About a Partial Differential Equation-Based Interpolator for Signal Envelope Computing: Existence Results and Applications

Abstract

This paper models and solves the mathematical problem of interpolating characteristic points of signals by a partial differential Equation-(PDE-) based approach. The existence and uniqueness results are established in an appropriate space whose regularity is similar to cubic spline one. We show how this space is suitable for the empirical mode decomposition (EMD) sifting process. Numerical schemes and computing applications are also presented for signal envelopes calculation. The test results show the usefulness of the new PDE interpolator in some pathological cases like input class functions that are not so regular as in the cubic splines case. Some image filtering tests strengthen the demonstration of PDE interpolator performance.

1. Introduction

Interpolators are widely used in signal and processing or data analysis. In particular for the empirical mode decomposition (EMD) algorithm [1, 2], the iterative estimation of the signal trend is based on the computing of the envelopes obtained by the cubic spline interpolation of local extrema. The spline interpolation has been recognized as being very effective for EMD. But for signals that have no local extremum, the cubic spline interpolation fails. We proposed a PDE-based model which overcomes this limit of classical EMD implementation, in the computing of the envelopes for signals that have no local extremum [2–4]. Recently, the Spectral Intrinsic Decomposition (SID) method [5], based on the spectral decomposition of the PDE interpolator, provides a new application of our model. This PDE interpolator contribute, to the mathematical modeling of the EMD and has provided various applications in signal and image processing [4, 6]. In this paper, we describe the mathematical modeling of the new PDE interpolator by variational methods. The resolution of the variational problem leads to existence and uniqueness results in appropriate spaces. The paper is organized as follows. In Sections 2 and 2.1 recalls some PDE models in signal and image processing, and in Section 2.2 some mathematical preliminary notions are set out. In Section 3, the mathematical modeling is exposed. In Sections 4 and 5, the resolution of the variational problem is dealt. Subsequently, numerical implementation and applications are presented in Section 6. At last, we finish by conclusions.

2. Some Preliminaries

2.1. PDE Models in Signal and Image Processing and EMD Principle

2.1.1. Some Diffusion Equations

This part consists of a brief and nonexhaustive presentation of classical nonlinear diffusion filters for 1D and 2D signal processing, with more focus on the 2D case. In this purpose, we can recall the model for nonlinear diffusion in image filtering proposed by Catte et al. [7]. This filter is a modified version of the well-know Perona and Malik model [8]. The basic equation that governs nonlinear diffusion filtering is 𝑢_𝑡(𝐱,𝑡)=div𝑔|∇𝑢(𝐱,𝑡)|²∇𝑢(𝐱,𝑡), (1) with 𝐱=(𝑥₁,𝑥₂), and where 𝑢(𝐱,𝑡) is a filtered version of the original image 𝑢(𝐱,𝑡)=𝑢₀(𝐱) as the initial condition, with reflecting boundary conditions. In (1), 𝑔(⋅) is the conductivity (or diffusivity) function, which is dependent (in space and time) on the image gradient magnitude. Several forms of diffusivity were introduced in the original paper of Perona and Malik [8]. All forms of diffusivity are chosen to be a monotonically decreasing function of the signal gradient. Possible expressions for conductivity are 𝑔₁()=1𝐱,𝑡1+|∇𝑢(𝐱,𝑡)|/𝛽²,𝑔₂(𝐱,𝑡)=exp−|∇𝑢(𝐱,𝑡)|/𝛽². (2) Parameter 𝛽 is a threshold one, which influences the anisotropic smoothing process. The nonlinear equation (1) acts as a forward parabolic equation smoothing regions while preserving edges. Other methods based on high-order PDE are provided for image restoration like in [9–11]. Efficient numerical schemes were introduced in [12] based on additive operator splitting (AOS) scheme, or based on alternating direction implicit (ADI) scheme. See [12–15] for a review and extensions of these methods. Unlike these methods of high-order PDE that are specially developed for denoising, our model was constructed to interpolate the characteristic points of a signal.

The major problem of nonlinear diffusion-based process is that it is generally difficult to correctly separate the high frequency components from the low frequency ones. In case of denoising applications, the objective of this process is to use the diffusivity function as a guide to retain useful data and suppress noise.

Numerous authors have proposed fourth-order PDEs for image smoothing and denoising with the hope that these methods would perform better than their second-order analogues [16–18]. Indeed, there are good reasons to consider fourth-order equations. The first reason that can be retained is the fact that fourth-order linear diffusion damps oscillations at high frequencies (e.g., noise) much faster than second-order diffusion. On the other hand, the theory of fourth order nonlinear PDEs is far less developed than the second order one. Also such equations often do not satisfy a maximum principle or comparison principle, and implementation of the equations could thus introduce artificial singularities or other undesirable behavior. In recent studies, Tumblin [19], Tumblin and Turk [16], and Wei [17] proposed the following form: 𝑢_𝑡(𝐱,𝑡)=−div𝑔(𝑚(𝑢))∇Δ𝑢(𝐱,𝑡), (3) where 𝑔(⋅)=𝑔₁(⋅) as in (2) and 𝑚 is some measurement of 𝑢(𝐱,𝑡). In [16], (3) is called a “low curvature image simplifier” (LCIS), and a good choice for 𝑚 is defined as 𝑚=Δ𝑢 to enforce isotropic diffusion [19]. These PDE tools for digital signal and image processing make more reachable the 2D extension of the 1D PDE-based method for characteristic points interpolation presented in this paper.

2.1.2. The Empirical Mode Decomposition Principle

The EMD [1] method decomposes iteratively a signal into amplitude modulation-frequency modulation (AM-FM) type components called intrinsic mode functions (IMF). The underlying principle of this decomposition is to locally identify in the signal, the most rapid oscillations defined as the waveform interpolating interwoven local maxima and minima. To do this, local extrema points are interpolated with a cubic spline, to yield the upper and lower envelopes. The mean envelope (half sum of upper and lower envelopes) is then subtracted from the initial signal, and the same interpolation scheme is reiterated on the remainder. The so-called sifting process stops when the mean envelope is reasonably zero everywhere, and the resulting signal is called the first IMF. The higher-order IMFs are iteratively extracted applying the same procedure to the initial signal after the previous IMFs have been removed.

2.2. Some Useful Mathematical Concepts

Throughout the paper, Ω denotes an open-bounded subset of ℝ^𝑛,  𝑛=1  or  2. In the sequel, we will need the following definitions and results.

Definition 1. we define one the spaces 𝑉 and 𝒱 as follows 𝑉=𝑣∈𝐻³(Ω)∣𝜕𝑣𝜕𝜈_𝜕Ω=0,𝒱=𝑣∈𝐻³(Ω)∣𝜕𝑣𝜕𝜈_𝜕Ω=0,_Ω,𝒱𝑣𝑑𝑥=0_𝑒=𝑣∈𝐻³(Ω)suchthat∃Γ_Ω⊂Ω,openandverifying_ΓΩ𝑣𝑑𝑥=0and𝜕𝑣𝜕𝜈_𝜕Ω=0. (4)

Definition 2. For a given function 𝑣∈𝐻¹(Ω), if 𝐷_𝑑(𝑣)(𝑥) and 𝐷_𝑔(𝑣)(𝑥) denote, respectively, the right and left derivative of 𝑣 at 𝑥; the minmod derivatives of 𝑣 is define by 𝜕𝑣(𝑥)𝐷𝜕𝑥=minmod_𝑔(𝑣)(𝑥),𝐷_𝑑(𝑣𝑥)=𝑝𝑝,𝑞)(whereminmod𝑝,𝑞signmin,if𝑝𝑞>0,0,if𝑝𝑞≤0. (5) A 𝐻¹(Ω) function being absolutely continuous admits right and left derivatives, then 𝑢₀∈𝐻¹(Ω) has obviously left and right derivatives, so that we can validate numerically computing of the diffusivity function 𝑔 defined in Section 3.

Definition 3. Let (𝑥_𝑛)_𝑛∈ℕ be a sequence of elements in a vectorial normed space (𝐸,‖⋅‖_𝐸); it is said that (𝑥_𝑛)_𝑛∈ℕ converge, weakly [20] in 𝐸, and noted by 𝑥_𝑛⇀𝑥, if exists an element 𝑥∈𝐸 such that ∀𝑓∈𝐸^′,  lim_𝑛→∞𝑓(𝑥_𝑛)=𝑓(𝑥), where 𝐸^′ denotes the set of continuous linear forms on 𝐸.

Definition 4 (the Green formula). Let 𝑢,𝑣∈𝐶²(Ω), then _ΩΔ𝑢𝑑𝑥=_𝜕Ω𝜕𝑢𝜕𝑛𝑑𝑠,_Ω∇𝑢∇𝑣𝑑𝑥=−_Ω𝑢Δ𝑣𝑑𝑥+_𝜕Ω𝜕𝑣𝜕𝑛𝑑𝑠,_Ω(𝑢Δ𝑣−𝑣Δ𝑢)𝑑𝑥=_𝜕Ω𝑢𝜕𝑣𝜕𝑛−𝑣𝜕𝑢𝜕𝑛𝑑𝑠Greenformula, (6) where 𝜕𝑢/𝜕𝑛 denotes the normal derivative of 𝑢 on the boundary 𝜕Ω of Ω.

Definition 5 (the Poincaré-Wirtinger inequality). Let Ω be an open-bounded set, and let 𝑢∈𝐻¹(Ω), then there exists a constant 𝐶>0 such that the norm of 𝑢 in 𝐻¹(Ω) and the norm of 𝑢 in 𝐿²(Ω) are linked by the following inequality: 1𝑢−|Ω|_Ω𝑢𝑑𝑠_𝐻1(Ω)≤𝐶‖∇𝑢‖_𝐿²_(Ω), (7) where |Ω| denotes the length of Ω. The best constant 𝐶 in the Poincaré-Wirtinger inequality is 1/𝜆, for example, the inverse of the first positive eigenvalue 𝜆 of the Laplace operator with homogenous Neumann boundaries conditions. In our case, Ω⊂ℝ, and |Ω| is the diameter of Ω.

Theorem 6   (regularity theorem). Let 𝑢∈𝑢∈𝐻³(Ω)𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝜕𝑢𝜕𝑛_𝜕Ω=0,_Ω𝑢𝑑𝑥=0, (8) then there exist two strictly positive reals 𝐶 and 𝐾 such that firstly,‖∇𝑢‖²_𝐿2(Ω)≤𝐶‖Δ𝑢‖²_𝐿2(Ω), (9) and secondly, ‖𝑢‖²_𝐻3(Ω)≤𝐾‖Δ𝑢‖²_𝐻1(Ω). (10)

The main existence and uniqueness result of the solution of our henceforth problem (29) is due to the application of the following Lions theorem [21].

Theorem 7 (the Lions theorem). Let 𝐻 and 𝑉 be two Hilbert spaces with 𝑉⊂𝐻. One considers a bilinear application 𝑎 on 𝑉×𝑉 and 𝐵 an operator on 𝑉. Under the following conditions: (1)∃𝜇 and 𝜌>0 such that |𝑎(𝑢,𝑣)|≥𝜌‖𝑣‖²_𝑉−𝜇|𝑣|²_𝐻,(2)∃𝑐₂>0 such that |(𝑣,𝐵𝑣)_𝑉|≤𝑐₂|𝑣|²_𝑉.
For 𝑢₀=𝑢(0)∈𝐻, the problem 𝑢∈𝐿²(0,𝑇;𝑉)∩𝐶(0,𝑇;𝐻), (11) such that ∀𝑣∈𝑉,𝑑𝑢𝑑𝑡,𝐵𝑣_𝐻+𝑎(𝑢,𝑣)=0 (12) and 𝑢₀=𝑢(0), has a unique solution.
Furthermore, 𝑑𝑢/𝑑𝑡∈𝐿²(0,𝑇;𝐻^′).

3. Mathematical Modeling of the New PDE-Interpolator

Recall that the PDE model aimed to contribute to the mathematical modeling of the EMD, as it is in [2]. In a first step, to be in line with the classical EMD, our goal is to model the upper or lower envelope as the asymptotic solution of a PDE system whose initial condition is the input signal that we want to interpolate local extrema (or more generally characteristic points). Initially, this envelope is obtained by cubic spline interpolation. Let Ω be the open domain of an given finite energy signal 𝑢₀∈𝐿²(Ω). We construct an operator 𝐴 of appropriate domain 𝐷(𝐴) as follows: 𝜕𝑢[]𝑢()𝜕𝑡+𝐴𝑢=0in0,𝑇×Ω,0,𝑥=𝑢₀inΩ,plusboundariesconditions. (13)

The need of asymptotic solution existence denoted by 𝑢_∞ implies (𝜕𝑢/𝜕𝑡)(∞,𝑥)=0, for example, according to (13), we have 𝐴𝑢(∞,𝑥)=0. Moreover, the requirement of the same regularity between the solution and a cubic spline leads to 𝜕⁴𝑢/𝜕𝑥⁴(∞,𝑥)=0. Thus, in the first analysis, we can choose 𝐴=𝜕⁴𝑢/𝜕𝑥⁴. To leave invariant features points during the diffusion process, simply multiply in the expression 𝐴𝑢 the term 𝜕⁴𝑢/𝜕𝑥⁴ by function 𝑔 depending on spatial variable 𝑥 and vanishing at characteristic points. That gives (𝑥)(𝑥)𝜕𝐴𝑢)(=𝑔⁴𝑢𝜕𝑥⁴. (14) Another reminiscent form of long-range diffusion [22] is the following: (𝑥)=𝜕𝐴𝑢)(𝑔(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³. (15) Function 𝑔 can be interpreted as a diffusivity function whose role is to control the diffusion process. We take it necessarily positive to have a direct diffusion and the existence of solution. Other forms of operators are possible [3] we present some ones as follows: for (14) (𝑥)(𝑥𝜕𝐴𝑢)(=𝑔)−𝜃²𝑢𝜕𝑥²+()𝜕1−𝜃⁴𝑢𝜕𝑥⁴, (16) or for (15), (𝑥)=𝜕𝐴𝑢)(𝑔(𝑥)𝜕𝑥−𝜃𝜕𝑢+()𝜕𝜕𝑥1−𝜃³𝑢𝜕𝑥³, (17) or accomplete diffusion involving a flow (𝑥)=𝜕𝐴𝑢)((𝑥)𝜕𝑥−𝜃𝑔𝜕𝑢+()𝜕𝜕𝑥1−𝜃𝑔(𝑥)𝜕𝜕𝑥²𝑢𝜕𝑥², (18) where 0≤𝜃≤1 is the tension factor which controls the regularity of the solution.

Both forms allow more freedom on the regularity of the solution 𝑢. To access the mathematical properties of the solution, we chose the second form in (17) with 𝜃=0 and the corresponding equation (15).

At the local scale, between two consecutive characteristic points—two local maxima, for example—the diffusion induces a smoothing phenomenon that deletes the local minimum. A simple form for 𝑔 to calculate the envelope is given by a positive piecewise function lower than 1 that is constant between two characteristic points of 𝑢₀ and zeroed only at these points. Characteristic points are often being defined by their values and the signs of first, second, and third local derivatives of 𝑢₀. We characterize the function 𝑔 as depending on sign(𝜕𝑢₀/𝜕𝑥),sign(𝜕²𝑢₀/𝜕𝑥²), and sign(𝜕³𝑢₀/𝜕𝑥³).

For the purpose of existence and regularity of the solution, we are led to work with the regularized version of sign: sign_𝜗(𝑧)=2𝜋arctan𝜋𝑧𝜗, (19) where 𝜗 is a regularization coefficient.

In the following, we define 𝑔^±_𝑒 for extrema detection, 𝑔_𝑓 for turning points detection, and 𝑔^±_mc for maximum and minimum curvature points detection: 𝑔^±_𝑒(𝑥)=19sign𝜕𝑢₀𝜕𝜕𝑥±sign²𝑢₀𝜕𝑥²+1²(+)(−)𝑔formaximaandforminima,_𝑓(𝑥)=sign_𝜗𝜕²𝑢₀𝜕𝑥²²𝑔forturningpoints,_±mc(𝑥)=19𝜕sign³𝑢₀𝜕𝑥³𝜕±sign²𝑢₀𝜕𝑥²+1², (20) (+) for maximum curvature points and (−) for minimum curvature points. All these functions are of the form 𝑔(𝑥)=[ℎ(𝑥)]²  with ℎ(𝑥)=0 at characteristic points.

So, we have 𝑔^′(𝑥)=0if 𝑔(𝑥)=0. This property formally allows us to cancel 𝐴𝑢 and 𝜕𝑢/𝜕𝑡 of the form (15) at the points where 𝑔 is null. The factor 1/9 is a normalization term and verifies 0≤𝑔(𝑥)≤1. A more general diffusivity function 𝑔 could be envisaged.

Let us consider the following differential operators 𝐴_𝑒 and 𝐴_𝑓 (𝑒 for upper and 𝑓 for inflexion or curvature): ∀𝑢∈𝐷𝐴_𝑒,𝐴_𝑒𝜕𝑢=𝑔𝜕𝑥^±𝑒(𝑥)𝜕³𝑢𝜕𝑥³, (21) or ∀𝑢∈𝐷𝐴_𝑓,𝐴_𝑓𝜕𝑢=𝑔𝜕𝑥^𝑓(𝑥)𝜕³𝑢𝜕𝑥³. (22)

If 𝑢₀∈𝐶²(Ω), as it is the case in the classical implementation EMD, there is no problem for interpolation by cubic spline method. But it is indeed a restriction on the regularity of 𝑢₀ which is due to the choice of the interpolation technique that requires a good detection of local extrema for the envelopes calculation. The basic EMD principle must apply for input functions that are not regular as functions in 𝐶²(Ω), for example, for signal 𝑢₀∈𝐻¹(Ω), as it is the case in reality [23]. But numerically, without even a regularization function sign, if the derivatives are taken in the sense of minmod called flux limiter or slope, the function 𝑔(𝑥) still has a meaning.

3.1. Interpretation of the Diffusivity Action in the Diffusion Process

Between two consecutive maxima 𝑋_max𝑖 and 𝑋_max𝑖+1, the function 𝑔⁺_𝑒 is piecewise constant and can be written as follows: 𝑔(𝑥)=0,at𝑋_max𝑖,𝑋_max𝑖+1,49,at𝑋_inf𝑖,𝑋_inf𝑖+1,𝑋_min𝑖,1,on𝑋_inf𝑖,𝑋_min𝑖∪𝑋_min𝑖,𝑋_inf𝑖+1,19,sur𝑋_max𝑖,𝑋_inf𝑖∪𝑋_inf𝑖+1,𝑋_max𝑖+1. (23)

We have a diffusion in the sense that 𝑢_𝑡=𝜕𝑢/𝜕𝑡 unhook to the maximum of the curve 𝑢₀. The reason is that the diffusion is more pronounced in local minimum because the smoothing effect tends to regularize curves. Thus, the upper envelope 𝑢(𝑡≅∞,⋅) of 𝑢₀ is less oscillating than 𝑢₀ is. The same behavior occurs for the mean envelope. We would then try to interpret locally the relationship 𝑢≥𝑢₀ by evoking the maximum principle [24]. But, this principle is not immediately checked for equations of the type (15). The justification lies in the fact that smoothing implies ‖∇𝑢(𝑡,⋅)‖_𝐿²_(Ω)≤‖∇𝑢₀‖_𝐿2(Ω).

In summary, we have just built a PDE system (1)having the input signal 𝑢₀ to decompose as an initial value condition, (2)such that the result of a diffusion process preserves characteristic points of 𝑢₀, (3)with a regularity similar to a cubic spline.

3.2. Formulation of the Mathematical Problem

Let 𝐴 be one of the above constructed interpolation operators, and 𝑇>0 be a diffusion time sufficiently long. To fix ideas, we solve the problem of upper envelope calculus. Indeed, for the other kinds of characteristic points, the same problem arises with analogous mathematical resolution methods.

The problem posed by our mathematical modeling is to find a solution 𝑢 of the system 𝜕𝑢[]𝑢()𝜕𝑡+𝐴𝑢=0in0,𝑇×Ω,0,𝑥=𝑢₀inΩ,plusboundariesconditions. (24) Then, the mean envelope (interpolating turning points) calculus is stated as follows.

For given 𝑢₀, find 𝑢 such that 𝜕𝑢+𝜕𝜕𝑡𝑔𝜕𝑥^𝑓(𝑥)𝜕³𝑢𝜕𝑥³[]𝑢()=0in0,𝑇×Ω,0,𝑥=𝑢₀inΩ,plusboundariesconditions. (25)

3.3. Comments on the Degeneracy in Problem (24)

To prevent the zero degeneracy case in the system (25) whose solution is not directly accessible by variational methods, we are led to solve an intermediate problem with 0<𝛼≤𝑔_𝑓,𝛼(𝑥)≤𝛽≤1. (26) That is the nondegenerate problem defined hereafter.

3.4. Formulation of the Zero Degeneracy Induced by Diffusivity Function

The diffusivity function in (26) is very close to zero at the characteristic points while refraining to cancel and to solve the system (25), now, just to move on to the limit when 𝛼 tends to zero to retrieve (25) and its solution.

Thus, in particular, we can define the function 𝑔_𝑓,𝑛(𝑥)=𝑔_𝑓(𝑥)+1𝑛,𝑛∈ℕ^∗ (27) which verifies the conditions of (26).

The new system that we call nondegenerate problem is fomulated as follows.

Find 𝑢_𝑛 such that 𝜕𝑢_𝑛+𝜕𝜕𝑡𝑔𝜕𝑥^𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³[]𝑢=0in0,𝑇×Ω,_𝑛(0,𝑥)=𝑢₀inΩ,plusboundariesconditions. (28)

3.5. Formulation of the Nondegenerate Problem

For 𝑛 sufficiently large, 𝑢_𝑛 is a close approximation of the envelope 𝑢, interpolating the characteristic points.

By a density technique, we can demonstrate that this sequence (𝑢_𝑛)_𝑛∈ℕ converges to the solution of the system (25). By posing for the convenience of notation 0<𝛼≤𝑔_𝑓,𝑛=ℎ≤𝛽≤1, the non-degenerate problem is reworded as.

Find 𝑢 such that 𝜕𝑢+𝜕𝜕𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³[]𝑢()=0in0,𝑇×Ω,0,𝑥=𝑢₀inΩ,plusboundariesconditions. (29)

Equation (29) is an operational differential equations type. The techniques for the resolution of this kind of problem are numerous we mainly refer to the variational formulation to find weak solutions [25] and the method of the approximation of evolutionary operators. The last allows to work on a new elliptic problem. The solution is obtained by passing to the limit of the approximate solution nstead of the equation.

In the next section, we solve (29) by a variational approach, according to the resolution methods for parabolic problems described in [7, 20].

4. Existence and Uniqueness of Solutions for PDE Interpolator

4.1. The Guideline for the Resolution of the Mathematical Problem

Now, we summarize the general procedure to solve the problem. (1)First step: resolution of the problem (29). This step includes (i)the variational formulation in Section 4.3, (ii)the resolution of the variational problem in appropriate functional space,(iii)the converse showing that the solution of the variational problem solves the departure problem in Section 4.3.4. (2)Second step: in Section 5, construction of the sequence (𝑢_𝑛)_𝑛∈ℕ of solutions of problem (29), where 𝑢_𝑛 is the solution obtained with the diffusivity. ℎ_𝑛(𝑥)=ℎ(𝑥)+(1/𝑛),  𝑛∈ℕ^∗, see Section 5. Finally, demonstrate that the sequence (𝑢_𝑛)_𝑛∈ℕ converges to a limit 𝑢, where 𝑢 is the solution of problem (25), in Section 5.

4.2. The Main Results Formulation

The main result which is going to be solves is the following.

Theorem 8. For 𝑢₀∈𝐻¹(Ω), exists a unique solution 𝑢∈𝐿²(0,𝑇;𝒱)∩𝐶([0,𝑇];𝐻¹(Ω)) of the variational problem 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥+_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥=0,𝑓𝑜𝑟𝑎𝑛𝑦𝑣∈𝒱 (30) such that 𝑢(0)=𝑢₀ and 𝑑𝑢/𝑑𝑡∈𝐿²(0,𝑇;𝐻). Reversely, let 𝑢 be a solution of the variational problem (30), then (𝜕𝑢/𝜕𝑡)+(𝜕/𝜕𝑥)(ℎ(𝑥)(𝜕³𝑢/𝜕𝑥³))=0.

And the complete follows result.

Theorem 9. For 𝑢₀∈𝐻¹(Ω), exists a unique solution 𝑢∈𝐿²(0,𝑇;𝒱)∩𝐶[0,𝑇];𝐻¹(Ω), (31) verifying 𝜕𝑢+𝜕𝜕𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³[]𝜕=0𝑖𝑛0,𝑇×Ω,₃𝑢𝜕𝑥³_𝜕Ω=0𝑜𝑛𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠𝑜𝑓Ω,𝜕𝑢𝜕𝑛_𝜕Ω,𝑢()=0𝑜𝑛𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠𝑜𝑓Ω0,𝑥=𝑢₀𝑖𝑛Ω. (32)

4.3. Variational Formulation

Let 𝑣∈𝐻³(Ω), and (to make sense to integrals) multiply the equation 𝜕𝑢+𝜕𝜕𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³=0 (33) of the system (29) by the test function 𝜕²𝑣/𝜕𝑥² after we integrate on Ω, then it comes out that _Ω𝑑𝑢𝜕𝑑𝑡²𝑣𝜕𝑥²𝑑𝑥+_Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝜕²𝑣𝜕𝑥²𝑑𝑥=0. (34) By using the Green formula (see Definition 4) and integration by parts, we get firstly _Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝜕²𝑣𝜕𝑥²=ℎ(𝑥)𝜕𝑑𝑥₃𝑢𝜕𝑥³𝜕²𝑣𝜕𝑥²_𝜕Ω−_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥, (35) and secondly, _Ω𝑑𝑢𝜕𝑑𝑡²𝑣𝜕𝑥²𝑑𝑑𝑥=𝑑𝑡_Ω𝑢𝜕²𝑣𝜕𝑥²𝑑𝑥,_Ω𝑢𝜕²𝑣𝜕𝑥²𝑑𝑥=_𝜕Ω𝑢𝜕𝑣𝜕𝑛𝑑𝜎−_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥. (36) As we are in one-dimension case, the normal derivative and the gradient on the edge in the use of the Green formula are the same, that is to say, _𝜕Ω𝑢𝜕𝑣𝜕𝑛𝑑𝜎=_𝜕Ω𝑢𝜕𝑣𝜕𝑥𝑑𝜎. (37) In the following, we adopt the notation 𝜕𝑣/𝜕𝑛.

Considering 𝑢∈𝐻³(Ω) such that 𝜕³𝑢𝜕𝑥³_𝜕Ω=0,𝑣∈𝑉=𝑣∈𝐻³(Ω)∣𝜕𝑣𝜕𝑛_𝜕Ω=0 (38) it comes out that 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥+_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥=0,∀𝑣∈𝑉. (39) Let us consider the following bilinear forms 𝑎(𝑢,𝑣)=_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑏()=𝑑𝑥,𝑢,𝑣_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥. (40) The addition of boundaries conditions transforms our initial problem into the system 𝜕𝑢+𝜕𝜕𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³[]𝜕=0in0,𝑇×Ω,₃𝑢𝜕𝑥³_𝜕Ω=0ontheedgeofΩ,𝜕𝑢𝜕𝑛_𝜕Ω,𝑢()=0ontheedgeofΩ0,𝑥=𝑢₀inΩ. (41)

The variational formulation of our problem consequently, subject to additional conditions that would cause a change of space, is the following 𝑑Find𝑢∈𝑉suchthat(𝑏(()𝑑𝑡𝑢,𝑣))+𝑎𝑢,𝑣=0,∀𝑣∈𝑉, (42) where 𝑎 and 𝑏 are like in Theorem 7.

This kind of problem is studied in [21]. However, before continuing the resolution, we explore some properties of the space 𝑉.

4.3.1. On the Quotient Space 𝑉/ℝ

The space 𝑉 is defined by 𝐷(𝑎)∩𝐷(𝑏), where 𝐷(𝑎) and 𝐷(𝑏) are the domains of 𝑎 and 𝑏. Let us consider the quotient space 𝑉ℝ={𝑣+𝑐,𝑐∈ℝ}suchthat𝑣∈𝐻³(Ω),and𝜕𝑣𝜕𝑛_𝜕Ω=0 (43) and the new space 𝒱  defined by 𝒱=𝑣∈𝐻³(Ω)∣𝜕𝑣𝜕𝑛_𝜕Ω=0,_Ω𝑣𝑑𝑥=0. (44) Let 𝑣^∼∈𝑉/ℝ, and let 𝑢 be a representative of 𝑣^∼ class, then (𝜕𝑣^∼/𝜕𝑛)|_𝜕Ω is defined by 𝜕𝑣^∼𝜕𝑛_𝜕Ω=𝜕(𝑢+𝑐)𝜕𝑛_𝜕Ω=𝜕𝑢𝜕𝑛_𝜕Ω=0with𝑐∈ℝ. (45) The nullity of ∫𝑣_Ω^∼𝑑𝑥 means ∃𝑐∈ℝ,  constant such that  ∫_Ω(𝑢+𝑐)𝑑𝑥=0,  with  𝑢  one representative of  𝑣^∼1if𝑐=−|Ω|_Ω𝑢𝑑𝑥,then_Ω∼𝑣𝑑𝑥=_Ω()=𝑢+𝑐𝑑𝑥_Ω1𝑢−|Ω|_Ω𝑢𝑑𝑥𝑑𝑥=0. (46)

So for all 𝑣^∼∈𝑉/ℝ,  we have  ∫𝑣_Ω^∼𝑑𝑥=0. Accordingly, any element of the quotient space 𝑉/ℝ is, on one hand, in 𝐻³(Ω) with normal derivative on the boundaries of Ω null and, on the other hand, is zero average on Ω. Thus, the quotient space 𝑉/ℝ is identified with the space 𝒱. We can work later in the space 𝒱 with zero mean. The final variational problem to solve is the following 𝑑find𝑢∈𝒱suchthat(𝑏(()𝑑𝑡𝑢,𝑣))+𝑎𝑢,𝑣=0,∀𝑣∈𝒱. (47)

4.3.2. Application of the Lions Theorem

To apply the Lions theorem, we need to identify the applications 𝑎,  𝑏, and 𝐵 and the spaces 𝐻 and 𝑉. And finally, we must prove that the norm defined in 𝒱, which derives from the scalar product given by the bilinear application 𝑎, is equivalent to the norm of 𝐻³(Ω).

Let us define 𝐵 as the identity, and let us define 𝑎 and 𝑏 by |𝑎(𝑢,𝑣)|=(√ℎ(𝜕³𝑢/𝜕𝑥³),√ℎ(𝜕³𝑣/𝜕𝑥³))_𝐿²_(Ω), and 𝑏(𝑢,𝑣)=∫_Ω(𝜕𝑢/𝜕𝑥)(𝜕𝑣/𝜕𝑥)𝑑𝑥, obtained from (39), (40); the spaces 𝑉=𝒱 and 𝐻 are define as follows.

(a) The linear operator 𝐵: in the variational problem (11) in the Lions theorem, (𝑑𝑢/𝑑𝑡,𝐵𝑣)_𝐻 denotes a scalar product on 𝐻. Indeed, we define 𝐻={𝑣∈𝐻¹(Ω)suchthat(𝜕𝑣/𝜕𝑛)|_𝜕Ω=0,∫_Ω𝑣𝑑𝑥=0}.

The operator 𝐵=𝐼, 𝐵𝑣=𝑣, is the identity which is continuous (verify the Lions conditions Theorem 7), and the term (𝑑𝑢/𝑑𝑡,𝐵𝑣)_𝐻=(𝑑/𝑑𝑡)(𝑏(𝑢,𝑣)) is equal to _Ω𝜕𝜕𝑥𝑑𝑢𝑑𝑡𝜕𝑣𝜕𝑥𝑑𝑥=_Ω𝑑𝑑𝑡𝜕𝑢𝜕𝑥𝜕𝑣=𝜕𝜕𝑥𝑑𝑥𝜕𝑥𝑑𝑢,𝑑𝑡𝜕𝑣𝜕𝑥_𝐿²_(Ω). (48) In this identification, ((𝜕/𝜕𝑥)(𝑑𝑢/𝑑𝑡),𝜕𝑣/𝜕𝑥)_𝐿²_(Ω) is a well scalar product in the subspace 𝐻 of 𝐻¹(Ω), formed by functions with zero mean. Indeed, according to Poincare-Wirtinger inequality (see Definition 5) in 𝐻¹(Ω), 1𝑢−|Ω|_Ω𝑢𝑑𝑥_𝐻1(Ω)≤1𝜆𝜕𝑢𝜕𝑥_𝐿²_(Ω), (49) where 𝜆 is the first positive eigenvalue of Laplace operator in Ω. But as 𝑢,𝑣∈𝒱,  then  ∫_Ω𝑢𝑑𝑥=0, consequently there exists 𝐶₂>0 such that: ‖𝑢‖_𝐻¹_(Ω)≤𝐶₂𝜕𝑢𝜕𝑥_𝐿²_(Ω), (50) that is to say the 𝐻¹(Ω) norm is equivalent to the gradient norm in 𝒱. Hence, one can replace the scalar product of the gradient by 𝐻¹(Ω) scalar product.

Hereinafter, we complete the proof that the precedent choices allow the application of the Lions theorem in our context.

(b) 𝒱is an Hilbert space with equivalent norm to 𝐻³(Ω) norm.

Theorem 10. The space 𝒱, resulting from (11) and 𝒱 analysis, defined by 𝒱=𝑣∈𝐻³(Ω)∣𝜕𝑣𝜕𝑛_𝜕Ω=0,_Ω𝑣𝑑𝑥=0, (51) is an Hilbert space with the a norm equivalent to 𝐻³(Ω) norm.

Proof. At first 𝒱={𝑣∈𝐻³(Ω)∣(𝜕𝑣/𝜕𝑛)∣_𝜕Ω=0,∫_Ω𝑣𝑑𝑥=0} is a closed vector subspace of 𝐻³(Ω). Note that the condition of the nullity of the mean in 𝒱 may be replaced by ∫_ΓΩ𝑢𝑑𝑥=0, where Γ_Ω is a nonempty open subset of Ω. In this case 𝒱 becomes rather 𝒱_𝑒=𝑣∈𝐻³(Ω)suchthat∃Γ_Ω⊂Ωverifying_ΓΩ𝑣𝑑𝑥=0,and𝜕𝑣𝜕𝑛_𝜕Ω=0. (52)
From the fact that 0<𝛼≤ℎ(𝑥)≤𝛽<1,∀𝑥∈Ω, (53) the norms built with the two bilinear applications 𝑎 and 𝑏 are equivalent to the norm constructed by the linear form, noted by |‖‖|²_𝒱 with 𝑣∈𝒱, and given by |‖𝑣‖|²_𝒱=𝜕𝑣𝜕𝑥²_𝐿²_(Ω)+𝜕³𝑣𝜕𝑥³²_𝐿2(Ω). (54)Indeed, |‖‖|²_𝒱 is as well a norm in 𝒱. There is no difficulty to formally verify the triangular inequality. If |‖𝑣‖|²_𝒱=0, then ‖𝜕𝑣/𝜕𝑥‖²_𝐿2(Ω)=0 and ‖𝜕³𝑣/𝜕𝑥³‖²_𝐿²_(Ω)=0. Consequently, 𝑣 is necessarily constant almost everywhere in Ω, for example, 𝑣 is equal to a constant real 𝐾. But as  ∫_Ω𝑣𝑑𝑥=0 implies 𝐾|Ω|=0, so 𝐾=𝑣=0. We have thus built a standard scalar product on 𝒱 from bilinear forms of the variational formulation. 𝒱 is a Hilbert space with the scalar product associated with the norm (54). In addition, this norm is equivalent to the 𝐻³(Ω) norm, as it shown in Lemma A.2. Now, we just need to prove that the correspondence (𝑢,𝑣)∈𝒱×𝒱↦𝑎(𝑢,𝑣) is continuous and coercive on the Hilbert space 𝒱.

4.3.3. The Application 𝑎(⋅,⋅) Is Continuous and Coercive

For 𝑢,𝑣∈𝒱, |𝑎(𝑢,𝑣)|=√ℎ𝜕³𝑢𝜕𝑥³,√ℎ𝜕³𝑣𝜕𝑥³_𝐿2(Ω)𝜕≤𝛽³𝑢𝜕𝑥³_𝐿2(Ω)⋅𝜕³𝑣𝜕𝑥³_𝐿2(Ω)≤𝛽|‖𝑢‖|_𝒱⋅|‖𝑣‖|_𝒱, (55) hence the continuity of 𝑎(⋅,⋅) is on 𝒱×𝒱. In order to complete the application conditions of Theorem 7, we only check the first condition of the same theorem.

For 𝑣∈𝒱,−‖𝑣‖²_𝐻1(Ω)≤−𝜕𝑣𝜕𝑥²_𝐿²_(Ω), (56) hence |‖𝑣‖|²_𝒱−‖𝑣‖²_𝐻1(Ω)≤|‖𝑣‖|²_𝒱−𝜕𝑣𝜕𝑥²_𝐿²_(Ω), (57) nevertheless 𝑎(𝑣,𝑣)=√ℎ𝜕³𝑣𝜕𝑥³²_𝐿2(Ω)𝜕≥𝛼³𝑣𝜕𝑥³²_𝐿2(Ω)=𝛼|‖𝑣‖|²_𝒱−𝜕𝑣𝜕𝑥²_𝐿²_(Ω). (58) Thus, 𝑎(𝑣,𝑣)≥𝛼|‖𝑣‖|²_𝒱−‖𝑣‖²_𝐻1(Ω),∀𝑣∈𝒱. (59) We have found thus (𝜌,𝜇)=(𝛼,𝛼)such that 𝑎(𝑣,𝑣)≥𝜌|‖𝑣‖|²_𝒱−𝜇‖𝑣‖²_𝐻1(Ω),∀𝑣∈𝒱. (60) All the conditions of Theorem 7 are acquired; now, the main existence result for solution of system (41) can be enunciated as follows.

Theorem 11. For 𝑢₀∈𝐻¹(Ω), exists a unique solution 𝑢∈𝐿²(0,𝑇;𝒱)∩𝐶([0,𝑇];𝐻¹(Ω)) of the problem 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥+_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑢(0)𝑑𝑥=0,𝑓𝑜𝑟𝑎𝑛𝑦𝑣∈𝒱𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡=𝑢₀,𝑑𝑢𝑑𝑡∈𝐿²(0,𝑇;𝐻). (61)

4.3.4. Equivalence with the Initial Problem

Reversely, let 𝑢 be a solution of the variational problem 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥+_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥=0∀𝑣∈𝒱. (62) As we must consider a test function 𝑣∈𝐻³(Ω)𝜕suchthat²𝑣𝜕𝑥²stillin𝐷(Ω),𝜕𝑣𝜕𝜈_𝜕Ω=0. (63)

Let us take a function test 𝜙∈𝐷(Ω) and 𝐾∈ℝ such that _Ω𝜙+𝐾𝑑𝑥=0. (64)

We pose the following Neumann problem: 𝜕²𝑣𝜕𝑥²=𝜙+𝐾,𝜕𝑣𝜕𝑛_𝜕Ω=0. (65)

Its variational formulation gives a unique solution in 𝐻={𝑤∈𝐻¹(Ω)∣∫_Ω𝑤𝑑𝑥=0}, and the regularity of 𝜙∈𝐻²(Ω) implies the regularity of the solution which is in 𝑣∈𝐻³(Ω), consequently 𝑣∈𝒱.

Returning to (62) with 𝑣, solution of the problem (65), the first term can be written as 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝑑𝜕𝑥𝑑𝑥=𝑑𝑡_Ω𝑢𝜕²𝑣𝜕𝑥²𝑑𝑥−_𝜕Ω𝑢𝜕𝑣,𝜕𝜕𝑛𝑑𝜎as²𝑣𝜕𝑥²=𝜙+𝐾,𝜕𝑣𝜕𝑛_𝜕Ω𝑑=0,anditcomesoutthat𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝑑𝜕𝑥𝑑𝑥=𝑑𝑡_Ω𝑢=𝑑𝜙+𝐾𝑑𝑥𝑑𝑡_Ω𝑢𝜙𝑑𝑥+𝐾_Ω𝑢𝑑𝑥. (66) However, ∫_Ω𝑢𝑑𝑥=0, subsequently 𝑑𝑑𝑡_Ω𝜕𝑢𝜕𝑥𝜕𝑣𝑑𝜕𝑥𝑑𝑥=𝑑𝑡_Ω𝑢𝜙𝑑𝑥=_Ω𝑑𝑢𝑑𝑡𝜙𝑑𝑥. (67) The second term of (62) is written as follows: _Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥=_Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕𝜙𝜕𝑥𝑑𝑥, (68) which gives in the distribution sense _Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕³𝑣𝜕𝑥³𝑑𝑥=_Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝜙𝑑𝑥, (69) from which _Ω𝑑𝑢𝑑𝑡𝜙𝑑𝑥+_Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝜙𝑑𝑥=0,∀𝜙∈𝐷(Ω). (70) We deduce that 𝑑𝑢+𝜕𝑑𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³=0. (71) We verify now that (𝜕³𝑢/𝜕𝑥³)|_𝜕Ω=0. Returning to (70) with 𝜙=𝜕²𝑣/𝜕𝑥² and according to the Green formula, we have _Ω𝑑𝑢+𝜕𝑑𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝜕²𝑣𝜕𝑥²−𝑑𝑥_𝜕Ωℎ(𝑥)𝜕³𝑢𝜕𝑥³𝜕𝜕𝑛𝜕𝑣𝜕𝑥⋅𝑛𝑑𝜎=0∀𝑣∈𝒱. (72) But as (𝜕/𝜕𝑛)(𝜕𝑣/𝜕𝑥)⋅𝑛=(𝜕/𝜕𝑥)(𝜕𝑣/𝜕𝑥)=𝜕²𝑣/𝜕𝑥², it gives according to (71) ∫_𝜕Ωℎ(𝑥)(𝜕³𝑢/𝜕𝑥³)(𝜕²𝑣/𝜕𝑥²)𝑑𝜎=0∀𝑣∈𝒱, and consequently ℎ(𝑥)(𝜕³𝑢/𝜕𝑥³)=0 on boundaries of Ω, or as ℎ(𝑥)≠0,  𝜕³𝑢/𝜕𝑥³=0  on boundaries of Ω.

In conclusion 𝑢 is solution of the initial equation (41), then we enunciate the following theorem.

Theorem 12. For 𝑢₀∈𝐻¹(Ω), exists a unique solution 𝑢∈𝐿²(0,𝑇;𝒱)∩𝐶([0,𝑇];𝐻¹(Ω)), verifying 𝜕𝑢+𝜕𝜕𝑡ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³[]𝜕=0𝑖𝑛0,𝑇×Ω,₃𝑢𝜕𝑥³_𝜕Ω=0𝑜𝑛𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠𝑜𝑓Ω,𝜕𝑢𝜕𝑛_𝜕Ω,𝑢()=0𝑜𝑛𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑖𝑒𝑠𝑜𝑓Ω0,𝑥=𝑢₀𝑖𝑛Ω. (73)

4.3.5. Some Remarks on the Modeling Space

The nullity condition of the average for functions in 𝒱 has been decisive in the construction of the norm on this space. This space is prima facie clearly not natural for EMD because of entry function 𝑢₀ that is not necessarily zero mean. But at the cost of working with an initial condition 𝑣₀∈𝐻¹(Ω), such that 𝑣₀=𝑢₀−(1/|Ω|)∫_Ω𝑢₀𝑑𝑥, we can consider 𝐻 as 𝐻¹(Ω) functions with null local mean.

The Nullity of the Mean Envelope: A Characteristic of 𝒱 and EMD Algorithm. This condition, far from being superfluous, was even expected because the space 𝒱 is the space of the mean envelopes which must be zero mean in EMD stifting process for mode extraction. If it were to calculate the upper or lower envelopes, it should be noted that the condition of the mean nullity raised a query. In EMD algorithm, this nullity condition is affected on the mean envelope and not on upper or lower envelopes. This question does not arise in the model giving the envelope interpolating inflection points of a signal. The elements of 𝒱_𝑒 are not zero mean on Ω, but on a non-empty subset Γ_Ω of Ω.

The nullity of the mean of the signal on a Γ_Ω is not immediately visible and warranty to decompose any function. We show in Lemma A.1 that each relevant function to be decomposed by EMD is an integrable function possessing at least three extrema belonging to 𝒱_𝑒. The modeling space of the extrema interpolation is 𝒱_𝑒 defined in (52).

Let 𝑓 be a function admitting at least three extrema we can consider that 𝑓 passes at least once a zero at a point 𝑥_𝑧 in its definition domain. Otherwise, we just work with the translatory (as our model is invariant by translation of the input function) vector which is equals to the half amplitude of 𝑓.

5. Convergence of the Sequence (𝑢_𝑛)_𝑛∈ℕ of Solutions of (41) to a Solution of the Called Degenerate Problem

In the following sections, we demonstrate that the sequence (𝑢_𝑛)_𝑛∈ℕ of solutions of the non degenerate problem is bounded. Next, we prove that there exists a subsequence of (𝑢_𝑛)_𝑛∈ℕ which converges weakly to an element 𝑢∈𝒱 and, finally, that this element is solution of the degenerate initial problem.

5.1. Some Estimations

From (28) we have in the distributions sense (multiplying by 𝜕²𝑢_𝑛/𝜕𝑥²): _Ω𝜕𝑢_𝑛𝜕𝜕𝑡²𝑢_𝑛𝜕𝑥²𝑑𝑥+_Ω𝜕ℎ𝜕𝑥^𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³𝜕²𝑢_𝑛𝜕𝑥²𝑑𝑥=0, (74) which is equivalent to 𝜕𝑢_𝑛,𝜕𝜕𝑡²𝑢_𝑛𝜕𝑥²_𝑉′𝑉+𝜕ℎ𝜕𝑥^𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³,𝜕²𝑢_𝑛𝜕𝑥²_𝑉′𝑉=0. (75) After integration by parts, it comes out that 12_Ω𝜕𝜕𝑡𝜕𝑢_𝑛𝜕𝑥²𝑑𝑥+_Ωℎ_𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³²𝑑𝑥=0, (76) then by integrating with respect to the variable 𝑡∈]0,𝑇[, it comes out that 12𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)−12𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω)+^𝑡_0Ωℎ_𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³²𝑑𝑥𝑑𝑠=0, (77) which implies that 𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2^𝑡_0Ωℎ_𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³²=𝑑𝑥𝑑𝑠𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (78) But as ℎ_𝑛(𝑥)≥𝛼, we have 𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2^𝑡_0Ωℎ_𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³²≥𝑑𝑥𝑑𝑠𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2𝛼^𝑡_0Ω𝜕³𝑢_𝑛𝜕𝑥³²𝑑𝑥𝑑𝑠 (79) or 𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2^𝑡_0Ωℎ_𝑛(𝑥)𝜕³𝑢_𝑛𝜕𝑥³²≥𝑑𝑥𝑑𝑠𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2𝛼^𝑡₀𝜕³𝑢_𝑛𝜕𝑥³²_𝐿2(Ω)𝑑𝑠, (80) from which 𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω)≥𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)+2𝛼^𝑡₀𝑢_𝑛²_𝒱−𝜕𝑢_𝑛(𝑠,⋅)𝜕𝑥²_𝐿²_(Ω)𝑑𝑠. (81) Thus, on one hand, 𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω)≥𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω). (82) But 𝜕𝑢_𝑛(𝑡,⋅)𝜕𝑥²_𝐿²_(Ω)≥𝜆²𝑢_𝑛²_𝐻1(Ω), (83) where 𝜆 comes from inequality (49), which implies that 𝑢_𝑛²_{𝐿∞0,𝑇;𝐻1(Ω)}≤1𝜆²𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω),𝑢_𝑛²_{𝐿20,𝑇;𝐻1(Ω)}≤𝑇𝜆²𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (84)(85)Then, the sequence (𝑢_𝑛)_𝑛∈ℕ  is bounded in 𝐿^∞(0,𝑇;𝐻¹(Ω)) and in 𝐿²(0,𝑇;𝐻¹(Ω)). And on the other hand 2𝛼^𝑇₀𝑢_𝑛²_𝒱−𝜕𝑢_𝑛(𝑠,⋅)𝜕𝑥²_𝐿²_(Ω)𝑑𝑠≤𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω), (86) which gives ^𝑇₀𝑢_𝑛²_𝒱𝑑𝑠≤^𝑇₀𝜕𝑢_𝑛(𝑠,⋅)𝜕𝑥²_𝐿²_(Ω)1𝑑𝑠+2𝛼𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (87) But as 𝜕𝑢_𝑛(𝑠,⋅)𝜕𝑥²_𝐿²_(Ω)≤𝑢_𝑛(𝑠,⋅)²_𝐻1(Ω), (88) therefore ^𝑇₀𝑢_𝑛²_𝒱𝑑𝑠≤^𝑇₀𝑢_𝑛(𝑠,⋅)²_𝐻1(Ω)1𝑑𝑠+2𝛼𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (89) Thus, ^𝑇₀𝑢_𝑛²_𝒱𝑑𝑠≤𝑇𝑢_𝑛²_{𝐿∞0,𝑇;𝐻1(Ω)}+12𝛼𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω), (90) which implies according to (84) that ^𝑇₀𝑢_𝑛²_𝒱𝑇𝑑𝑠≤𝜆²𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω)+12𝛼𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (91) We finally deduced the second estimate 𝑢_𝑛²_{𝐿2()0,𝑇;𝒱}=𝑢_𝑛²_{𝐿20,𝑇;𝒱𝑒}≤𝑇𝜆²+12𝛼𝜕𝑢₀𝜕𝑥²_𝐿²_(Ω). (92)Then, the sequence (𝑢_𝑛)_𝑛∈ℕ is bounded in 𝐿²(0,𝑇;𝒱).

5.2. Weak Convergence of the Subsequence of (𝑢_𝑛)_𝑛∈ℕ

The sequence (𝑢_𝑛)_𝑛∈ℕ is bounded in the space 𝐿²(0,𝑇;𝒱). Then there exists a subsequence (𝑢_𝑛𝑘)_𝑘∈ℕ that converge weakly to 𝑢 in 𝐿²(0,𝑇;𝒱) with 𝑑𝑢/𝑑𝑡∈𝐿²(0,𝑇;𝑉^′).

Leting 𝑣∈𝒱 be a test function, we have the following.

Firstly. As 𝑢_𝑛𝑘→𝑢 weakly in 𝐿²(0,𝑇;𝑉), therefore, 𝑢_𝑛𝑘→𝑢 weakly in 𝐿²(0,𝑇;𝐻¹(Ω)).

Thus, in the distributions sense, _Ω𝑑𝑢_𝑛𝑘𝜕𝑑𝑡²𝑣𝜕𝑥²𝑑𝑑𝑥=𝑑𝑡_Ω𝜕𝑢_𝑛𝑘𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥, (93) by integrating on [0,𝑡], we have ^𝑡_0Ω𝑑𝑢_𝑛𝑘𝜕𝑑𝑠²𝑣𝜕𝑥²𝑑𝑥𝑑𝑠=−^𝑡₀𝑑𝑑𝑠_Ω𝜕𝑢_𝑛𝑘𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥𝑑𝑠=−^𝑡₀𝑑𝑢_𝑛𝑘,𝜕𝑑𝑠²𝑣𝜕𝑥²_𝑉′𝑉𝑑𝑠=−_Ω𝑢_𝑛𝑘(𝑡)−𝑢_𝑛𝑘(0)𝜕²𝑣𝜕𝑥²𝑑𝑥. (94) By integrating by parts again the last equation, we have _Ω𝑢_𝑛𝑘(𝑡)−𝑢_𝑛𝑘(0)𝜕²𝑣𝜕𝑥²𝑑𝑥=_Ω𝜕𝑢_𝑛𝑘(𝑡)𝜕𝑥𝜕𝑣−𝜕𝑥𝑑𝑥_Ω𝜕𝑢_𝑛𝑘(0)𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥. (95) This expression converges to 𝜕𝑢(_Ω𝑡)𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥−_Ω𝜕𝑢₀𝜕𝑥𝜕𝑣𝜕𝑥𝑑𝑥=_Ω𝜕𝑢(𝑡)−𝜕𝑥𝜕𝑢₀𝜕𝑥𝜕𝑣=𝜕𝑥𝑑𝑥_Ω𝑢(𝑡)−𝑢₀𝜕²𝑣𝜕𝑥²=𝑑𝑥_𝑡0𝑑𝑢,𝜕𝑑𝑠²𝑣𝜕𝑥²_𝑉′𝑉𝑑𝑠. (96) Therefore,∫^𝑡₀∫_Ω(𝑑𝑢_𝑛𝑘/𝑑𝑠)(𝜕²𝑣/𝜕𝑥²)𝑑𝑥𝑑𝑠converges when 𝑛_𝑘  tends to infinity to ^𝑡_0Ω𝑑𝑢𝜕𝑑𝑠²𝑣𝜕𝑥²𝑑𝑥𝑑𝑠. (97) Consequently, 𝜕𝑢_𝑛𝑘/𝜕𝑡→𝜕𝑢/𝜕𝑡 weakly in 𝐿²(0,𝑇;𝒱^′).

Secondly. As ℎ_𝑛(𝑥)=ℎ(𝑥)+(1/𝑛),𝜕wehaveℎ𝜕𝑥^𝑛𝑘𝜕³𝑢_𝑛𝑘𝜕𝑥³(𝑥)𝜕−ℎ³𝑢𝜕𝑥³=𝜕ℎ(𝑥)𝜕𝜕𝑥𝜕𝑥³³𝑢_𝑛𝑘+1−𝑢𝑛_𝑘𝜕⁴𝑢_𝑛𝑘𝜕𝑥⁴.However,_Ω𝜕⁴𝑢_𝑛𝑘𝜕𝑥⁴𝑣𝑑𝑥=_𝜕Ω𝜕³𝑢𝜕𝑥³−𝑣⋅𝑛𝑑𝜎_Ω𝜕𝑣𝜕𝜕𝑥³𝑢_𝑛𝑘𝜕𝑥³𝜕𝑑𝑥.Andbecause³𝑢𝜕𝑥³_𝜕Ω𝜕=0,³𝑢_𝑛𝑘𝜕𝑥³𝜕convergesto³𝑢𝜕𝑥³in𝐿²0,𝑇;𝐿²(Ω), (98) then, 1𝑛_𝑘Ω𝜕⁴𝑢_𝑛𝑘𝜕𝑥⁴=1𝑣𝑑𝑥𝑛_𝑘Ω𝜕³𝑢_𝑛𝑘𝜕𝑥³𝜕𝑣≤1𝜕𝑥𝑑𝑥𝑛_𝑘𝜕𝑣𝜕𝑥_𝐿²_(Ω)𝜕³𝑢_𝑛𝑘𝜕𝑥³_𝐿2(Ω) (99) converges to zero when 𝑛_𝑘 tends to infinity.

On the other hand, from (98), we have _Ω𝜕ℎ(𝑥)𝜕𝜕𝑥𝜕𝑥³³𝑢_𝑛𝑘−𝑢𝑣𝑑𝑥≤𝛽_Ω𝜕³𝜕𝑥³𝑢_𝑛𝑘−𝑢𝜕𝑣𝜕𝑥𝑑𝑥 (100) which converges to zero in 𝐿²(0,𝑇;𝐿²(Ω)) according to weak convergence (see Definition 3) in 𝐻³(Ω)  of  𝜕³𝑢_𝑛𝑘/𝜕𝑥³  to  𝜕³𝑢/𝜕𝑥³. Thus, we have shown that in the distributions sense of the one hand _Ω𝑑𝑢_𝑛𝑘𝜕𝑑𝑡²𝑣𝜕𝑥²𝑑𝑥 (101) converges to _Ω𝑑𝑢𝜕𝑑𝑡²𝑣𝜕𝑥²𝑑𝑥, (102) and on the other hand, _Ω𝜕ℎ𝜕𝑥^𝑛𝑘𝜕³𝑢_𝑛𝑘𝜕𝑥³𝑣𝑑𝑥 (103) converges to _Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝑣𝑑𝑥. (104) Consequently, for each 𝑣∈𝐿²(0,𝑇;𝑉^′), _Ω𝑑𝑢_𝑛𝑘𝑑𝑡𝑣𝑑𝑥+_Ω𝜕ℎ𝜕𝑥^𝑛𝑘𝜕³𝑢_𝑛𝑘𝜕𝑥³𝑣𝑑𝑥 (105) converges in the distributions sense to _Ω𝑑𝑢𝑑𝑡𝑣𝑑𝑥+_Ω𝜕ℎ(𝑥)𝜕𝜕𝑥³𝑢𝜕𝑥³𝑣𝑑𝑥, (106) that is to say, ⟨(𝑑𝑢_𝑛𝑘/𝑑𝑡),𝑣⟩_𝑉′𝑉+⟨(𝜕/𝜕𝑥)(ℎ_𝑛𝑘(𝑥)(𝜕³𝑢_𝑛𝑘/𝜕𝑥³)),𝑣⟩_𝑉′𝑉 converges when 𝑛_𝑘  tends to infinity to ⟨(𝑑𝑢/𝑑𝑡),𝑣⟩_𝑉′𝑉+⟨(𝜕/𝜕𝑥)(ℎ(𝑥)(𝜕³𝑢/𝜕𝑥³)),𝑣⟩_𝑉′𝑉. It remains to show that the limit 𝑢 is a solution of the degenerate problem. But ∫_Ω(𝑑𝑢_𝑛𝑘/𝑑𝑡)𝑣𝑑𝑥+∫_Ω(𝜕/𝜕𝑥)(ℎ_𝑛𝑘(𝜕³𝑢_𝑛𝑘/𝜕𝑥³))𝑣𝑑𝑥=0, which implies in the distributions sense