Singular limiting solutions for elliptic problem involving exponentially dominated nonlinearity and convection term

  • Sami Baraket1Email author,

    Affiliated with

    • Imed Abid2,

      Affiliated with

      • Taieb Ouni2 and

        Affiliated with

        • Nihed Trabelsi2

          Affiliated with

          Boundary Value Problems20112011:10

          DOI: 10.1186/1687-2770-2011-10

          Received: 22 March 2011

          Accepted: 12 August 2011

          Published: 12 August 2011

          Abstract

          Given Ω bounded open regular set of ℝ2 and x1, x2, ..., x m ∈ Ω, we give a sufficient condition for the problem

          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equa_HTML.gif

          to have a positive weak solution in Ω with u = 0 on ∂Ω, which is singular at each x i as the parameters ρ, λ > 0 tend to 0 and where f(u) is dominated exponential nonlinearities functions.

          2000 Mathematics Subject Classification: 35J60; 53C21; 58J05.

          Keywords

          singular limits Green's function nonlinear Cauchy-data matching method

          1 Introduction and statement of the results

          We consider the following problem
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ1_HTML.gif
          (1)
          where ∇ is the gradient and Ω is an open smooth bounded subset of ℝ2. The function a is assumed to be positive and smooth. In the following, we take a(u) = e λu and f(u) = e λu (e u + e γu ), for λ > 0 and γ ∈(0, 1), then problem (1) take the form
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ2_HTML.gif
          (2)
          Using the following transformation
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equb_HTML.gif
          then the function w satisfies the following problem
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ3_HTML.gif
          (3)

          with ϱ = (λρ2)1-λ. So when λ → 0+, the exponent http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq1_HTML.gif tends to infinity while the exponent http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq2_HTML.gif tends to -∞. For ϱ ≡ 0, problem (3) has been studied by Ren and Wei in [1]. See also [2].

          We denote by ε the smallest positive parameter satisfying
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ4_HTML.gif
          (4)
          Remark that ρ ~ ε as ε → 0. We will suppose in the following
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equc_HTML.gif
          In particular, if we take http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq3_HTML.gif , then the condition (A λ ) is satisfied. Under the assumption (A λ ), we can treat equation (2) as a perturbation of the following:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equd_HTML.gif

          for γ ∈ (0, 1).

          Our question is: Does there exist vε,λa sequence of solutions of (2) which converges to some singular function as the parameters ε and λ tend to 0?

          In [3], Baraket et al. gave a positive answer to the above question for the following problem
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ5_HTML.gif
          (5)

          with a regular bounded domain Ω of ℝ2. They give a sufficient condition for the problem (5) to have a weak solution in Ω which is singular at some points (x i )1≤imas ρ and λ a small parameters satisfying (A λ ), where the presence of the gradient term seems to have significant influence on the existence of such solutions, as well as on their asymptotic behavior.

          In case λ = 0 the authors in [4] gave also a positive answer for the following problem
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ6_HTML.gif
          (6)
          for γ ∈ (0, 1) as ρ tends to 0. When λ = 0 and γ = 0, problem (2) reduce to
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ7_HTML.gif
          (7)

          The study of this problem goes back to 1853 when Liouville derived a representation formula for all solutions of (7) which are defined in ℝ2, see [5]. It turns out that, beside the applications in geometry, elliptic equations with exponential nonlinearity also arise in modeling many physical phenomenon, such as thermionic emission, isothermal gas sphere, gas combustion, and gauge theory [6]. When ρ tends to 0, the asymptotic behavior of nontrivial branches of solutions of (7) is well understood thanks to the pioneer work of Suzuki [7] which characterizes the possible limit of nontrivial branches of solutions of (7). His result has been generalized in [8] to (6) with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq4_HTML.gif , and finally by Ye in [9] to any exponentially dominated nonlinearity f(u). The existence of nontrivial branches of solutions with single singularity was first proved by Weston [10] and then a general result has been obtained by Baraket and Pacard [11]. These results were also extended, applying to the Chern-Simons vortex theory in mind, by Esposito et al. [12] and Del Pino et al. [13] to handle equations of the form -Δu = ρ2V(x)e u where V is a nonconstant positive potential. See also [1416] wherever this rule is applicable. where the Laplacian is replaced by a more general divergence operator and some new phenomena occur. Let us also mention that the construction of nontrivial branches of solutions of semilinear equations with exponential nonlinearities allowed Wente to provide counter examples to a conjecture of Hopf [17] concerning the existence of compact (immersed) constant mean curvature surfaces in Euclidean space. Another related problem is the higher dimension problem with exponential nonlinearity. For example, the 4-dimensional semilinear elliptic problem with bi-Laplacian is treated in [18] and the problem with an additional singular source term given by Dirac masses is treated in [19] in the radial case. The results in [18, 19] are generalized to noncritical points of the reduced function, see [20].

          We introduce now the Green's function G(x, x') defined on Ω × Ω, to be solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Eque_HTML.gif
          and let H(x, x') = G(x, x') + 4log |x - x'|, its regular part. Let m ∈ ℕ, we set
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ8_HTML.gif
          (8)

          which is well defined in (Ω) m for x i x j for ij. Our main result is the following

          Theorem 1 Given β ∈ (0, 1). Let Ω an open smooth bounded set of2, λ > 0 satisfying the condition (A λ ), γ ∈ (0, 1) and S = {x1, ... x m } ⊂ Ω be a nonempty set. Assume that, the point (x1, ..., x m ) is a nondegenerate critical point of the function
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equf_HTML.gif
          then there exist ε0 > 0, λ0 > 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq5_HTML.gif a family of solutions of (2), such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equg_HTML.gif

          One of the purpose of the present paper is to present a rather efficient method: nonlinear Cauchy-data matching method to solve such singularly problems. This method has already been used successfully in geometric context (constant mean curvature surfaces, constant scalar curvature metrics, extremal Kähler metrics, manifolds with special holonomy, ...) and appeared in the study [18] in the context of partial differential equations.

          2 Construction of the approximate solution

          We first describe the rotationally symmetric approximate solutions of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ9_HTML.gif
          (9)
          in ℝ2 which will play a central role in our analysis. Given ε > 0, we define
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ10_HTML.gif
          (10)
          which is clearly a solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ11_HTML.gif
          (11)
          in ℝ2. Let us notice that equations (11) is invariant under dilation in the following sense: If v is a solution of (11) and if τ > 0, then v(τ ·) + 2logτ is also a solution of (11). With this observation in mind, we define for all τ > 0
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ12_HTML.gif
          (12)

          2.1 A linearized operator on ℝ2

          For all ε, τ, λ > 0, we set
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ13_HTML.gif
          (13)
          for δ ∈ (0, 1). We define the linear second order elliptic operator
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ14_HTML.gif
          (14)
          which corresponds to the linearization of (11) about the solution u1 (= uε = τ = 1) given by (10) which has been defined in the previous section. We are interested in the classification of bounded solutions of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif in ℝ2. Some solutions are easy to find. For example, we can define
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equh_HTML.gif
          where r = |x|. Clearly http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq7_HTML.gif and this reflects the fact that (11) is invariant under the group of dilations τu(τ ·) + 2 logτ. We also define, for i = 1, 2
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equi_HTML.gif

          which are also solutions of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq8_HTML.gif . Since, these solutions correspond to the invariance of the equation under the group of translations au(· + a). We recall the following result which classifies all bounded solutions of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif which are defined in ℝ2.

          Lemma 1 [11] Any bounded solution of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq6_HTML.gif defined in2 is a linear combination of ϕ i for i = 0, 1, 2.

          Let B r denote the ball of radius r centered at the origin in ℝ2.

          Definition 1 Given k ∈ ℕ, β ∈ (0, 1) and μ ∈ ℝ, we introduce the Hölder weighted spaces http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq9_HTML.gif as the space of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq10_HTML.gif for which the following norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equj_HTML.gif

          is finite.

          We define also
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equk_HTML.gif

          As a consequence of the result of Lemma 1, we recall the surjectivity result of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq11_HTML.gif given in [11].

          Proposition 1 [11]
          1. (i)
            Assume that μ > 1 and μ ∉ ℕ, then
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equl_HTML.gif
             
          is surjective.
          1. (ii)
            Assume that δ > 0 and δ ∉ ℕ then
            http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equm_HTML.gif
             

          is surjective.

          We set http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq12_HTML.gif , we define

          Definition 2 Given k ∈ ℕ, β ∈ (0, 1) and μ ∈ ℝ, we introduce the Hölder weighted spaces http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq13_HTML.gif as the space of functions in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq14_HTML.gif for which the following norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equn_HTML.gif

          is finite.

          Then, we define the subspace of radial functions in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq15_HTML.gif by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equo_HTML.gif
          We would like to find a solution u of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ15_HTML.gif
          (15)
          in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq16_HTML.gif . By using the transformation, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq17_HTML.gif then Eq. (15) is equivalent to
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ16_HTML.gif
          (16)
          in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We look for a solution of (16) of the form v(x) = u1(x) + h(x), this amounts to solve
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ17_HTML.gif
          (17)

          In http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We will need the following:

          Definition 3 Given http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq19_HTML.gif , k∈ ∞, β ∈ (0, 1) and μ ∈ ℝ, the weighted space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq20_HTML.gif is defined to be the space of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq21_HTML.gif endowed with the norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equp_HTML.gif
          For all σ ≥ 1, we denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq22_HTML.gif the extension operator defined by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ18_HTML.gif
          (18)
          where t α χ(t) is a smooth non-negative cutoff function identically equal to 1 for t ≤ 1 and identically equal to 0 for t ≥ 2. It is easy to check that there exists a constant c = c(μ) > 0, independent of σ ≥ 1, such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ19_HTML.gif
          (19)
          We fix δ ∈ (0, 1) and denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq23_HTML.gif to be a right inverse of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq24_HTML.gif provided by Proposition 1. To find a solution of (17), it is enough to find a fixed point h, in a small ball of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq25_HTML.gif , solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ20_HTML.gif
          (20)
          We have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equq_HTML.gif
          This implies that given κ > 0, there exist c κ > 0 (only depend on κ), such that for δ ∈ (0,1) and |x| = r, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equr_HTML.gif
          Making use of Proposition 1 together with (19), we conclude that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ21_HTML.gif
          (21)
          Now, let h1, h2 such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq26_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq25_HTML.gif , then for δ ∈ (0, 1 - r] we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equs_HTML.gif
          Similarly, making use of Proposition 1 together with condition (A λ ) and (19), we conclude that given κ > 0, there exist ε κ > 0, λ κ > 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq27_HTML.gif (only depend on κ) such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ22_HTML.gif
          (22)

          Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq28_HTML.gif for all λ ∈ (0, λ κ ) and ε ∈ (0, ε κ ). Then, (21) and (22) are enough to show that h ↦ ℵ is a contraction from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq29_HTML.gif into itself and hence has a unique fixed point h in this set. This fixed point is solution of (20) in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . We summarize this in the:

          Proposition 2 Given δ ∈ (0, 1 - γ] and κ > 1, then there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (independent of ε and λ) and a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq31_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq32_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equt_HTML.gif

          solves (16) in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif .

          2.2 Analysis of the Laplace operator in weighted spaces

          In this section, we study the mapping properties of the Laplace operator in weighted Hölder spaces. Given x1, ..., x m ∈ Ω, we define x := (x1, ..., x m )
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equu_HTML.gif
          and we choose r0 > 0 so that the balls http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq33_HTML.gif of center x i and radius r0 are mutually disjoint and included in Ω. For all r ∈ (0, r0), we define
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equv_HTML.gif

          With these notations, we have:

          Definition 4 Given k ∈ ℝ, β ∈ (0,1) and ν ∈ ℝ, we introduce the Hölder weighted space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq34_HTML.gif as the space of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq35_HTML.gif for with the following norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equw_HTML.gif

          is finite.

          When k ≥ 2, we denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq36_HTML.gif be the subspace of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq37_HTML.gif satisfying w = 0 on ∂Ω. We recall the

          Proposition 3 [21] Assume that ν < 0 and ν ∉ ℤ, then
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equx_HTML.gif

          is surjective. Denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq38_HTML.gif a right inverse of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq39_HTML.gif .

          Remark 1 Observe that, when ν < 0, ν ∉ ℤ, the right inverse even though is not unique and can be chosen to depend smoothly on the points x1, ..., x m , at least locally. Once a right inverse is fixed for some choice of the points x1, ..., x m , a right inverse which depends smoothly on some points close to x1, ..., x m can be obtained using a simple perturbation argument. This argument will be used later in the nonlinear exterior problem, since we will move a little bit the points (x i ).

          2.3 Harmonic extensions

          We study the properties of interior and exterior harmonic extensions. Given http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq41_HTML.gif and define H i (=H i (φ; ·)) to be the solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ23_HTML.gif
          (23)

          We denote by e1, e2 the coordinate functions on S1.

          Lemma 2 [21] If we assume that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ24_HTML.gif
          (24)
          then there exists c > 0 such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equy_HTML.gif
          Given http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq42_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq43_HTML.gif to be the solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ25_HTML.gif
          (25)

          which decays at infinity.

          Definition 5 Given k ∈ ℕ, β ∈ (0,1) and ν ∈ ℝ, we define the space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq44_HTML.gif as the space of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq45_HTML.gif for which the following norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equz_HTML.gif

          is finite.

          Lemma 3 [21] If we assume that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ26_HTML.gif
          (26)
          Then there exists c > 0 such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaa_HTML.gif

          If FL2(S1) is a space of functions defined on S1, we define the space F to be the subspace of functions F of which are L2(S1) -orthogonal to the functions 1, e1,e2. We will need the:

          Lemma 4 [21] The mapping
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equab_HTML.gif

          where H i (= H i (ψ; ·)) and H e = H e (ψ; ·), is an isomorphism.

          3 The nonlinear interior problem

          We are interested in studying equations of type
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ27_HTML.gif
          (27)

          In http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif .

          Given http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq46_HTML.gif satisfying (24), we define
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equac_HTML.gif
          Then, we look for a solution of (27) of the form w = v + v and using the fact that H i is harmonic, this amounts to solve
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ28_HTML.gif
          (28)
          We fix μ ∈ (1,2) and denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq47_HTML.gif to be a right inverse of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq48_HTML.gif provided by Proposition 1. To find a solution of (28), it is sufficient to find http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq49_HTML.gif solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ29_HTML.gif
          (29)

          We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq50_HTML.gif , the nonlinear operator appearing on the right-hand side of equation (29).

          Given κ > 0 (whose value will be fixed later on), we further assume that the functions φ satisfy
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ30_HTML.gif
          (30)

          Then, we have the following result

          Lemma 5 Given κ > 0. There exist ε κ > 0, λ κ > 0, c κ > 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (only depend on κ) such that for all λ ∈ (0, λ κ ) and ε ∈ (0, ε κ )
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equad_HTML.gif
          and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equae_HTML.gif

          provided http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq51_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq52_HTML.gif .

          Proof. The proof of the first estimate follows from the asymptotic behavior of H i together with the assumption on the norm of boundary data φ given by (30). Indeed, let c κ be a constant depending only on κ (provided ε and λ are chosen small enough) it follows from the estimate of H i , given by lemma 2, that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaf_HTML.gif
          Since for each http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq53_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equag_HTML.gif
          where δ ∈ (0, 1 - γ]. Then
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equah_HTML.gif
          On the other hand, using the condition (A λ ), we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equai_HTML.gif
          and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaj_HTML.gif
          Making use of Proposition 1 together with (20), we get
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ31_HTML.gif
          (31)
          In order to derive the second estimate, we use the fact that, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq51_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq54_HTML.gif for i = 1,2, μ ∈ (1,2) and the condition (A λ ), then there exist c κ > 0 (only depend on κ) such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equak_HTML.gif
          Similarly, making use of Proposition 1 together with (19), we conclude that there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (only depend on κ) such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ32_HTML.gif
          (32)

          Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq55_HTML.gif for all λ ∈ (0, λ κ ) and ε ∈ (0, ε κ ). Then, (31) and (32) are enough to show that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq56_HTML.gif is a contraction from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq57_HTML.gif into itself and hence has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq58_HTML.gif in this set. This fixed point is solution of (20) in ℝ2. We summarize this in the following:

          Proposition 4 Given κ > 0, there exist ε κ > 0, λ κ > 0 and c κ > 0 (only depending on κ) such that for all ε ∈ (0, ε κ ), λ ∈ (0, λ κ ) satisfying (A), for all τ in some fixed compact subset of [τ -, τ+] ⊂ (0, ∞) and for a given φ satisfying (24)-(30), then there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq59_HTML.gif solution of (29) such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equal_HTML.gif
          Solve (27) in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq18_HTML.gif . In addition,
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equam_HTML.gif

          Observe that the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq59_HTML.gif being obtained as a fixed point for contraction mappings, it depends continuously on the parameter τ.

          4 The nonlinear exterior problem

          Recall that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq60_HTML.gif denote the unique solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equan_HTML.gif
          in Ω, with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq61_HTML.gif on ∂Ω. In addition, the following decomposition holds
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equao_HTML.gif
          where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq62_HTML.gif is a smooth function. Here, we give an estimate of the gradient of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq63_HTML.gif without proof (see [14], Lemma 2.1), there exists a constant c > 0, so that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equap_HTML.gif
          Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq64_HTML.gif close enough to x := (x1, ..., x m ), http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq65_HTML.gif close to 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq66_HTML.gif satisfying (26). We define
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ33_HTML.gif
          (33)

          where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq67_HTML.gif is a cutoff function identically equal to 1 in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq68_HTML.gif and identically equal to 0 outside http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq69_HTML.gif .

          We would like to find a solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ34_HTML.gif
          (34)
          in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq70_HTML.gif which is a perturbation of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq71_HTML.gif . Writing http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq72_HTML.gif . This amounts to solve
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaq_HTML.gif

          We need to define some auxiliary weighted spaces:

          Definition 6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq73_HTML.gif , k ∈ ℝ, β ∈ (0, 1) and ν ∈ ℝ, we define the Hölder weighted space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq74_HTML.gif as the set of functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq75_HTML.gif for which the following norm
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equar_HTML.gif

          is finite

          For all σ ∈ (0, r0/2) and all Y = (y1, ..., y m ) ∈ Ω m such that ||X - Y || ≤ r0/2, where X = (x1, ..., x m ), we denote by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equas_HTML.gif
          the extension operator defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq76_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq77_HTML.gif
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equat_HTML.gif
          for each i = 1, ..., m and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq78_HTML.gif in each Bσ/2(y i ), where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq79_HTML.gif is a cutoff function identically equal to 1 for t ≥ 1 and identically equal to 0 for t ≤ 1/2. It is easy to check that there exists a constant c = c(ν) > 0 only depending on ν such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ35_HTML.gif
          (35)
          We fix
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equau_HTML.gif
          and denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq80_HTML.gif a right inverse of Δ provided by Proposition 3 with http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq81_HTML.gif . Clearly, it is enough to find http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq82_HTML.gif solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ36_HTML.gif
          (36)
          where
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equav_HTML.gif
          We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq83_HTML.gif the nonlinear operator which appears on the right hand side of Eq.(36). Given κ > 0 (whose value will be fixed later on), we assume that the points http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq84_HTML.gif , the functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq85_HTML.gif and the parameters http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq86_HTML.gif to satisfy
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ37_HTML.gif
          (37)
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ38_HTML.gif
          (38)
          and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ39_HTML.gif
          (39)

          Then, the following result holds

          Lemma 6 Given κ > 0, there exist ε κ > 0, λ κ > 0, c κ > 0 and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq30_HTML.gif (depending on κ) such that for all ε ∈ (0, ε κ ), λ ∈ (0, λ κ )
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaw_HTML.gif
          and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equax_HTML.gif

          provided http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq87_HTML.gif and satisfy http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq88_HTML.gif .

          Proof: Recall that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq89_HTML.gif , we will estimate http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq90_HTML.gif in different subregions of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq91_HTML.gif .

          * In http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq92_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq93_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq94_HTML.gif and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ40_HTML.gif
          (40)
          so that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equay_HTML.gif
          Hence, for ν ∈ (- 1, 0) and for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif small enough, we get
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equaz_HTML.gif
          * In http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq96_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq94_HTML.gif . Thus
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equba_HTML.gif
          So, for ν ∈ (- 1, 0), we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbb_HTML.gif
          * In http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq97_HTML.gif , using the estimat (40), then we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbc_HTML.gif
          where
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbd_HTML.gif
          Then
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Eqube_HTML.gif
          So,
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ41_HTML.gif
          (41)
          Making use of Proposition 3 together with (34), we conclude that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ42_HTML.gif
          (42)
          For the proof of the second estimate, let http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq99_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq100_HTML.gif for i = 1,2, we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbf_HTML.gif
          Then for γ ∈ (0,1), we get
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbg_HTML.gif
          So, for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif small enough and using the estimate (35), there exist http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq101_HTML.gif (depending on κ ), such that:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ43_HTML.gif
          (43)

          Reducing λ κ > 0 and ε κ > 0 if necessary, we can assume that, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq55_HTML.gif for all λ ∈ (0, λ κ ) and ε ∈ (0, ε κ ). Then, (42) and (43) are enough to show that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq102_HTML.gif is a contraction from http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq103_HTML.gif into itself and hence has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq104_HTML.gif in this set. This fixed point is solution of (35). We summarize this in the following

          Proposition 5 Given κ > 0, there exists ε κ > 0 and λ κ > 0 (depending on κ) such that for all ε ∈ (0, ε κ ) and λ ∈ (0, λ κ ), for all set of parameter http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq95_HTML.gif satisfying (39) and function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq105_HTML.gif satisfying (26), there exists a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq106_HTML.gif solution of (36) such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbh_HTML.gif

          As in the previous section, observe that the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq107_HTML.gif being obtained as a fixed point for contraction mapping, depends smoothly on the parameters http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq108_HTML.gif and the points http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq109_HTML.gif .

          5 The nonlinear Cauchy-data matching

          Keeping the notations of the previous sections, we gather the results of Proposition 4 and 5. Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq110_HTML.gif ∈ Ω m are given close to x := (x1, ..., x m ) and satisfy (37). Assume also that τ := (τ1, ..., τ m ) ∈ [τ - , τ +] m ⊂ (0, ∞) m are given (the values of τ- and τ + will be fixed shortly). First, we consider some set of boundary data http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq111_HTML.gif satisfying (24). We set
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbi_HTML.gif
          According to the result of Proposition 4, we can find http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq112_HTML.gif a solution of
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ44_HTML.gif
          (44)
          in each http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq113_HTML.gif that can be decomposed as
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbj_HTML.gif
          where the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq114_HTML.gif satisfies
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ45_HTML.gif
          (45)
          Similarly, given some boundary data http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq115_HTML.gif satisfying (26), some parameters http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq116_HTML.gif satisfying (38), provide ε ∈ (0, ε κ ) and λ ∈ (0, λ κ ), we use the result of Proposition 5, to find a solution v ext of (43) which can be decomposed as
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbk_HTML.gif
          in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq117_HTML.gif where, the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq118_HTML.gif satisfies
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ46_HTML.gif
          (46)
          It remains to determine the parameters and the functions in such a way that the function which is equal to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq119_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq120_HTML.gif and that is equal to vext in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq121_HTML.gif is a smooth function. This amounts to find the boundary data and the parameters so that, for each i = 1 ..., m
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ47_HTML.gif
          (47)
          on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif . Assuming we have already done so, this provides for each ε and λ small enough a function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq123_HTML.gif (which is obtained by patching together the functions http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq119_HTML.gif and the function vext) solution of -Δv - λ |∇v|2 = ρ2 (e v + e γv ) and elliptic regularity theory implies that this solution is in fact smooth. This will complete the proof of our result since, as ε and λ tend to 0, the sequence of solutions we have obtained satisfies the required properties, namely, away from the points x i the sequence vε,λconverges to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq124_HTML.gif . Before we proceed, the following remarks are due. First, it will be convenient to observe that the function http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq125_HTML.gif can be expanded as
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ48_HTML.gif
          (48)
          near http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq126_HTML.gif . The function
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbl_HTML.gif
          which appear in the expression of v ext can be expanded as
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ49_HTML.gif
          (49)
          Near http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif . Here, we have defined
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbm_HTML.gif
          Thus for x near http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq126_HTML.gif , we have
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ50_HTML.gif
          (50)

          where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq127_HTML.gif .

          Next, in (47), all functions are defined on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif , but it will be convenient to solve the following equations
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ51_HTML.gif
          (51)

          on S1. Here, all functions are considered as functions of yS1 and we have simply used the change of variables http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq128_HTML.gif to parameterize http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq122_HTML.gif .

          Since the boundary data, we have chosen satisfy (24) and (26), we can decompose
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbn_HTML.gif
          where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq129_HTML.gif are constant functions on S1, http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq130_HTML.gif belong to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq131_HTML.gif and where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq132_HTML.gif are L2(S1) orthogonal to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif . Projecting the equations (51) over http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq133_HTML.gif will yield the system
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ52_HTML.gif
          (52)
          Let us comment briefly on how these equations are obtained. They simply come from (51) when expansions (48) and (49) are used, together with the expression of H i and H e given in Lemma 2 and Lemma 3, and also the estimates (45) and (46). The system (52) can be readily simplified into
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbo_HTML.gif
          We are now in a position to define τ - and τ + since, according to the above, as ε and λ tend to 0 we expect that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq135_HTML.gif will converge to x i and that τ i will converge to http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq136_HTML.gif satisfying
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbp_HTML.gif
          and hence, it is enough to choose τ - and τ + in such a way that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbq_HTML.gif
          We now consider the L2-projection of (51) over http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif . Given a smooth function f defined in Ω, we identify its gradient http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq137_HTML.gif with the element of http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq134_HTML.gif
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbr_HTML.gif
          With these notations in mind, we obtain the equations
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ53_HTML.gif
          (53)
          Finally, we consider the L2-projection onto L2(S1). This yields the system
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ54_HTML.gif
          (54)
          Thanks to the result of Lemma 4, this last system can be re-written as
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbs_HTML.gif
          If we define the parameters t = (t i ) ∈ ℝ m by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equbt_HTML.gif
          then, the system we have to solve reads
          http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_Equ55_HTML.gif
          (55)

          where as usual, the terms http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq138_HTML.gif depend nonlinearly on all the variables on the left side, but is bounded (in the appropriate norm) by a constant (independent of ε and λ) time http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq139_HTML.gif , provide ε ∈ (0, ε κ ) and λ ∈ (0, λ κ ). Then, the nonlinear mapping which appears on the right-hand side of (55) is continuous and compact. In addition, reducing ε κ and λ κ if necessary, this nonlinear mapping sends the ball of radius http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq140_HTML.gif (for the natural product norm) into itself, provided κ is fixed large enough. Applying Schauder's fixed Theorem in the ball of radius http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2011-10/MediaObjects/13661_2011_Article_10_IEq140_HTML.gif in the product space where the entries live yields the existence of a solution of Eq. (55) and this completes the proof of our Theorem 1. □

          Declarations

          Acknowledgements

          The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group project No RGP-VPP-087.

          Authors’ Affiliations

          (1)
          Department of Mathematics, College of Science, King Saud University
          (2)
          Département de Mathématiques, Faculté des Sciences de Tunis Campus Universitaire

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          Copyright

          © Baraket et al; licensee Springer. 2011

          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.