A Priori Bounds in L p and in W 2 , p for Solutions of Elliptic Equations

and Applied Analysis 3 Theorem 1. If g ∈ M(Ω), with q > 2 and λ = 0 if n = 2, and q ∈ ]2, n] and λ = n − q if n > 2, then the operator in (17) is bounded from ∘ W 1,2 (Ω) to L(Ω). Moreover, there exists a constant C ∈ R+ such that 󵄩󵄩󵄩󵄩gu 󵄩󵄩󵄩󵄩 L(Ω) ≤ C 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Mq,λ(Ω)‖u‖W1,2(Ω) ∀u ∈ ∘ W 1,2 (Ω) , (18) with C = C(n, q). Let p > 1 and r, t ∈ [p, +∞[. If Ω is an open subset of R having the cone property and g ∈ M(Ω), with r > p if p = n, then the operator in (17) is bounded from W(Ω) to L(Ω). Moreover, there exists a constant c ∈ R+ such that 󵄩󵄩󵄩󵄩gu 󵄩󵄩󵄩󵄩 L(Ω) ≤ c 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Mr(Ω)‖u‖W1,p(Ω) ∀u ∈ W 1,p (Ω) , (19) with c = c (Ω, n, p, r). If g ∈ M(Ω), with t > p if p = n/2, then the operator in (17) is bounded fromW(Ω) to L(Ω). Moreover, there exists a constant c ∈ R+ such that 󵄩󵄩󵄩󵄩gu 󵄩󵄩󵄩󵄩 L(Ω) ≤ c 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Mt(Ω)‖u‖W2,p(Ω) ∀u ∈ W 2,p (Ω) (20) with c = c(Ω, n, p, t). 3. The Variational Problem Consider, in an unbounded open subset Ω of R, n ≥ 2, the second-order linear differential operator in divergence form


Introduction
The aim of this work is to give an overview on some recent results dealing with the study of a certain kind of the Dirichlet problem in the framework of unbounded domains.To be more precise, given an unbounded open subset Ω of R  ,  ≥ 2, we are concerned with the elliptic second-order linear differential operator in variational form with coefficients   ∈  ∞ (Ω) and with the associated Dirichlet problem
In [1][2][3], the authors also provide the bound giving explicit description of the dependence of the constant  on the data of the problem.
In two recent works, [4,5], considering a more regular set Ω and supposing that the lower order terms coefficients are as in [3] for  ≥ 3 and as in [2] for  = 2, we prove that if  ∈  2 (Ω) ∩  ∞ (Ω), then there exists a constant , whose dependence is completely described, such that for any bounded solution  of (2) and for every  > 2.
This can be done taking into account two different sign hypotheses, namely, (5) and the less common Successively, in [6], we deepen the study begun in [4,5] showing that to a bounded datum  ∈  2 (Ω) it corresponds a bounded solution .This allows us to prove, by means of an approximation argument, that if  belongs to  2 (Ω) ∩   (Ω),  > 2, then the solution is in   (Ω) too and verifies (7).Putting together the two preliminary   -estimates,  > 2, obtained under the different sign assumptions and adding the further hypothesis that the   are also symmetric, by means of a duality argument, we finally obtain (7) for  > 1, for each sign hypothesis, assuming no boundedness of the solution and for  ∈  2 (Ω) ∩   (Ω).
To conclude, we provide two applications of our final   -bound,  > 1, recalling the results of [7,8] where our estimate plays a fundamental role in the study of certain weighted and non-weighted non-variational problems with leading coefficients satisfying hypotheses of Miranda's type (see [9]).The nodal point in this analysis is the existence of the derivatives of the leading coefficients that allows us to rewrite the involved operator in variational form and avail ourselves of the above-mentioned a priori bound.
Always in the framework of unbounded domains, the study of different variational problems can be found in [10,11].Quasilinear elliptic equations with quadratic growth have been considered in [12].In [13][14][15] a very general weighted case, with principal coefficients having vanishing mean oscillation, has been taken into account.

A Class of Spaces of Morrey Type
In this section we recall the definitions and the main properties of a certain class of spaces of Morrey type where the coefficients of our operators belong.These spaces generalize the classical notion of Morrey spaces to unbounded domains and were introduced for the first time in [3]; see also [16] for some details.Thus, from now on, let Ω be an unbounded open subset of R  ,  ≥ 2. By Σ(Ω) we denote the -algebra of all Lebesgue measurable subsets of Ω.For  ∈ Σ(Ω),   is its characteristic function, || its Lebesgue measure, and equipped with the norm defined in (9).

The Variational Problem
Consider, in an unbounded open subset Ω of R  ,  ≥ 2, the second-order linear differential operator in divergence form Assume that the leading coefficients satisfy the hypotheses For the lower order terms coefficients suppose that Furthermore, let one of the following sign assumptions hold true: in the distributional sense on Ω, with  positive constant.
It is natural to associate to  the bilinear form , and observe that, in view of Theorem 1, the form  is continuous on (Ω) and so the operator  : Let us start collecting some preliminary results concerning the existence and uniqueness of the solution of problem (22), as well as some a priori estimates.For the case where assumptions (ℎ 1 )-(ℎ 3 ) are taken into account and for  = 2, we refer to [2] while for  ≥ 3 details can be found in [3].If (ℎ 1 ), (ℎ 2 ), and (ℎ 4 ) hold true, the results are proved in the more recent [5].
The next step in our analysis is to achieve an   -estimate,  > 2, for the solution of (22) (see Theorem 8).This requires some additional hypotheses on the regularity of the set and on the datum , and some preparatory results that essentially rely on the introduction of certain auxiliary functions   , used for the first time by Bottaro and Marina in [1] and employed in the framework of Morrey type spaces in [3].Let us give their definition and recall some useful properties.
In order to prove a fundamental preliminary estimate, obtained for  > 2 (see Theorem 7), we need to take products involving the above defined functions   as test functions in the variational formulation of our problem (23).To be more precise, in the first set of hypotheses ((ℎ 1 )-(ℎ 3 )), the test functions needed are || −2   .The following result ensures that these functions effectively belong to Lemma 4, whose rather technical proof can be found in [4], is a generalization of a known result by Stampacchia (see [18], or [19] 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  to some problems arising for harmonic or subharmonic functions in the theory of potential.
Once achieved (31), always in [4], we could prove the next lemma.Let   be the functions of Lemma 3 obtained in correspondence of a given  ∈ and of a positive real number  specified in the proof of Lemma 4.1 of [4].One has the following.Lemma 5. Let  be the bilinear form defined in (23).If Ω has the uniform  1 -regularity property, under hypotheses (ℎ 1 )-(ℎ 3 ), there exists a constant  ∈ R + such that where  depends on , ], .
If we consider the second set of hypotheses ((ℎ 1 ), (ℎ 2 ), and (ℎ 4 )), the test functions required in (23) are the products and of a positive real number  specified in the proof of Lemma 4.1 of [5].In this last case and if Ω has the uniform  1 -regularity property, a result of [20] where  depends on , , ], .
The proof, which is different according to hypothesis (ℎ 3 ) or (ℎ 4 ), is essentially performed into two steps.In the first step, we show some regularity results, exploiting a technique introduced by Miranda in [21].Namely, we prove that if  ∈ ∘  1,2 (Ω) is the solution of ( 22) with  ∈  2 (Ω)∩ ∞ (Ω), then, the datum  being more regular, one also has  ∈  ∞ (Ω).Thus Theorem 7 applies giving that  ∈   (Ω) and satisfies (34).The second step consists in considering a datum  ∈  2 (Ω) ∩   (Ω) and then one can conclude by means of some approximation arguments; see also [16].Finally, in [6], we prove the main result, that is, the claimed   -bound,  > 1.To this aim, a further assumption on the leading coefficients is required: Then one has the following.
Proof.For  ≥ 2, Theorems 2 and 8 already prove the result.
It remains to show it for 1 <  < 2.
As a consequence of Theorem 2 (in the second set of hypotheses) the solution  of (46) exists and is unique.Furthermore, by Theorem 8 (in the second set of hypotheses) one also has Hence, if we denote by  the solution of Finally, taking  = || −1 sign  in (49), we get the claimed result.

Non-Variational Problems
In this section, we show two applications of our main estimate (43).
To this aim, let  > 1 and assume that Ω has the uniform  1,1 -regularity property.
The first application is contained in Theorem 3.2 and Corollary 3.3 of [7] (see also [22] where the case  = 2 is considered) and reads as follows.Moreover, the problem is uniquely solvable.
The nodal point in achieving these results consists in the existence of the derivatives of the   .Indeed, this consents to rewrite the operator  in divergence form and exploit (43) in order to obtain an estimate as that in (51) but for more regular functions.Then, one can prove (51) by means of an approximation argument.Estimate (51) immediately takes to the solvability of problem (52) via a straightforward application of the method of continuity along a parameter, see, for instance, [23], and by the already known solvability of an opportune auxiliary problem.
As second application of (43), we obtain, in [8], an analogous of Theorem 10, in a weighted framework.Namely, we consider a weight function   that is a power of a function  of class   One of the main tools in the proof of Theorem 11 is given by the existence of a topological isomorphism from  ,  (Ω) to  , (Ω) and from ∘  ,  (Ω) to ∘  , (Ω).This isomorphism consents to deduce by the non-weighted bound in (51) the corresponding weighted estimate in (56), taking into account also the imbedding results of Theorem 1.The existence and uniqueness of the solution of problem (57) follow then, as in the previous case, from a direct application of the method of continuity along a parameter by the solvability of a suitable auxiliary problem.