Optimal Interior Partial Regularity for Nonlinear Elliptic Systems for the Case under Natural Growth Condition

Abstract

We consider the interior regularity for weak solutions of second-order nonlinear elliptic systems with subquadratic growth under natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particularly the regularity we obtained is optimal.

1. Introduction

In this paper we consider optimal interior partial regularity for the weak solutions of nonlinear elliptic systems with subquadratic growth under natural growth condition of the following type:

(1.1)

where is a bounded domain in , and taking values in , and has value in . , stand for the of and . To define weak solution to (1.1), one needs to impose certain structural and regularity conditions on and the inhomogeneity , as well as to restrict to a particular class of functions as follows, for ,

(E1) are differentiable functions in and there exists such that

(1.2)

(E2) is uniformly strongly elliptic, that is, for some , we have

(1.3)

(E3) There exists and monotone nondecreasing such that

(1.4)

for all , and ; without loss of generality, we take .

Furthermore (E1) allows us to deduce the existence of a function with for all such that is monotone nondecreasing for fixed , is concave and monotone nondecreasing for fixed , and such that for all and , we have

(1.5)

(E4) there exist constants and , such that

(1.6)

or

()

(1.7)

Definition 1.1.

By a weak solution of (1.1) with structure assumptions (E1)–(E4) (or (E4)), we mean a vector valued function such that

(1.8)

for all .

Even under reasonable assumptions on and , in the case of systems (i.e., ) one cannot, in general, expect that weak solutions of (1.1) will be classical, that is, -solutions. This was first shown by De Giorgi [1, 2]. The goal, then, is to establish partial regularity theory. We refer the reader to monographs of Giaquinta [3, 4] for an extensive treatment of partial regularity theory for systems of the form (1.1), as well as more general elliptic systems.

In the class direct proofs, one "freezes the coefficients" with constant coefficients. The solution of the Dirichlet problem associated to these coefficients with boundary data and the solution itself can then be compared. This procedure was first carried out by Giaquinta and Modica [5].

But the technique of harmonic approximation is to show that a function which is "approximately-harmonic" lies close to some harmonic function. This technique has its origins in Simon's proof [6] of the regularity theorem of Allard [7]. Which also be used in [8] to find a so-called -regularity theorem for energy minimizing harmonic maps. The technique of harmonic approximation allows the author to simplify the original -regularity theorem due to Schoen and Uhlenbeck [9].

In the remarkable proof when given by Duzaar and Grotowski in [10], the key difference is that the solution is compared not to the solution of the Dirichlet problem for the system with frozen coefficients, but rather to an A-harmonic function which is close to in , where is a function corresponding from weak solutions. In particular, the optimal regularity result can be obtained. In [11, 12], we deal with the optimal partial regularity of the weak solution to (1.1) for the case by the method of A-harmonic approximation technique, which is advantage to the result of [13]. The extension of A-harmonic approximation technique also can be found in [14, 15].

The purpose of this paper is to establish the optimal partial regularity of weak solution to (1.1) under natural growth condition with subquadratic growth, that is, the case of , directly. Indeed the main difficulty in our setting is that the exponent of the integral function is negative (), which means we cannot use the amplify technique as usually. Motivated by the technique used in [16], where the authors considered the minimizers of nonquadratic functional, we removed the hinder at last. And then with the help of -harmonic approximation technique, one can find a -harmonic function, which is close to a function in sense of , the function is which we defined in Lemma 4.2 and which is a corresponding function from the weak solution . Thanks to the standard results of linear theory presented in Section 2 and the elementary inequalities, we obtain the decay estimate of

(1.9)

and the optimal regularity. Now we may state the main result.

Theorem 1.2.

Let be a weak solution of (1.1) with . Suppose that the natural growth conditions (E1)–(E4) (or (E4)) and hold. Then there exists that is open in and for is defined in (E3). Furthermore,

(1.10)

where

(1.11)

In particular, .

2. The -Harmonic Approximation Technique and Preliminary Lemmas

In this section, we present the -harmonic approximation lemma, the key ingredient in proving our regularity result, and some useful preliminaries will be need in later. At first, we introduce two new functions.

Throughout the paper we will use the functions and defined by

(2.1)

for each and for any . From the elementary inequality

(2.2)

applied to the vector we deduce that

(2.3)

which immediately yields

(2.4)

The purpose of introducing is the fact that in contrast to , the function is a convex function on . This can easily be shown as follows. Firstly a direct computation yields that is convex and monotone increasing on with . Secondly we have

(2.5)

for any .

We use a number of properties of which can be found in [17, Lemma ].

Lemma 2.1.

Let and be the functions defined in (2.1). Then for any and there holds:

(i)

(ii)

(iii)

(iv)

(v)

(vi)for allwith

The inequalities (i)–(iii) also hold if we replace by .

For later purposes we state the following two simple estimates which can easily be deduced from Lemma 2.1(i) and (vi). For with we have for the estimate

(2.6)

as for we have

(2.7)

The next result we would state is the -harmonic approximation lemma, which is prove in [18].

Lemma 2.2 (-harmonic approximation lemma).

Let be positive constants. Then for any there exist with the following property. For any bilinear form on which is elliptic in the sense of Legendre-Hadamard with ellipticity constant and upper bound , for any satisfying

(2.8)

for all , there exists an -harmonic function satisfying

(2.9)

Definition 2.3.

Here a function is called -harmonic if it satisfies

(2.10)

for all .

Then we would recall a simple consequence of the a prior estimates for solutions of linear elliptic systems of second order with constant coefficients; see [17, Proposition ] for a similar result.

Lemma 2.4.

Let be such that

(2.11)

for any , where is elliptic in the sense of Legendre-Hadamard with ellipticity constant and upper bound . Then and

(2.12)

where the constant depends only on , and .

The next lemma is a more general version of [17, Lemma ], which itself is an extension of [3, Lemma , Chapter V]. The proof in which can easily be adapted to the present situation by replacing the condition of homogeneity by Lemma 2.1(ii).

Lemma 2.5.

Let , and be a nonnegative bounded function satisfying

(2.13)

for all . Then there exists a constant such that

(2.14)

And then we state a Poincare type inequality involving the function , which have been found in [17] and, in a sharp way, in [18].

Lemma 2.6 (Poincare-type inequality).

Let and , then

(2.15)

where . In particular, the previous inequality is valid with replaced by .

We conclude the section with an algebraic fact can be retrieved again from [16], Lemma 2.1.

Lemma 2.7.

For every and , one has

(2.16)

for any , not both zero if .

3. A Caccioppoli Second Inequality

For , we define and we simply write .

In order to prove the main result, our first aim is to establish a suitable Caccioppoli inequality.

Lemma 3.1 (Caccioppoli second inequality).

Let be a weak solution of (1.1) with and hold under natural growth conditions (E1)–(E4) (or (E4)). Then for every , and arbitrary with , one has

(3.1)

for

(3.2)

where and the constant .

Proof.

Let . Choose and a standard cut off function with on , which satisfies . For and , let

(3.3)

and define

(3.4)

Then

(3.5)

and further there holds

(3.6)

Using hypothesis (E2), from Lemma 2.7, and as the elementary inequality

(3.7)

we can get

(3.8)

A simple calculation yields

(3.9)

By (E1), Lemma 2.7 and (3.7), there holds

(3.10)

Noting that supp and , one can take the domain into , and four parts, and then by Young inequality and the estimations (2.6) and (2.7), thus there is

(3.11)

From the structure condition (E3) yields

(3.12)

Similar to , we split the domain of integration into four parts as follows. And on the part , we see

(3.13)

as on the set , there are

(3.14)

and on the case , one can get

(3.15)

Finally, noting that , then for the case , there exists a constant such that

(3.16)

Combining these estimations on , we have

(3.17)

for

And noting that , and that , and similarly as , we see

(3.18)

and for positive to be fixed later, we have

(3.19)

On the part , argue anginous as and , by Young's inequality and (2.6) and (2.7), we have

(3.20)

Similarly, on the part , we see

(3.21)

and on the part ,

(3.22)

and on the part ,

(3.23)

Combining these estimates in , and noting that and , we have

(3.24)

Finally, on we use Lemma 2.1(iv) and (vi) to bound the integrand of the left-hand side of (3.9) from below:

(3.25)

Using this in (3.9) together with the estimates , , , and we finally arrive at

(3.26)

The proof is now completed by applying Lemma 2.5.

4. The Proof of the Main Theorem

In this section we proceed to the proof of the partial regularity result and hence consider to be a weak solution of (1.1). Then we have the following.

Lemma 4.1.

Consider and with . Furthermore fixed in and set and . Then for the weak solution to systems (1.1) with and being hold, there holds

(4.1)

for and where one defines

(4.2)

for and

Proof.

We assume initially that . Applying Lemma 2.7 and noting that the definition of the weak solution of (1.1), for , we deduce

(4.3)

Rearranging this, we find

(4.4)

Using the structure condition (E1) and the estimate (1.5) for the modulus of continuity of , by Lemma 2.7 and let

(4.5)

we can derive

(4.6)

Noting that the estimates (2.6) and (2.7), using first Hölder's inequality and then Jensen's inequality:

(4.7)

here we have used for .

By (E3), Young inequality, (2.6), (2.7), and noting the function monotone nondecreasing and and that , we can estimate as follows

(4.8)

for .

Similar to (3.11), to estimate , one can divide the domain as previously mentioned. On the set , for

(4.9)

while on the part and noting that ,

(4.10)

On

(4.11)

Finally, on the case , there exists a constant such that

(4.12)

for

Whereas, Lemma 2.1 yields

(4.13)

where is defined in Lemma 3.1.

Noting that , and by Young's inequality, we see

(4.14)

On , by (2.7) and Young inequality, we have

(4.15)

On the other hand, on , using (2.6) and Young inequality, we have

(4.16)

Thus

(4.17)

Combining these estimates and noting that definition of , we derive

(4.18)

By Lemma 2.6, there is

(4.19)

Combining the above of with (4.4) and noting the definition of , we can get the lemma immediately.

We next establish an initial excess-improvement estimate, assuming that the excess is initially sufficient small. We also define , and , where stands for the constants form Lemma 2.1(vi). The precise statement is the following.

Lemma 4.2 (excess-improvement).

Consider weak solution satisfying the conditions of Theorem 1.2 and fixed in (E3). Then we can find positive constants , and and (with depends only on , , , and and with , and depending only on these quantities as well as ) such that the smallness condition :

(4.20)

together imply the growth condition

(4.21)

Here one uses the abbreviate .

Proof.

For to be determined later, we take to be corresponding constant from the -harmonic approximation lemma, that is, Lemma 2.2, and set

(4.22)

where stands for the constant from Lemma 2.1(vi).

Then, from (2.4) and Lemma 2.1(vi), we have

(4.23)

And by Lemma 4.1 and the smallness condition

(4.24)

we can deduce

(4.25)

Inequalities (4.23) and (4.25) fulfill the condition of -harmonic approximation lemma, which allow us to apply Lemma 2.2. Therefore we can find a function which is -harmonic such that

(4.26)

With the help of Lemma 2.1(iii) and (v), we have

(4.27)

where the constant depends only on , and .

We proceed to estimate the right-hand side of (4.27). Decomposing into the set with and that with , that using Lemma 2.1(i) and Hölder inequality, we obtain

(4.28)

where we have abbreviated

(4.29)

Now, since and is monotone increasing, we deduce from (4.27), also by using Lemma 2.1(i) and (ii), that there holds

(4.30)

where depends only on , and . Therefore it remains for us to estimate the quantity . By considering the cases and seperately and keeping in mind (4.26), we have (using Lemma 2.1(i)):

(4.31)

Using the assumption and Lemma 2.4, this shows

(4.32)

Lemma 3.1 applied on with , respectively , instead of , respectively, ; note that the constant depends only on :

(4.33)

for

(4.34)

Lemma 2.1(iii) yields

(4.35)

where the constant is given by . To estimate the right-hand side of (4.33) we use (2.4), Lemma 2.1(ii) (note that ) and (4.26) to infer

(4.36)

Using Lemma 2.1(i), Taylor's theorem applied to on , Lemma 2.4 and (4.31), we obtain

(4.37)

Using the smallness condition and (4.32) together with the definition of yields

(4.38)

Combining all the above estimates with (4.33), and let for , we get

(4.39)

where the constant depends only on , and (the dependency from occurs due to the fact that depends on ). Choose suitable such that , and inserting this into (4.30) we easily find (recalling also that ):

(4.40)

where the constant has the same dependencies as .

The regularity result then follows from the fact that this excess-decay estimate for any in a neighborhood of . From this estimate we conclude (by Campanato's characterization of Hölder continuous functions [19, 20]) that has the modulus of continuity by a constant times . By Lemma 2.1(iv) this modulus of continuity carries over to .