Dirichlet problem for divergence form elliptic equations with discontinuous coefficients

  • Sara Monsurrò1 and

    Affiliated with

    • Maria Transirico1Email author

      Affiliated with

      Boundary Value Problems20122012:67

      DOI: 10.1186/1687-2770-2012-67

      Received: 27 February 2012

      Accepted: 15 June 2012

      Published: 28 June 2012

      Abstract

      We study the Dirichlet problem for linear elliptic second order partial differential equations with discontinuous coefficients in divergence form in unbounded domains. We establish an existence and uniqueness result and we prove an a priori bound in L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif, p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq2_HTML.gif.

      MSC:35J25, 35B45, 35R05.

      Keywords

      elliptic equations discontinuous coefficients a priori bounds

      1 Introduction

      We are interested in the Dirichlet problem
      { u W i 1 , 2 ( Ω ) , L u = f , f W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ1_HTML.gif
      (1.1)
      where Ω is an unbounded open subset of R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq3_HTML.gif n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq4_HTML.gif, and L is a linear uniformly elliptic second order differential operator with discontinuous coefficients in divergence form
      L = i , j = 1 n x j ( a i j x i + d j ) + i = 1 n b i x i + c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ2_HTML.gif
      (1.2)

      If Ω is bounded, this problem is classical in literature and has been deeply analyzed taking into account various kinds of hypotheses on the coefficients (for more details see, for instance, [16]).

      Considering unbounded domains, as far as we know, the first work on this subject goes back to [7], where Bottaro and Marina provide, for n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq5_HTML.gif, an existence and uniqueness result for the solution of problem (1.1) assuming that
      a i j L ( Ω ) , i , j = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ3_HTML.gif
      (1.3)
      b i , d i L n ( Ω ) , i = 1 , , n , c L n / 2 ( Ω ) + L ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ4_HTML.gif
      (1.4)
      c i = 1 n ( d i ) x i μ , μ R + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ5_HTML.gif
      (1.5)

      In this order of ideas, various generalizations have been performed still maintaining hypotheses (1.3) and (1.5) but weakening the condition (1.4). Indeed in [8], where the case n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq4_HTML.gif is considered, b i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq6_HTML.gif d i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq7_HTML.gif and c are supposed to satisfy assumptions as those in (1.4), but just locally. Successively in [9], for n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq5_HTML.gif, further improvements have been carried on since b i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq6_HTML.gif d i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq7_HTML.gif and c are in suitable Morrey-type spaces with lower summabilities.

      In [79] we also find the bound
      u W 1 , 2 ( Ω ) C f W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ6_HTML.gif
      (1.6)

      where the dependence of the constant C on the data of the problem is fully determined.

      More recently, in [10], supposing that the coefficients of lower-order terms are as in [9] for n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq8_HTML.gif and as in [8] for n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq9_HTML.gif, we showed that, for a sufficiently regular set Ω, and if f L 2 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq10_HTML.gif, then there exists a constant C, whose dependence is completely described, such that
      u L p ( Ω ) C f L p ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ7_HTML.gif
      (1.7)

      for any bounded solution u of (1.1) and for every p ] 2 , + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq11_HTML.gif.

      Here, in the same framework but replacing the classical hypothesis of sign (1.5) by the less common one
      c i = 1 n ( b i ) x i μ , μ R + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ8_HTML.gif
      (1.8)

      we establish two kinds of results for the solution of (1.1). First of all, we provide an existence and uniqueness theorem, then, taking into account an additional assumption on the regularity of the boundary of Ω, we prove the analogue of (1.7).

      Let us briefly survey the way these results are achieved. In Section 2, we introduce the tools needed in the sequel. The definitions and some features of the Morrey-type spaces are given and some functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, related somehow to the solution of the problem and to the coefficients of the operator, are described, together with some specific properties. Section 3 is devoted to the solvability of problem (1.1). We start proving, by means of the above mentioned functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, the estimate in (1.6) that leads also to the uniqueness at once. Then, in view of well-known results of the operator theory, we get the existence verifying that L is a Fredholm operator with zero index. In the last section, we prove the claimed L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif-estimate. This is done by means of a technical lemma, exploiting again the functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, which allows us to conclude.

      Considering the case p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq13_HTML.gif, we notice that, as a consequence of (1.6), the bound (1.7) is true under both sign hypotheses even supposing no regularity on the boundary of Ω.

      We believe that the two estimates (1.7), obtained under the different sign assumptions, combined together should permit to prove, by means of a duality argument, that (1.7) holds true actually for any p ] 1 , + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq14_HTML.gif, considering one of the hypotheses (1.5) or (1.8) at a time.

      For further studies of the Dirichlet problem for linear elliptic second order differential equations with discontinuous coefficients in divergence form in unbounded domains we refer the reader also to [1113].

      2 Tools

      This section is devoted to the definitions and to some fundamental properties of the Morrey-type spaces where the coefficients of lower-order terms of our operator belong, and of some functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif related to the solution of the problem and to all the coefficients of the operator (see the proofs of Theorem 3.1 and Lemma 4.1 for more details on this aspect) that are indispensable tools in the sequel.

      Given an unbounded open subset Ω of R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq3_HTML.gif, n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq4_HTML.gif, we denote by Σ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq15_HTML.gif the σ-algebra of all Lebesgue measurable subsets of Ω. For any E Σ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq16_HTML.gif, χ E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq17_HTML.gif is its characteristic function and E ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq18_HTML.gif is the intersection E B ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq19_HTML.gif ( x R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq20_HTML.gif, r R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq21_HTML.gif), where B ( x , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq22_HTML.gif is the open ball centered in x and with radius r.

      For q [ 1 , + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq23_HTML.gif and λ [ 0 , n [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq24_HTML.gif, the space of Morrey type M q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq25_HTML.gif is the set of all the functions g in L l o c q ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq26_HTML.gif such that
      g M q , λ ( Ω ) = sup τ ] 0 , 1 ] x Ω τ λ / q g L q ( Ω ( x , τ ) ) < + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equa_HTML.gif

      endowed with the norm above defined. Moreover, M q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq27_HTML.gif denotes the closure of C ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq28_HTML.gif in M q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq25_HTML.gif. These functional spaces generalize the classical notion of Morrey spaces to the case of unbounded domains and were introduced in [9] (we refer also to [14] where further characteristics are considered).

      For the reader’s convenience, in the next lemma we recall some results of [15] and [8, 9] concerning the multiplication operator
      u W i 1 , 2 ( Ω ) g u L 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ9_HTML.gif
      (2.1)

      where the function g belongs to suitable spaces of Morrey type.

      Lemma 2.1 If g M q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq29_HTML.gif, with q > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq30_HTML.gifand λ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq31_HTML.gifif n = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq9_HTML.gif, and q ] 2 , n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq32_HTML.gifand λ = n q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq33_HTML.gifif n > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq34_HTML.gif, then the operator in (2.1) is bounded and there exists a constant c R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq35_HTML.gifsuch that
      g u L 2 ( Ω ) c g M q , λ ( Ω ) u W 1 , 2 ( Ω ) u W i 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ10_HTML.gif
      (2.2)

      with c = c ( n , q ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq36_HTML.gif.

      Moreover, if g M q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq37_HTML.gif, then the operator in (2.1) is also compact.

      Now, let us deal with the above mentioned functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif. They were employed for the first time in [7] and were studied in the framework of Morrey-type spaces in [9].

      For h > k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq38_HTML.gif, we define the functions of the real variable t
      G k ( t ) = { t k if t > k , 0 if k t k , t + k if t < k , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ11_HTML.gif
      (2.3)
      and
      G k h ( t ) = G k ( t ) G h ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ12_HTML.gif
      (2.4)
      Lemma 2.2 Let g M o q , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq39_HTML.gif, u W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq40_HTML.gifand ε R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq41_HTML.gif. Then there exist r N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq42_HTML.gifand k 1 , , k r R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq43_HTML.gif, with 0 = k r < k r 1 < < k 1 < k 0 = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq44_HTML.gif, such that set
      u s = G k s k s 1 ( u ) , s = 1 , , r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ13_HTML.gif
      (2.5)
      one has u 1 , , u r W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq45_HTML.gif and
      u = u 1 + + u r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ14_HTML.gif
      (2.6)
      u s 2 u u s , s = 1 , , r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ15_HTML.gif
      (2.7)
      | u s | | u | , s = 1 , , r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ16_HTML.gif
      (2.8)
      u x i ( u s ) x j = ( u s ) x i ( u s ) x j , s = 1 , , r , i , j = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ17_HTML.gif
      (2.9)
      u ( u s ) x i = ( u s + + u r ) ( u s ) x i , s = 1 , , r , i = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ18_HTML.gif
      (2.10)
      g χ supp ( u s ) x M q , λ ( Ω ) ε , s = 1 , , r , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ19_HTML.gif
      (2.11)
      r c , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ20_HTML.gif
      (2.12)

      with c = c ( ε , q , g M q , λ ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq46_HTML.gifpositive constant.

      Proof The proofs of the properties (2.6), (2.7), (2.9), (2.11) and (2.12) can be found in [9].

      Inequality (2.8) is an immediate consequence of (2.7).

      Considering (2.10), observe that in the case s = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq47_HTML.gif it is a trivial consequence of (2.6).

      Thus let us fix s N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq48_HTML.gif and such that 2 s r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq49_HTML.gif. As already proved in [16] and in [7], in the case of unbounded domains, one has
      ( G k s k s 1 ( u ) ) x i = G k s k s 1 ( u ) u x i , a.e. in Ω , i = 1 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equb_HTML.gif
      This, together with (2.3) and (2.4), gives
      supp ( u s ) x i { x Ω s.t. k s < | u | < k s 1 , u x i 0 } ¯ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ21_HTML.gif
      (2.13)

      i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq50_HTML.gif.

      On the other hand, by definition,
      supp u h { x Ω s.t. | u | k h } , h = 1 , , r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ22_HTML.gif
      (2.14)
      Combining (2.14) and (2.13), we conclude that
      supp u h supp ( u s ) x i = , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equc_HTML.gif

      h = 1 , , s 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq51_HTML.gif, i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq50_HTML.gif. Hence by (2.6) we get (2.10). □

      3 Existence and uniqueness result

      Let Ω be an unbounded open subset of R n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq3_HTML.gif, n 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq4_HTML.gif.

      We are interested in the study of the following Dirichlet problem in Ω:
      { u W i 1 , 2 ( Ω ) , L u = f , f W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ23_HTML.gif
      (3.1)
      where L is a second order linear differential operator in divergence form
      L = i , j = 1 n x j ( a i j x i + d j ) + i = 1 n b i x i + c , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ24_HTML.gif
      (3.2)
      satisfying the following hypotheses on the leading coefficients: Considering the coefficients of lower-order terms, we suppose that We associate to L the bilinear form
      a ( u , v ) = Ω ( i , j = 1 n ( a i j u x i + d j u ) v x j + ( i = 1 n b i u x i + c u ) v ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ25_HTML.gif
      (3.3)
      u , v W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq52_HTML.gif.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equd_HTML.gif

      As a consequence of Lemma 2.1, a is continuous on W i 1 , 2 ( Ω ) × W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq53_HTML.gif; and therefore, the operator L : W i 1 , 2 ( Ω ) W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq54_HTML.gif is continuous too.

      Theorem 3.1 Under hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif)-( h 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq56_HTML.gif), problem (3.1) is uniquely solvable and its solution u satisfies the estimate
      u W 1 , 2 ( Ω ) C f W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ26_HTML.gif
      (3.4)

      where C is a constant depending on n, t, ν, μ, d i b i M 2 t , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq57_HTML.gif, i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq58_HTML.gif.

      Proof We start proving estimate (3.4) that yields also to the uniqueness of the solution at once. Successively, in view of classical results concerning operator theory, to get the existence, it will be enough to verify that L is a Fredholm operator with zero index.

      Let u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, for s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq59_HTML.gif, be the functions of Lemma 2.2 corresponding to a solution u of (3.1), to g = i = 1 n | d i b i | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq60_HTML.gif and to a positive real number ε that will be specified in the sequel.

      By a well-known characterization of the space W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq61_HTML.gif, we have
      f = f 0 i = 1 n ( f i ) x i , f k L 2 ( Ω ) , k = 0 , , n . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Eque_HTML.gif
      Thus, if we take u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif as a test function in the variational formulation of problem (3.1), by simple calculations and (2.9) and (2.10), we obtain
      Ω f 0 u s d x + i = 1 n Ω f i ( u s ) x i d x = a ( u , u s ) = Ω [ i , j = 1 n a i j u x i ( u s ) x j + i = 1 n ( d i u ( u s ) x i + b i u x i u s ) + c u u s ] d x = Ω [ i , j = 1 n a i j u x i ( u s ) x j + i = 1 n b i ( u u s ) x i + c u u s + i = 1 n ( d i b i ) u ( u s ) x i ] d x = Ω [ i , j = 1 n a i j ( u s ) x i ( u s ) x j + i = 1 n b i ( u u s ) x i + c u u s + i = 1 n ( d i b i ) ( h = s r u h ) ( u s ) x i ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equf_HTML.gif
      Hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif) and ( h 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq56_HTML.gif) together with (2.7) give then
      Ω f 0 u s d x + i = 1 n Ω f i ( u s ) x i d x ν Ω ( u s ) x 2 d x + μ Ω ( u s ) 2 d x Ω h = s r | u h | i = 1 n | d i b i | | ( u s ) x i | d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ27_HTML.gif
      (3.5)
      On the other hand, by the Hölder inequality, the embedding results contained in Lemma 2.1 and using hypothesis ( h 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq62_HTML.gif) and (2.11), one has that there exists a constant c 0 R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq63_HTML.gif such that
      Ω h = s r | u h | i = 1 n | d i b i | | ( u s ) x i | d x h = s r | u h | g χ supp ( u s ) x L 2 ( Ω ) ( u s ) x L 2 ( Ω ) c 0 h = s r u h W 1 , 2 ( Ω ) g χ supp ( u s ) x M 2 t , λ ( Ω ) u s W 1 , 2 ( Ω ) ε c 0 u s W 1 , 2 ( Ω ) h = s r u h W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equg_HTML.gif

      with c 0 = c 0 ( n , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq64_HTML.gif.

      Hence, set
      μ 0 = min { ν , μ } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equh_HTML.gif
      by (3.5) we get
      μ 0 u s W 1 , 2 ( Ω ) 2 f 0 L 2 ( Ω ) u s L 2 ( Ω ) + i = 1 n f i L 2 ( Ω ) ( u s ) x i L 2 ( Ω ) + ε c 0 u s W 1 , 2 ( Ω ) h = s r u h W 1 , 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equi_HTML.gif
      Thus, choosing ε = μ 0 2 c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq65_HTML.gif we have
      u s W 1 , 2 ( Ω ) 1 μ 0 f W 1 , 2 ( Ω ) + 1 2 h = s r u h W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equj_HTML.gif

      for s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq66_HTML.gif.

      If we rewrite the last inequality for s = r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq67_HTML.gif and we estimate u r W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq68_HTML.gif, then for s = r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq69_HTML.gif and we estimate u r 1 W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq70_HTML.gif and so on, we get by substituting that
      u s W 1 , 2 ( Ω ) 2 r s + 1 μ 0 f W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equk_HTML.gif

      for s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq66_HTML.gif.

      Therefore, taking into account (2.6), we conclude that
      u W 1 , 2 ( Ω ) s = 1 r u s W 1 , 2 ( Ω ) ( 2 r 1 ) 2 μ 0 f W 1 , 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equl_HTML.gif

      This, together with (2.12), ends the proof of the bound in (3.4).

      Now, as it was already mentioned, it only remains to show that the operator
      L : u W i 1 , 2 ( Ω ) L u W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equm_HTML.gif

      is a Fredholm operator with zero index.

      To this aim, set γ = i = 1 n ( d i b i ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq71_HTML.gif and denote by γu, u W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq40_HTML.gif, the element of W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq61_HTML.gif given by
      γ u : v W i 1 , 2 ( Ω ) Ω γ u v d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equn_HTML.gif

      which is well defined in view of Lemma 2.1.

      Then, consider the problem
      { u W i 1 , 2 ( Ω ) , L u + 1 2 ν γ u = f , f W 1 , 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ28_HTML.gif
      (3.6)

      Clearly, if we show that (3.6) has a unique solution, we end our proof, since in this case the operator L can be seen as a sum between a Fredholm operator with zero index and a compact operator; and therefore, it is a Fredholm operator with zero index itself.

      Indeed, we explicitly observe that the operator
      u W i 1 , 2 ( Ω ) γ u W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equo_HTML.gif
      is compact, since, by hypothesis ( h 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq62_HTML.gif) and Lemma 2.1, it is obtained as a composition between the compact operator
      u W i 1 , 2 ( Ω ) γ 1 / 2 u L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equp_HTML.gif
      and the bounded one
      v L 2 ( Ω ) γ 1 / 2 v W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equq_HTML.gif
      where γ 1 / 2 v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq72_HTML.gif, v L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq73_HTML.gif, is the element of W 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq61_HTML.gif defined by
      γ 1 / 2 v : w W i 1 , 2 ( Ω ) Ω γ 1 / 2 v w d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equr_HTML.gif
      To get the existence and uniqueness of the solution of problem (3.6), we want to make use of Lax-Milgram Lemma. Thus let us consider the bilinear form associated to it
      a ( u , v ) + 1 2 ν Ω γ u v d x , u , v W i 1 , 2 ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ29_HTML.gif
      (3.7)
      The continuity of the form (3.7) can be easily obtained by Lemma 2.1. Considering the coercivity, for every u W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq40_HTML.gif, in view of hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif) and ( h 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq56_HTML.gif), one has
      a ( u , u ) = Ω i , j = 1 n a i j u x i u x j d x + Ω i = 1 n ( b i ( u 2 ) x i + c u 2 ) d x + Ω i = 1 n ( d i b i ) u u x i d x ν u x L 2 ( Ω ) 2 + μ u L 2 ( Ω ) 2 + Ω i = 1 n ( d i b i ) u u x i d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equs_HTML.gif
      On the other hand, Hölder and Young inequalities give that
      Ω i = 1 n | d i b i | | u | | u x i | d x ν 2 u x L 2 ( Ω ) 2 + 1 2 ν i = 1 n ( d i b i ) u L 2 ( Ω ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equt_HTML.gif
      and therefore,
      a ( u , u ) + 1 2 ν Ω γ u 2 d x min { ν 2 , μ } u W 1 , 2 ( Ω ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equu_HTML.gif

      This concludes the proof of Theorem 3.1. □

      4 An a priori bound in L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif

      Here we want to prove, for a sufficiently regular datum f, a L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif-a priori estimate, p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq2_HTML.gif, for a bounded solution of problem (3.1).

      To this aim, we require a further assumption on the boundary of Ω:

      Moreover, a technical lemma below is needed. We note that the proof of Lemma 4.1 follows the idea of the one of the estimate (3.4). However, in this case, there are some specific arguments that need to be explicitly treated.

      Let u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif be the functions of Lemma 2.2 corresponding to a fixed u W i 1 , 2 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq74_HTML.gif, to g = i = 1 n | d i b i | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq60_HTML.gif and to a positive real number ε to be specified in the proof of Lemma 4.1. The following result holds true:

      Lemma 4.1 Let a be the bilinear form in (3.3). Under hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif)-( h 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq75_HTML.gif), there exists a constant C R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq76_HTML.gifsuch that
      Ω | u s | p 2 ( ( u s ) x 2 + u s 2 ) d x C h = s r a ( u , | u h | p 2 u h ) , s = 1 , , r , p ] 2 , + [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ30_HTML.gif
      (4.1)

      where C depends on s, r, ν, μ.

      Proof Let u g ε and u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, for s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq66_HTML.gif, be as above specified. Since u L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq77_HTML.gif, by definition of u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif and by Lemma 2.2, the functions u s W i 1 , 2 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq78_HTML.gif. Therefore, in view of hypothesis ( h 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq75_HTML.gif), Lemma 3.2 in [17] applies giving that | u s | p 2 u s W i 1 , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq79_HTML.gif for any p > 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq2_HTML.gif.

      Thus, we can take | u s | p 2 u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq80_HTML.gif as a test function in (3.3), obtaining by (2.9) that
      a ( u , | u s | p 2 u s ) = Ω [ i , j = 1 n a i j u x i ( | u s | p 2 u s ) x j + i = 1 n b i ( | u s | p 2 u s u ) x i + c | u s | p 2 u s u + i = 1 n ( d i b i ) u ( | u s | p 2 u s ) x i ] d x = Ω [ ( p 1 ) | u s | p 2 i , j = 1 n a i j ( u s ) x i ( u s ) x j + i = 1 n b i ( | u s | p 2 u s u ) x i + c | u s | p 2 u s u + ( p 1 ) | u s | p 2 u i = 1 n ( d i b i ) ( u s ) x i ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equv_HTML.gif
      If we set
      μ 0 = min { ν , μ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equw_HTML.gif
      and
      H s ( u ) = | u s | p 2 ( ( u s ) x 2 + ( u s ) 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ31_HTML.gif
      (4.2)
      by hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif) and ( h 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq56_HTML.gif) and in view of (2.7), one has
      μ 0 Ω H s ( u ) d x a ( u , | u s | p 2 u s ) + ( p 1 ) Ω g | u s | p 2 | u | ( u s ) x d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ32_HTML.gif
      (4.3)
      On the other hand, by (2.6), (2.8) and (2.10), using the Hölder inequality, we get that there exists a constant c 0 R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq63_HTML.gif, such that
      Ω g | u | | u s | p 2 ( u s ) x d x Ω g | u | | u | p / 2 1 | u s | p / 2 1 ( u s ) x d x c 0 h = s r Ω g | u h | p / 2 | u s | p / 2 1 ( u s ) x d x c 0 | u s | p / 2 1 ( u s ) x L 2 ( Ω ) h = s r g | u h | p / 2 χ supp ( u s ) x L 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equx_HTML.gif

      with c 0 = c 0 ( r , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq81_HTML.gif.

      Thus, using hypothesis ( h 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq62_HTML.gif), by Lemma 2.1 and (2.11), we obtain
      Ω g | u | | u s | p 2 ( u s ) x d x c 1 | u s | p / 2 1 ( u s ) x L 2 ( Ω ) g χ supp ( u s ) x M 2 t , λ ( Ω ) h = s r | u h | p / 2 W 1 , 2 ( Ω ) c 1 ε | u s | p / 2 1 ( u s ) x L 2 ( Ω ) h = s r | u h | p / 2 W 1 , 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ33_HTML.gif
      (4.4)

      with c 1 = c 1 ( r , p , n , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq82_HTML.gif.

      Now, we observe that explicit calculations give
      | u h | p / 2 W 1 , 2 ( Ω ) p 2 ( Ω H h ( u ) d x ) 1 / 2 , h = s , , r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ34_HTML.gif
      (4.5)
      Hence, putting together (4.3), (4.4) and (4.5), we get
      Ω H s ( u ) d x 1 μ 0 a ( u , | u s | p 2 u s ) + c 2 μ 0 ε ( Ω H s ( u ) d x ) 1 / 2 h = s r ( Ω H h ( u ) d x ) 1 / 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equy_HTML.gif

      with c 2 = c 2 ( r , p , n , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq83_HTML.gif.

      Thus, by Young inequality,
      Ω H s ( u ) d x 1 μ 0 a ( u , | u s | p 2 u s ) + c 3 μ 0 ε ( Ω H s ( u ) d x ) 1 / 2 ( h = s r Ω H h ( u ) d x ) 1 / 2 1 μ 0 a ( u , | u s | p 2 u s ) + c 3 μ 0 ( η 2 Ω H s ( u ) d x + ε 2 2 η h = s r Ω H h ( u ) d x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equz_HTML.gif

      with c 3 = c 3 ( r , p , n , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq84_HTML.gif.

      Choosing η = μ 0 c 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq85_HTML.gif and ε = μ 0 c 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq86_HTML.gif we have
      Ω H s ( u ) d x 2 μ 0 a ( u , | u s | p 2 u s ) + 1 2 h = s r Ω H h ( u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equ35_HTML.gif
      (4.6)

      s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq66_HTML.gif.

      If we rewrite the last inequality for s = r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq67_HTML.gif, then for s = r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq69_HTML.gif and take into account the estimate of Ω H r ( u ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq87_HTML.gif obtained in the previous step, and so on, we conclude our proof. Indeed, we get
      Ω H s ( u ) d x C h = s r a ( u , | u h | p 2 u h ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equaa_HTML.gif

      with C = C ( s , r , μ 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq88_HTML.gif. □

      We are finally in position to prove the above mentioned L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif-bound.

      Theorem 4.2 Assume that the hypotheses ( h 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq55_HTML.gif)-( h 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq75_HTML.gif) are satisfied. If f is in L 2 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq89_HTML.gifand the solution u of (3.1) is in W i 1 , 2 ( Ω ) L ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq90_HTML.gif, then
      u L p ( Ω ) C f L p ( Ω ) p ] 2 , + [ , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equab_HTML.gif

      where C is a constant depending on n, t, p, ν, μ, d i b i M 2 t , λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq57_HTML.gif, i = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq58_HTML.gif.

      Proof Fix p ] 2 , + [ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq91_HTML.gif. If we consider the functions u s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq12_HTML.gif, s = 1 , , r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq92_HTML.gif, corresponding to the solution u, to g and ε as in Lemma 4.1, easy computations together with (2.6) give that
      Ω | u | p d x c 0 s = 1 r Ω | u s | p d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equac_HTML.gif

      with c 0 = c 0 ( r , p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq93_HTML.gif.

      Thus, by (4.1), one has
      Ω | u | p d x c 0 s = 1 r C s h = s r a ( u , | u h | p 2 u h ) c 1 s = 1 r a ( u , | u s | p 2 u s ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equad_HTML.gif

      with C s = C s ( s , r , ν , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq94_HTML.gif and c 1 = c 1 ( r , p , ν , μ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq95_HTML.gif.

      Hence by (2.8) and Hölder inequality, we get
      u L p ( Ω ) p c 1 s = 1 r Ω f | u s | p 2 u s d x r c 1 Ω | f | | u | p 1 d x r c 1 f L p ( Ω ) u L p ( Ω ) p 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_Equae_HTML.gif

      This concludes the proof, in view of (2.12). □

      Author’s contributions

      The authors conceived and wrote this article in collaboration and with the same responsibility. Both of them read and approved the final manuscript.

      Declarations

      Acknowledgement

      The authors would like to thank anonymous referees for a careful reading of this article and for valuable suggestions and comments.

      Authors’ Affiliations

      (1)
      Dipartimento di Matematica, Università di Salerno

      References

      1. Chicco M: An a priori inequality concerning elliptic second order partial differential equations of variational type. Matematiche 1971, 26: 173-182.MathSciNet
      2. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order. Springer, Berlin; 1983.MATHView Article
      3. Ladyzhenskaja OA, Ural’tzeva NN: Equations aux Derivèes Partielles de Type Elliptique. Dunod, Paris; 1966.
      4. Miranda C:Alcune osservazioni sulla maggiorazione in L ν http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq96_HTML.gif delle soluzioni deboli delle equazioni ellittiche del secondo ordine. Ann. Mat. Pura Appl. 1963, 61: 151-169.MATHMathSciNetView Article
      5. Stampacchia G: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 1966, 15: 151-169.MathSciNet
      6. Trudinger NS: Linear elliptic operators with measurable coefficients. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 1973, 27: 265-308.MATHMathSciNet
      7. Bottaro G, Marina ME: Problema di Dirichlet per equazioni ellittiche di tipo variazionale su insiemi non limitati. Boll. Unione Mat. Ital. 1973, 8: 46-56.MATHMathSciNet
      8. Transirico M, Troisi M: Equazioni ellittiche del secondo ordine a coefficienti discontinui e di tipo variazionale in aperti non limitati. Boll. Unione Mat. Ital, B 1988, 2: 385-398.MATHMathSciNet
      9. Transirico M, Troisi M, Vitolo A: Spaces of Morrey type and elliptic equations in divergence form on unbounded domains. Boll. Unione Mat. Ital, B 1995, 9: 153-174.MATHMathSciNet
      10. Monsurrò S, Transirico M:A L p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq1_HTML.gif-estimate for weak solutions of elliptic equations. Abstr. Appl. Anal. 2012.
      11. Chicco M, Venturino M: Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain. Ann. Mat. Pura Appl. 2000, 178: 325-338. 10.1007/BF02505902MATHMathSciNetView Article
      12. Lions PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1985, 78: 205-212.MATH
      13. Lions PL: Remarques sur les équations linéaires elliptiques du second ordre sous forme divergence dans les domaines non bornés II. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1985, 79: 178-183.MATH
      14. Caso L, D’Ambrosio R, Monsurrò S: Some remarks on spaces of Morrey type. Abstr. Appl. Anal. 2010.
      15. Cavaliere P, Longobardi M, Vitolo A: Imbedding estimates and elliptic equations with discontinuous coefficients in unbounded domains. Matematiche 1996, 51: 87-104.MATHMathSciNet
      16. Stampacchia G: Equations elliptiques du second ordre à coefficients discontinus. Les presses de l’Université de Montréal, Montreal; 1966.MATH
      17. Caso L, Cavaliere P, Transirico M:Solvability of the Dirichlet problem in W 2 , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-67/MediaObjects/13661_2012_Article_178_IEq97_HTML.gif for elliptic equations with discontinuous coefficients in unbounded domains. Matematiche 2002, 57: 287-302.MATHMathSciNet

      Copyright

      © Monsurrò and Transirico; licensee Springer 2012

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.