(μ, ε_{1}, p, L, n, N).(27)

It is worth pointing out that this positive constant depends also on Λ(ε),i. e. the speed of convergence in the limit (13), through the constant σ^{2}(ε_{1}). If R_{1} = R_{0}

|V (Du)(x_{0})| ≤ c(∫_{B_{Ro}(x_{o})} |Du|^{p}dx)^{\frac{1}{p}} ,

with c = c(n, N, L, p). In conclusion, we proved that

|Du(x_{o})| ≤ c(1 + (∫_{B_{Ro}(x_{o})} |Du|^{p}dx)^{\frac{1}{p}} ),

where c depends on p, L, n, N and on Λ(ε).

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