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ON THE RELATION BETWEEN GENERALIZED MORREY SPACESAND MEASURE DATA PROBLEMS

PAOLO BARONI

ABSTRACT. We consider measure data problems of p-Laplacian type. The measure onthe right-hand side has the property that the total variation of a generic ball decays in termsof generic functions of the radius; we show that this condition has a natural relation withgradient integrability properties and we get, as corollary, borderline cases of classic results.

To Carlo Sbordone, mathematician and neapolitan gentleman,on the occasion of his 70th birthday.

1. INTRODUCTION

We study measure data problems of p-Laplacian type:

div a(x,Du) = in , (1.1)

where is a bounded open set in Rn, n 2. The Carathedory vector field a() satisfiesthe growth and monotonicity assumptionsa(x, 1) a(x, 2), 1 2

(s+ |1|+ |2|

)p2|1 2|2,|a(x, )| L

(s+ ||)p1

(1.2)

for almost every x and for all 1, 2, Rn, with 0 < 1 L and s 0. Theright-hand side is a signed Borel measure with finite total total variation, and we supposein general that it does not belong to the dual of the energy space naturally associated to theoperator on the left-hand side. For this reason in the paper we shall always assume p n.

The goal of this study is the analysis of some borderline cases for the integrability of thegradient of solutions to (1.1), when the measure on the right-hand side is known to havecertain decay properties. In particular we suppose

L1,()() := supBR(x0)

R1

(R)||(BR(x0))

Rn

2 PAOLO BARONI

way to encode such regularity is in terms of Marcinkiewicz spaces: this is to say, we locallyestimate the decay of the measure of the super level set of Du{x : |Du(x)| > }in terms of functions related to . Integrability properties of the gradient of solutions tomeasure data problems are usually formulated in terms of Marcinkiewicz spaces, whichare the optimal ones in view of the behavior of the fundamental solutions; see for instance[2, 10, 20, 22] and [4, 5]. Clearly, generalized Marcinkiewicz spaces must be consideredhere, due to the generality of the situation we are considering.

In this first part of the paper we just want to propose significant corollaries to the mainestimate in generalized Marcinkiewicz spaces we are going to describe in Theorem 2.5.We think that the most interesting corollary is the following

Corollary 1.1. Let [p, n]. If u W 1,p() is a solution to (1.1), where Lloc()satisfies the assumption (1.3) with

(R) = R log(1/R) with > 1 (1.4)or

(R) = R log1(1/R) log(log(1/R)) with > 1 (1.5)for all R R0, for some R0 (0, 1], then

|Du|p1 L1loc () (1.6)

and the local estimate(BR(x0)

|Du|(p1)1 dx

) 1(p1)

cB2R(x0)

(|Du|+ s+ 1

)dx

+ c

[R

(R)

] 1p1

1p1L1,()()

(1.7)

holds for every ball such that B2R(x0) , R 1, with the constant depending onn, p, , L, (or ), () and R0.

Let us stress that in the case = p, the previous result gives that if the measure decaysas

||(BR(x0))Rn

.1

Rp log(1/R)for some > p 1

then |Du| Lploc(). On the other hand, we notice that in this case a classic result by Wolff(see the forthcoming (2.1)) states that the measure belongs to the dual of the energy space,and it is therefore natural to solve (1.1) obtaining an energy solution (that are, in particular,SOLAs). Notice moreover that once fixed an exponent q [n(p 1)/(n 1), p], theprevious result, choosing

=q

q (p 1)implies that is sufficient to set

(R) = Rq

q(p1) log(1/R), >p 1

q (p 1)or

(R) = Rq

q(p1) logp1

q(p1) (1/R) log(log(1/R)), >p 1

q (p 1)close to zero in order to have the gradient in Lq locally.

GENERALIZED MORREY SPACES AND MEASURE DATA PROBLEMS 3

One could also consider conditions ensuring Orlicz regularity, in the particular class ofZygmund spaces, for the gradient:

Corollary 1.2. Let [p, n) and R or = n and 1. If u W 1,p() is as inCorollary 1.1 and Lloc() satisfies the assumption (1.3) with

(R) = R log(1/R) with > ( + 1)( 1), (1.8)or

(R) = R log(+1)(1)(1/R) log(log(1/R)) with > 1, (1.9)for R R0 for some R0 (0, 1]; then

|Du|p1 L1 log L (1.10)

locally in and the estimate

DuL

1 log L(BR(x0))

cB2R(x0)

(|Du|+ s+ 1

)dx

+ c

[R

(R)

] 1p1

1p1L1,()()

(1.11)

holds for a constant depending on n, p, , L, (respectively, ), () and R0.

We recall the reader that a measurable function g : A Rn R belongs to the Orliczspace L

1 log L(A) = L

log L(A) for q > 1, R, if

A

|g|log(e+ |g|) dx 0 :

1

A

|g|log

(e+|g|

)dx 1

}.

Remark 1.3. Note that the previous results are stated as a priori estimates for energysolutions; it is standard to extend them to SOLAs, that is, solutions of (1.1) obtained asa limit of energy solutions with regularized data; see [3, 8, 15, 18] for details and for theprecise form of the local estimates in this case.

The result contained in Corollary 1.1 is the borderline (in terms of integrability of Du)case of the results in [20, 23] stating the following:

L1,(), [p, n] = |Du|p1 M1loc (). (1.12)

The space L1,() is a classic Morrey space, that in our case corresponds to the simple

choice (R) = R; on the other hand, the spaceM1loc () is the localized version of the

classic Marcinkiewicz, or weak Lebesgue, space (see (2.7)-(2.8) for () = ). Note

that for < p the aforementioned classic result of Wolff (see (2.1)) states that L1, embedsinto the dual space of W 1,p. In the case = n, the results in [20] give back the classic,sharp result

L1,n() Mb() = |Du| Mn(p1)n1 (),

for which we refer to [11]; notice that the class of measures satisfying (1.3) for = n isnothing else than the full space of signed Borel measures with finite total total variation.We stress that a different assumption implying the regularity in (1.10), dealing with betterintegrability instead of better decay properties of the datum, is contained in [21, Theorems3 & 12]:

L1,() L logL() or L logL()

4 PAOLO BARONI

imply

|Du|p1 L1loc ();

note that again the improvement should be of logarithmic type. We recall the reader thatL logL is the subspace of the Orlicz-Zygmund space L logL made of the functions such that their L logL norm on balls decays in terms of powers of the radius:

L logL(BR) . R.

As a final remark in the elliptic setting, we stress that we are able to reproduce anothersubtle phenomenon typical of elliptic equations with measure data: density information onthe measure transfer into density information for the solutions. In particular, a more refinedversion of the result depicted in (1.12), which can still be found in [20, 23], states that notonly Du M(BR(x0)) for every ball B2R(x0) , but also gives a description of thedecay of this norm in a way perfectly consistent with the decay of the measure. Indeed onehas, for every ball as before

||(BR(x0)) . Rn, [p, n] =|Du|p1M (BR(x0)) . Rn,

see [23, Theorem 4.3]. We refer to the forthcoming Remark 2.7 for a suitable version inour setting.

Result in the setting of classic Morrey spaces are also available in the more difficultparabolic setting: here the condition to be considered involves standard parabolic cylinders

supQR(z0)(0,T )

||(QR(z0))RN

GENERALIZED MORREY SPACES AND MEASURE DATA PROBLEMS 5

the open ball with center x0 and radius r > 0; when not important, or clear from thecontext, we shall omit denoting the center as follows: Br Br(x0). Very often, when nototherwise stated, different balls in the same context will share the same center. We shallalso denoteB1 = B1(0) if not differently specified. Finally, withB being a given ball withradius r and being a positive number, we denote by B the concentric ball with radiusr. With B Rn being a measurable subset with finite and positive measure |B| > 0, andwith g : B Rk, k 1, being a measurable map, we shall denote by

(g)B Bg(x) dx :=

1

|B|

Bg(x) dx

its integral average. We set

p := min{p 1, 1}.

and often we shall denote in short L1,() for L1,()(). For R, we shall also usethe notation [log t] , when log t is nonnegative, for log t. N is the set {1, 2, . . . } whileN0 = N {0}.

2.2. Properties of . A classic result by Wolff (see [14, Corollary of Theorem 1] and [1]too) implies that the a measure belongs to the dual space of W 1,p(Rn) if and only if itsWolff potential W1,p belongs to L

1(Rn, d). The local version of this results says that ifthe measure satisfies

10

((B(x))

np

) 1p1 d

d(x) = + (2.1)

then it does not belong to the dual of W 1,p(). Since in our case

10

(||(B(x))np

) 1p1 d

d||(x)

1p1L1,()()

||() 1

0

(p

()

) 1p1 d

once we want to treat the measure data setting, it is natural to suppose that 10

(p

()

) 1p1 d

=

10

(

()

) 1p1

d = +. (2.2)

We shall use the notation

f() =

(

()

) 1p1

(2.3)

and we take : [1,) (0, 1] as

() := f1();

recall that f(1) = 1 and notice that is decreasing. Our main assumption on will bethe following one: There exist constants H0 and 0, both larger than one, and a functionh : [H0,) [0,) such that

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