An Lp-Estimate for Weak Solutions of Elliptic Equations

and Applied Analysis 3 2. A Generalization of a Result by Stampacchia Let G : t ∈ R −→ G t 2.1 be a uniformly Lipschitz real function, such that there exists a positive constant K such that for every t′, t′′ ∈ R one has ∣G ( t′ ) −Gt′′∣ ≤ K∣t′ − t′′∣, 2.2 and suppose that G| −k,k 0, for a k ∈ R 2.3 and that its derivative G′ has a finite number of discontinuity points. A known result by Stampacchia, see Lemma1.1 in 1 or in 13 , for details , guarantees that given a function u, defined in an open bounded subset of R and belonging to ◦ W1,2, also the composition between G and u is in ◦ W1,2 and gives an explicit expression for the derivative of this composition, up to sets of null Lebesgue measure. Later on, in 7 , Bottaro and Marina explicitly observed that, up to few modifications, the proof of these results remains valid also for an unbounded open subset Ω of R, n ≥ 2. More precisely, u ∈ ◦ W1,2 Ω ⇒ G u G ◦ u ∈ ◦ W1,2 Ω , 2.4 and moreover G u xi G ′ u uxi, a. e. in Ω, i 1, . . . , n. 2.5 In Lemma 2.2 below, we show a further generalization of 2.4 , always in the case of unbounded domains. In order to prove Lemma 2.2, we need the following convergence results. Lemma 2.1. IfΩ has the uniform C1-regularity property, then for every u ∈ ◦ W1,2 Ω ∩L∞ Ω , there exists a sequence Φh h∈ N of functions such that Φh ∈ C∞ o Ω , Φh −→ u in ◦ W1,2 Ω , sup h∈ N ‖Φh‖L∞ Ω ≤ ‖u‖L∞ Ω . 2.6 If G is a uniformly Lipschitz function as in 2.1 , 2.2 , and 2.3 and such that its derivative G′ has a finite number of discontinuity points, G Φh −→ G u in L2 Ω , 2.7 4 Abstract and Applied Analysis G Φh ⇀ G u weakly in ◦ W1,2 Ω . 2.8 Moreover, there exists a sequence gh h∈ N such that gh −→ G u in ◦ W1,2 Ω , 2.9 where gh ∑h j 1 cj G Φj with cj ≥ 0 and ∑h j 1 cj 1. Proof. The statement in 2.6 has been proved in 14 . The L2-convergence in 2.7 easily follows by 2.2 and by the convergence in 2.6 . The ◦ W1,2-convergences in 2.8 and 2.9 can be obtained as in the proof of Lemma 1.1 of 13 , with opportune modifications due to the fact that the set Ω is unbounded see also 7 . We point out that next lemma is a fundamental tool in our analysis since it is the core of the proof of Lemma 3.3 that will allow us to take some specific test functions in the variational formulation of our problem. This will consent to show a technical result see Lemma 4.1 , which is the main point in the proof of our L-a priori bound. Lemma 2.2. Let G be a uniformly Lipschitz function as in 2.1 , 2.2 , and 2.3 and such that its derivative G′ has a finite number of discontinuity points. IfΩ has the uniform C1-regularity property, then for every u ∈ ◦ W1,2 Ω ∩ L∞ Ω one has |u|p−2G u ∈ ◦ W1,2 Ω , ∀p ∈ 2, ∞ . 2.10 Proof. Fix u ∈ ◦ W1,2 Ω ∩ L∞ Ω ; to show 2.10 we need different arguments according to different values of p. For 2 < p < 3 we need to verify that there exists a positive constant c such that ∣∣∣∣ ∫ Ω |u|p−2G u φxidx ∣∣∣∣ ≤ c ∥φ ∥∥ L2 Ω , ∀φ ∈ C1 o R , ∀i 1, . . . , n, 2.11 this ends the proof of our lemma as a consequence of a characterization of the space ◦ W1,2 Ω see, e.g., Proposition IX.18 of 15 . In order to prove 2.11 , we consider the sequence Φh h∈ N introduced in Lemma 2.1 and observe that, given φ ∈ C1 o R , one has ∫ Ω |u|p−2G u φxidx lim h→ ∞ ∫ Ω |Φh|G Φh φxidx, 2.12


Introduction
The Dirichlet problem for second order linear elliptic partial differential equations in divergence form and with discontinuous coefficients in bounded open subsets of R n , n ≥ 2, is a classical problem that has been widely studied by several authors we refer, e.g., to 1-6 .
In this paper, we want to analyze certain aspects of the same kind of problem, but in the framework of unbounded domains.
More precisely, given an unbounded open subset Ω of R n , n ≥ 2, we are interested in the study of the elliptic second order linear differential operator in variational form with coefficients a ij ∈ L ∞ Ω , and in the following associated Dirichlet problem

2 Abstract and Applied Analysis
Starting from a work of Bottaro and Marina see 7 , who proved an existence and uniqueness theorem for the solution of problem 1.2 , for n ≥ 3, assuming that analogous results have been successively obtained weakening the hypotheses on the lower order terms coefficients.First generalizations in this direction have been carried on in 8 , where n ≥ 2 and b i , d i , and c satisfy assumptions similar to those in 1.3 , but only locally.While in 9 , for n ≥ 3, these results have been further improved, since b i , d i , and c are assumed to belong to opportune Morrey type functional spaces with lower summability.
In the above-mentioned works 7-9 , the authors also provide the estimate where the dependence of the constant C on the data of the problem is completely described.
Here we suppose that the lower order terms coefficients are as in 9 for n ≥ 3 and as in 8 for n 2 and we prove an L p -a priori bound, p > 2.More precisely, for a sufficiently regular set Ω and given a datum f ∈ L 2 Ω ∩ L ∞ Ω , we show that there exists a constant C such that for any bounded solution u of 1.2 and for every p ∈ 2, ∞ .We point out that also in our analysis the dependence of the constant C is fully determined.We note that bound 1.5 can be also useful when dealing with certain nonvariational problems that, by means of the existence of the derivatives of the a ij , can be rewritten in variational form.
Among the authors who studied the Dirichlet problem for second order linear elliptic equations in divergence form with discontinuous coefficients in unbounded domains, we quote here also Lions in 10, 11 and Chicco and Venturino in 12 .
The proof of 1.5 is developed as follows.In Section 2 we extend a known result by Stampacchia see 1 , or 13 for details , obtained within the framework of the generalization of the study of certain elliptic equations in divergence form with discontinuous coefficients on a bounded open subset of R n to some problems arising for harmonic or subharmonic functions in the theory of potential.This is done in order to obtain a preliminary lemma, proved in Section 3, that permits to consider some particular test functions in the variational formulation of our problem.This allows us to prove a technical result Lemma 4.1 , that is the main point in the proof of the claimed L p -estimate.

A Generalization of a Result by Stampacchia
Let be a uniformly Lipschitz real function, such that there exists a positive constant K such that for every t , t ∈ R one has and suppose that and that its derivative G has a finite number of discontinuity points.
A known result by Stampacchia, see Lemma 1.1 in 1 or in 13 , for details , guarantees that given a function u, defined in an open bounded subset of R n and belonging to • W 1,2 , also the composition between G and u is in 2 and gives an explicit expression for the derivative of this composition, up to sets of null Lebesgue measure.
Later on, in 7 , Bottaro and Marina explicitly observed that, up to few modifications, the proof of these results remains valid also for an unbounded open subset If G is a uniformly Lipschitz function as in 2.1 , 2.2 , and 2.3 and such that its derivative G has a finite number of discontinuity points, Moreover, there exists a sequence g h h∈ N such that where g h h j 1 c j G Φ j with c j ≥ 0 and h j 1 c j 1.
Proof.The statement in 2.6 has been proved in 14 .
The L 2 -convergence in 2.7 easily follows by 2.2 and by the convergence in 2.6 .

The
• W 1,2 -convergences in 2.8 and 2.9 can be obtained as in the proof of Lemma 1.1 of 13 , with opportune modifications due to the fact that the set Ω is unbounded see also 7 .
We point out that next lemma is a fundamental tool in our analysis since it is the core of the proof of Lemma 3.3 that will allow us to take some specific test functions in the variational formulation of our problem.
This will consent to show a technical result see Lemma 4.1 , which is the main point in the proof of our L p -a priori bound.Lemma 2.2.Let G be a uniformly Lipschitz function as in 2.1 , 2.2 , and 2.3 and such that its derivative G has a finite number of discontinuity points.If Ω has the uniform C 1 -regularity property, then for every u ∈

we need different arguments according to different values of p.
For 2 < p < 3 we need to verify that there exists a positive constant c such that this ends the proof of our lemma as a consequence of a characterization of the space In order to prove 2.11 , we consider the sequence Φ h h∈ N introduced in Lemma 2.1 and observe that, given for i 1, . . ., n.
Indeed, by H ölder inequality we get and this quantity vanishes letting h → ∞, as a consequence of 2.6 and 2.7 .
On the other hand,

2.14
Having in mind 2.12 , we want to pass to the limit as h → ∞ in the right-hand side of this equality.
Concerning the first term, by 2.2 , Hölder inequality, and using the last relation in 2.6 , we obtain Thus, by the convergence in 2.6 , the quantity on the left-hand side goes to zero, letting h → ∞, and therefore For the last term we have

2.18
Moreover, by the weak convergence in 2.8 the first term on the right-hand side vanishes letting h → ∞.Concerning the second one, we get and, by 2.6 and 2.8 , also this quantity is null passing to the limit as h → ∞.
It remains to treat the second term of the right-hand side of 2.14 .To this aim let us introduce the sets where k is that of 2.3 .We observe that, in view of 2.6 , there exists h 0 ∈ N such that, up to sets of null Lebesgue measure, 2.21 and we can assume, without loss of generality, that h 0 1.Therefore, by 2.3 and 2.21 , one has

2.22
On the other hand, always using 2.6 , we can also deduce, with no loss of generality, that , for a. e. x ∈ D, ∀h ∈ N.

2.23
This, together with 2.6 and 2.7 , and by definition of D, gives, up to a subsequence,

2.24
Moreover, by 2.2 and 2.6 , Therefore, 2.24 and 2.25 being true, the bounded convergence theorem applies giving, up to a subsequence, for i 1, . . ., n, that is 2.11 .
For p ≥ 3, let us consider the sequence g h h∈ N introduced in Lemma 2.1 and put

2.28
Simple calculations give with c 1 positive constant depending only on p.We want to pass to the limit in the right-hand side of this inequality.For the first term it is easily seen that it goes to zero, in view of 2.9 .
For the last one, again from 2.9 , we get, up to a subsequence, x −→ 0, for a. e. x ∈ Ω.

2.30
Moreover, by 2.2 it follows that x , for a. e. x ∈ Ω, ∀h ∈ N.

2.31
Hence, from these last considerations and using the bounded convergence theorem we obtain, up to a subsequence, Therefore, by 2.29 , up to a subsequence, we have Now, observe that |u| p−2 g h ∈ • W 1,2 Ω , because of its compact support, then for any h ∈ N there exists a sequence this means that there exists m h ∈ N such that

2.35
By 2.33 and 2.35 we deduce that this ends the proof of our lemma.

Tools
We recall the definitions of the Morrey type spaces where the lower order terms coefficients of the operator will be chosen.These functional spaces were introduced for the first time in 9 in order to generalize to the case of unbounded domains of the classical notion of Morrey spaces.We start with some notation.Given any Lebesgue measurable subset F of R n , we denote by Σ F the σ-algebra of all Lebesgue measurable subsets of F. For any E ∈ Σ F , χ E is its characteristic function and E x, r is the intersection E ∩ B x, r x ∈ R n , r ∈ R , where B x, r is the open ball centered in x and with radius r.
For q ∈ 1, ∞ and λ ∈ 0, n , the space of Morrey type M q,λ Ω is the set of all the functions g in L q loc Ω such that For reader's convenience, we state here a result of 16 , adapted to our needs, providing the boundedness and an embedding estimate for the multiplication operator where the function g belongs to a suitable space of Morrey type M q,λ Ω .
Lemma 3.1.If g ∈ M q,λ Ω , with q > 2, λ 0 if n 2, and q ∈ 2, n , λ n − q if n > 2, then the operator in 3.2 is bounded.Moreover, there exists a constant c ∈ R such that Now, we recall a lemma, proved in 9 , describing the main properties of some functions u s , introduced in 7 , that will be of crucial relevance in the proof of our main result.
Let h ∈ R ∪ { ∞} and k ∈ R, with 0 ≤ k ≤ h.For each t ∈ R we set u x i u s x j u s x i u s x j , s 1, . . ., r, i, j 1, . . ., n, 3.9 12 with c c ε, q, g M q,λ Ω positive constant.
As already mentioned, the next lemma will allow us, in the last section, to take the products |u| p−2 u s as test functions in the variational formulation of our problem.

Lemma 3.3. If Ω has the uniform C 1 -regularity property, then for every
where u s , for s 1, . . ., r, are the functions of Lemma 3.2.
, by means of Lemma 2.2.In view of these considerations and 3.11 being true, we also get

An A Priori Bound
Let Ω be an unbounded open subset of R n , n ≥ 2, such that Ω has the uniform C 1 -regularity property.
h 0 We consider in Ω the second order linear differential operator in variational form with the following conditions on the coefficients: in the distributional sense on Ω.
We also associate to L the bilinear form We point out that, as a consequence of Lemma 3.1, a is continuous on Ω and so the operator L : Ω is continuous as well.We start showing a technical lemma.Let u s be the functions of Lemma 3.2 obtained in correspondence of a given u ∈

4.4
From this last equality, 3.8 , and hypotheses 21 and 24 we get

4.5
Hence, setting On the other hand, by the H ölder inequality, Lemmas 3.2 and 3.3, the embedding results contained in Lemma 3.1 and using hypothesis 23 and 3.7 , one has that there exists a constant c 0 ∈ R , such that with c 0 c 0 n, t .Now, we observe that explicit computations give   The linearity of a together with 3.11 and 3.12 then give 4.17 .Now, using 4.17 and H ölder inequality we end the proof, since with C C n, t, p, ν, μ, b i − d i M 2t,λ Ω .Indeed, if we consider the functions u s , s 1, . . ., r, obtained in correspondence with the solution u, of g and ε as in Lemma 4.1, by 3.11 we get Ω c 0 c 0 r .Thus, taking into account 4.3 , |u| p−2 u h ≤ C r s 1 a u, |u| p−2 u s , 4.19 with C s C s s, ν, μ and C C r, ν, μ .

2 x u 2
dx ≤ C a u, |u| p−2 u C Ω f|u| p−2 u dx ≤ C Ω f |u| p−1 dx ≤ C f L p Ω u and of a positive real number ε specified in the proof of Lemma 4.1.The following result holds true.