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Regularization method and a posteriori error estimates for the two membranes problem

  • Mohammed Bouchlaghem EMAIL logo , El Bekkaye Mermri and Zhor Mellah
Published/Copyright: December 31, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

This study presents a regularization method for the two membranes problem with non-homogeneous boundary conditions. We establish both convergence results and a priori estimates for this method. Using duality theory from convex analysis, we identify the dual problem related to the two membranes problem. Based on this dual formulation, we give a posteriori error estimates for both the continuous and discrete versions of the problem. These a posteriori error estimates are crucial for the practical implementation of the regularized problem.

Keywords: variational inequalities; two membranes problem; a posteriori error estimates; regularization method; duality theory
MSC 2010: 65K15; 49J40; 49M29

1 Introduction

Let Ω be an open and bounded subset of R n with a smooth boundary ∂ Ω . Let ( f 1 , f 2 ) ∈ [ L 2 ( Ω ) ] 2 and ( g 1 , g 2 ) ∈ [ H 1 ∕ 2 ( ∂ Ω ) ] 2 , with g 1 ⩾ g 2 on ∂ Ω . For i = 1 , 2 , we denote

H g i 1 ( Ω ) ≔ { ν ∈ H 1 ( Ω ) : ν = g i on ∂ Ω } ,

and we define

K ≔ { ( v 1 , v 2 ) ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) : v 1 ⩾ v 2 a.e. in Ω } .

We take into consideration the following problem

(1) Find ( u 1 , u 2 ) ∈ K such that ∑ i = 1 2 ∫ Ω ∇ u i ⋅ ∇ ( v i − u i ) d x + ∑ i = 1 2 ∫ Ω f i ( v i − u i ) d x ⩾ 0 ∀ ( v 1 , v 2 ) ∈ K .

This variational inequality is named two membranes problem [1,2], where the solution describes the position of the two membranes, constrained by each other and subjected to external forces. The regularity of the free boundary in the two membranes problem was studied in [3]. Addou and Mermri [4] reformulated the problem as a set of independent equations derived from the sub-differential of a continuous convex function. Later Bouchlaghem and Mermri [5] introduced a new mixed formulation of problem (1) and presented an iterative method to solve the problem numerically. The existence and uniqueness of the solution of the m -membrane problem has been explored in [6]. Additionally, Bozorgnia and Franek [7] introduced a numerical approximation of the m -membrane problem, where the problem is reformulated as a bound constraint quadratic minimization problem.

Problem (1) can be reformulated as an unconstrained variational inequality problem (refer to [5]). The challenge in solving this problem arises from the presence of a non-differentiable term Ψ (11). Our objective of this study is to introduce a regularization approach to address the non-differentiable minimization problem associated with the two membranes problem (14). The proposed regularization method involves replacing the non-differentiable function Ψ with a sequence of differentiable functions denoted by Ψ ξ , where ξ goes to zero [8–10]. We provide two examples of regularization and demonstrate the convergence of the approximate solutions along with a priori error estimates. Furthermore, by applying duality methods, using conjugate functions (as detailed in [11]), we establish a posteriori error estimates for both the continuous and discrete problems. Finally, we provide some concluding remarks.

Assume that we have evaluated both the approximate solution ( u h , 1 , u h , 2 ) of the discrete problem and the approximate solution ( u h , ξ , 1 , u h , ξ , 2 ) of the discrete regularized problem. Theoretically, obtaining more accurate approximations requires using smaller values for ξ . However, in practical applications, setting ξ too small is not feasible because it degrades the conditioning of the regularized problem and can lead to inaccuracies in the numerical solution. Therefore, practical implementation of the discrete regularized problem necessitates a posteriori error bounds or quantitative estimates once solutions to the regularized problems are obtained. These a posteriori error estimates can be used as stopping criteria in actual computations [12].

Huang et al. [9] and Addou et al. [13] proposed a regularization approach for a unilateral obstacle problem and derived a posteriori error estimates using the duality theory. This work extends the regularization method to the two membranes problem (1).

2 Background on variational inequality theory

The theory of variational inequalities is a relatively young mathematical discipline. The foundation of this theory was established by Lions and Stampacchia in 1967 [14]. The diverse applications of the variational inequalities theory are detailed by Duvaut and Lions in their 1972 work [15], and by Trémolières et al. [16]. In 1980, Kinderlehrer and Stampacchia published an excellent monograph titled “An Introduction to variational inequalities and their applications” [2]. The literature also includes several recent works on the analysis and numerical methods for variational inequality problems [17–19].

Before proceeding with the reformulation and regularization of the two membrane problem (1), we aim to present the background of the theory of elliptic variational inequalities ([2] Chap. 2 and [8], Chap. 1). First, we will state a fundamental theorem regarding the existence, uniqueness, and characterization of solutions to convex function minimization problems. Let

  • V be a reflexive Banach space and V * its dual space. We denote the duality pairing between V and V * by ⟨ ⋅ , ⋅ ⟩ .

  • L be a closed convex subset of V .

  • h 0 : V → R be a convex function, and we additionally assume that it is Gâteaux-differentiable.

  • φ : V → R ¯ be a convex, lower semi-continuous, and proper function, such that

    Dom ( φ ) ≔ { v ∈ V : φ ( v ) ∈ R } .

  • h : V → R ¯ be the function defined by h ≔ h 0 + φ , . We assume that h satisfies the following coercivity condition:

    (2) h ( v ) → + ∞ , as ‖ v ‖ V → + ∞ , v ∈ L .

Under the above assumptions on L , h 0 , φ , and h , we have the following theorem ([8], Theorem 2.1, page 141):

Theorem 2.1

The minimization problem

(3) F i n d u ∈ L s u c h t h a t h ( u ) = inf v ∈ L h ( v )

has a solution u, and furthermore, u is a solution to the problem:

(4) F i n d u ∈ L s u c h t h a t ⟨ h 0 ′ ( u ) , v − u ⟩ + φ ( v ) − φ ( u ) ≥ 0 ∀ v ∈ L .

This solution is unique if h is strictly convex.

This theorem shows that the minimization problem (3) can be reduced to solving a problem in the form of an inequality. This observation leads us to introduce the concept of variational inequality. In the following, we retain the same notations and assumptions stated at the beginning of this section and additionally assume that V is a Hilbert space. We introduce a continuous bilinear form a : V × V → R , and we assume that a ( ⋅ , ⋅ ) is coercive, in the sense that

∃ α > 0 , ∀ v ∈ V , a ( v , v ) ≥ α ‖ v ‖ 2 .

Let f be an element of V * . We consider the following two problems:

(5) Find u ∈ K such that a ( u , v − u ) + ⟨ f , v − u ⟩ ≥ 0 ∀ v ∈ L

and

(6) Find u ∈ V such that a ( u , v − u ) + φ ( v ) − φ ( u ) + ⟨ f , v − u ⟩ ≥ 0 ∀ v ∈ V .

Problem (5) is called a variational inequality of the first kind, while problem (6) is called a variational inequality of the second kind.

Theorem 2.2

([8] Theorem 3.1 page 3 and Theorem 4.1 page 5) The variational inequalities (5) and (6) have unique solutions.

We consider the following variational inequality

(7) Find u ∈ L such that a ( u , v − u ) + φ ( v ) − φ ( u ) + ⟨ f , v − u ⟩ ≥ 0 ∀ v ∈ L .

We assume that the bilinear form a ( ⋅ , ⋅ ) is symmetric and we define

  1. h 0 : V → R by h 0 ( v ) ≔ 1 2 a ( v , v ) + ⟨ f , v ⟩ and

  2. h : V → R ¯ by h ( v ) ≔ h 0 ( v ) + φ ( v ) .

It is clear that J 0 is Gâteaux-differentiable and that for all elements u , v ∈ V , we have

⟨ h 0 ′ ( u ) , v ⟩ = a ( u , v ) + ⟨ f , v ⟩ .

The coercivity of a ( ⋅ , ⋅ ) ensures that h satisfies the coercivity condition (2). Moreover, it can be seen that h is strictly convex. Therefore, by Theorem 2.1, we obtain the following result.

Proposition 2.3

The variational inequality (7) has a unique solution; moreover, it is equivalent to the following minimization problem:

(8) F i n d u ∈ L s u c h t h a t h ( u ) = min v ∈ L h ( v ) .

3 Reformulation and regularization of the two membranes problem (1)

In the following sections, we will use the same notation g i , i = 1 , 2 , to designate at the same time a function in H 1 ( Ω ) and its trace on the boundary ∂ Ω . We denote

• f ≔ ( f 1 , f 2 ) and g ≔ ( g 1 , g 2 ) , • a ( u , v ) ≔ ∫ Ω ∇ u 1 ⋅ ∇ v 1 d x + ∫ Ω ∇ u 2 ⋅ ∇ v 2 d x , ∀ u = ( u 1 , u 2 ) , v = ( v 1 , v 2 ) ∈ [ H 1 ( Ω ) ] 2 , • ( ( u , v ) ) ≔ ∫ Ω u 1 v 1 d x + ∫ Ω u 2 v 2 d x , ∀ u = ( u 1 , u 2 ) , v = ( v 1 , v 2 ) ∈ [ L 2 ( Ω ) ] 2 , • ν + ≔ max ( ν , 0 ) , ∀ ν ∈ L 2 ( Ω ) , • ν − ≔ min ( ν , 0 ) , ∀ ν ∈ L 2 ( Ω ) .

Problem (1) can be formulated as

(9) Find u ∈ K such that a ( u , v − u ) + ( ( f , v − u ) ) ⩾ 0 ∀ v ∈ K .

Set w ≔ u − g and

K 0 ≔ { ( v 1 , v 2 ) ∈ [ H 0 1 ( Ω ) ] 2 : v 1 + g 1 ⩾ v 2 + g 2 a.e. in Ω } .

Hence, problem (1) can be stated as follows:

(10) Find w ∈ K 0 such that a ( w + g , v − w ) + ( ( f , v − w ) ) ⩾ 0 ∀ v ∈ K 0 ,

where K 0 is a closed convex set of [ H 0 1 ( Ω ) ] 2 . Additionally, the bilinear form a ( ⋅ , ⋅ ) defines an inner product on [ H 0 1 ( Ω ) ] 2 ; thus, it is continuous and coercive. The function h : v ↦ h ( v ) = a ( g , v ) + ( ( f , v ) ) is a continuous linear form on [ H 0 1 ( Ω ) ] 2 . Therefore, problem (10) is a variational inequality of the first kind (5) which admits a unique solution, refer Theorem 2.2.

For the couple of the external forces f = ( f 1 , f 2 ) ∈ [ L 2 ( Ω ) ] 2 , we denote

f ˜ ≔ f 1 + − f 2 − 2 .

Let Ψ be the function defined from [ L 2 ( Ω ) ] 2 to R by

Ψ ( v ) ≔ ∫ Ω f ˜ ( v 2 − v 1 ) + d x , v = ( v 1 , v 2 ) ∈ [ L 2 ( Ω ) ] 2 .

By using the projection techniques onto the cone and its polar cone presented in [5] (pp. 2–4), we transform problem (10), which is a variational inequality of the first kind, into a variational inequality of the second kind (6). As a direct consequence of Theorem 3.1 of [5] (page 3), we have

Theorem 3.1

The two membranes problem (10) is equivalent to the following unconstrained variational inequality:

(11) F i n d w ∈ [ H 0 1 ( Ω ) ] 2 s u c h t h a t a ( w + g , v − w ) + Ψ ( v + g ) − Ψ ( w + g ) + ( ( f , v − w ) ) ⩾ 0 ∀ v ∈ [ H 0 1 ( Ω ) ] 2 .

Proof

We denote F 1 ≔ ( f 1 + , f 2 − ) , F 2 ≔ ( f 1 − , f 2 + ) , and for v = ( v 1 , v 2 ) ∈ [ L 2 ( Ω ) ] 2 ,

v + ≔ v 1 + ( v 2 − v 1 ) + 2 , v 2 − ( v 2 − v 1 ) + 2 .

According to Theorem 3.1 of [5] (page 3), problem (10) is equivalent to the following problem:

Find w ∈ [ H 0 1 ( Ω ) ] 2 such that a ( w + g , v − w ) + φ ( v ) − φ ( w ) + ( ( F 2 , v − w ) ) ≥ 0 ∀ v ∈ [ H 0 1 ( Ω ) ] 2 ,

where φ : [ L 2 ( Ω ) ] 2 → R , v ↦ φ ( v ) = ( ( F 1 , ( v + g ) + ) ) . We have

φ ( v ) − φ ( w ) = ∫ Ω f 1 + v 1 − w 1 + ( v 2 − v 1 + g 2 − g 1 ) + − ( w 2 − w 1 + g 2 − g 1 ) + 2 + … f 2 − v 2 − w 2 − ( v 2 − v 1 + g 2 − g 1 ) + − ( w 2 − w 1 + g 2 − g 1 ) + 2 d x

and

( ( F 2 , v − w ) ) = ∫ Ω [ f 1 − ( v 1 − w 1 ) + f 2 + ( v 2 − w 2 ) ] d x .

Since f 1 = f 1 + + f 1 − , f 2 = f 2 + + f 2 − , and f ˜ = f 1 + − f 2 − 2 ,

φ ( v ) − φ ( w ) + ( ( F 2 , v − w ) ) = ∫ Ω f ˜ ( ( v 2 − v 1 + g 2 − g 1 ) + − ( w 2 − w 1 + g 2 − g 1 ) + ) d x + … ∫ Ω [ f 1 ( v 1 − w 1 ) + f 2 ( v 2 − w 2 ) ] d x = Ψ ( v + g ) − Ψ ( w + g ) + ( ( f , v − w ) ) ,

which implies that problem (10) is equivalent to the following unconstrained variational inequality:

□ Find w ∈ [ H 0 1 ( Ω ) ] 2 such that a ( w + g , v − w ) + Ψ ( v + g ) − Ψ ( w + g ) + ( ( f , v − w ) ) ⩾ 0 ∀ v ∈ [ H 0 1 ( Ω ) ] 2 .

Due to the symmetry of the bilinear form a ( ⋅ , ⋅ ) , problem (11) is equivalent to the following minimization problem (Proposition 2.3):

(12) Find w ∈ [ H 0 1 ( Ω ) ] 2 : J 0 ( w ) = min v ∈ [ H 0 1 ( Ω ) ] 2 J 0 ( v ) ,

where J 0 is the function defined on [ H 0 1 ( Ω ) ] 2 by

J 0 ( v ) ≔ 1 2 a ( v , v ) + Ψ ( v + g ) + ( ( f , v ) ) + a ( g , v ) .

If we define u ≔ w + g , problem (11) can be written as

(13) Find u ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) such that a ( u , v − u ) + Ψ ( v ) − Ψ ( u ) + ( ( f , v − u ) ) ⩾ 0 ∀ v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) ,

which corresponds to the subsequent minimization problem as follows:

(14) Find u ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) : J ( u ) = min v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) J ( v ) ,

where J denotes the function defined on H g 1 1 ( Ω ) × H g 2 1 ( Ω ) by

J ( v ) ≔ 1 2 a ( v , v ) + Ψ ( v ) + ( ( f , v ) ) .

The functional J is not differentiable because Ψ is not. To address this, we approximate Ψ using a sequence of differentiable functions Ψ ξ as follows:

Ψ ξ ( v ) ≔ ∫ Ω f ˜ ψ ξ ( v 2 − v 1 ) d x ,

where ψ ξ is a differentiable and convex function from R into R , satisfying

lim ξ → 0 ψ ξ ( t ) = t + ∀ t ∈ R .

The regularized problem corresponding to problem (11) is given by

(15) Find w ξ ∈ [ H 0 1 ( Ω ) ] 2 such that a ( w ξ + g , v − w ξ ) + Ψ ξ ( v + g ) − Ψ ξ ( w ξ + g ) + ( ( f , v − w ξ ) ) ⩾ 0 ∀ v ∈ [ H 0 1 ( Ω ) ] 2 .

If we set u ξ ≔ w ξ + g , then the regularized problem corresponding to (13) is given by

(16) Find u ξ ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) such that a ( u ξ , v − u ξ ) + Ψ ξ ( v ) − Ψ ξ ( u ξ ) + ( ( f , v − u ξ ) ) ⩾ 0 ∀ v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) .

Since Ψ ξ is a convex, continuous, and proper functional, problem (15) (resp. problem (16)) has a unique solution w ξ (resp. u ξ ), [8].

We provide here examples of ψ ξ :

(17) ψ ξ 1 ( t ) = t − ξ 2 if t ⩾ ξ , t 2 2 ξ if 0 ⩽ t ⩽ ξ , 0 if t ⩽ 0 .

(18) ψ ξ 2 ( t ) = t 2 + ξ 2 if t ⩾ 0 , ξ if t ⩽ 0 .

Let w and w ξ be solutions of problems (11) and (15), respectively. To demonstrate the a priori error, we substitute v = w ξ in the inequality of problem (11) and v = w in the inequality of problem (15). Adding the resulting inequalities yields

(19) a ( w − w ξ , w − w ξ ) ⩽ Ψ ( w ξ + g ) − Ψ ξ ( w ξ + g ) + Ψ ξ ( w + g ) − Ψ ( w + g ) .

We note that the functions ψ ξ j , j = 1 , 2 , verify

∣ ψ ξ j ( t ) − t + ∣ ⩽ ξ , ∀ t ∈ R .

Let ‖ ⋅ ‖ [ H 0 1 ( Ω ) ] 2 denote the norm in [ H 0 1 ( Ω ) ] 2 associated with the inner product a ( ⋅ , ⋅ ) . Hence, from (19), we derive the following a priori error estimate:

(20) ‖ w − w ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ξ ‖ f ˜ ‖ L 1 ( Ω ) .

Consequently, w ξ converges to w in [ H 0 1 ( Ω ) ] 2 .

Theoretically, the a priori error estimate given by (20) guarantees that the solution of the regularized problem converges to the solution of the initial problem. However, in practice, this estimate cannot be used directly as a stopping criterion for the discrete case. Specifically, a stopping criterion of 1 0 − 6 would correspond to choosing ξ on the order of 1 0 − 12 , which is difficult to manage computationally. Therefore, the practical implementation of the regularized problem (15) necessitates the use of a posteriori error estimates.

4 Dual problem associated with the minimization problem (14)

Based on Section 3, the two membranes problem (1) is equivalent to the following minimization problem (refer (14)):

(21) Find u ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) : J ( u ) = min v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) J ( v ) ,

where J is the function defined on H g 1 1 ( Ω ) × H g 2 1 ( Ω ) as follows:

J ( v ) ≔ 1 2 a ( v , v ) + Ψ ( v ) + ( ( f , v ) ) .

In this section, we will identify the dual problem related to the primal problem (21). To do so, we require the following abstract frame.

We denote:

  • X a topological vector space, X * its dual, and ⟨ ⋅ , ⋅ ⟩ X the duality pairing between X and X * .

  • h : X → R ¯ a function and h * : X * → R ¯ its conjugate function defined by

    h * ( v * ) ≔ sup v ∈ X { ⟨ v * , v ⟩ X − h ( v ) } , v * ∈ X * .

  • Q a separable topological vector space, Q * its dual, and ⟨ ⋅ , ⋅ ⟩ Q the duality pairing between Q and Q * .

  • ℛ : X × Q → R ¯ a function and ℛ * : X * × Q * → R ¯ its conjugate function defined by

    ℛ * ( v * , q * ) ≔ sup ( v , q ) ∈ X × Q { ⟨ v * , v ⟩ X + ⟨ q * , q ⟩ Q − ℛ ( v , q ) } , ( v * , q * ) ∈ X * × Q * .

We assume that there exists a continuous linear operator Λ : X → Q , Λ ∈ ℒ ( X , Q ) , with transpose Λ * ∈ ℒ ( Q * , X * ) . We consider the following primal problem ([11], pp. 58–62):

(22) Find u ∈ X such that ℛ ( u , Λ u ) = inf v ∈ X ℛ ( v , Λ v ) .

Theorem 4.1

([9], Theorem 3.1) Assume that X is a reflexive Banach space and Q is a normed vector space. Let ℛ : X × Q → R ¯ be a proper, lower semi-continuous, and strictly convex function satisfying

  1. ∃ ν ∈ X , such that ℛ ( ν , Λ ν ) < + ∞ and the function q ↦ ℛ ( ν , q ) is continuous at Λ ν .

  2. The following coercivity condition: ℛ ( v , Λ v ) → + ∞ , as ‖ v ‖ X → + ∞ , v ∈ X .

Then, primal problem (22) admits a unique solution u, and we have

(23) ℛ ( u , Λ u ) = sup q * ∈ Q * − ℛ * ( Λ * q * , − q * ) .

Proposition 4.2

([11], Chap. 4, Proposition 1.2) Let l : Ω × R m → R be a Carathéodory function, and let L be the function defined by

L ( ν ) = ∫ Ω l ( x , ν ( x ) ) d x , ν ∈ [ L 2 ( Ω ) ] m .

The conjugate function of L is given by

(24) L * ( ν * ) = ∫ Ω l * ( x , ν * ( x ) ) d x , ∀ ν * ∈ [ L 2 ( Ω ) ] m ,

where

l * ( x , z ) = sup s ∈ R m { z ⋅ s − l ( x , s ) } ,

and the expression z ⋅ s denotes the inner product of z and s in R m .

To apply Theorem 4.1 to the minimization problem (14), we proceed by taking

• X = [ H 1 ( Ω ) ] 2 • Q = Q * = [ [ L 2 ( Ω ) ] n ] 2 × [ L 2 ( Ω ) ] 2 • Λ : X → Q , Λ v = [ ( ∇ v 1 , ∇ v 2 ) , ( v 1 , v 2 ) ] , v = ( v 1 , v 2 ) ∈ X • ℛ : X × Q → R , ℛ ( v , q ) = G ( v ) + L ( q ) , where • G ( v ) = ( ( f , v ) ) if v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) , + ∞ otherwise . • L ( q ) = ∫ Ω 1 2 ∣ q 1 , 1 ∣ 2 + 1 2 ∣ q 1 , 2 ∣ 2 + f ˜ ( q 2 , 2 − q 2 , 1 ) + d x ,

where q = ( q 1 , q 2 ) , with q 1 = ( q 1 , 1 , q 1 , 2 ) ∈ [ [ L 2 ( Ω ) ] n ] 2 and q 2 = ( q 2 , 1 , q 2 , 2 ) ∈ [ L 2 ( Ω ) ] 2 . The same notation is used to represent q * ∈ Q * . Hence, the formulation of problem (14) can be restated as

Find u ∈ X , ℛ ( u , Λ u ) = inf v ∈ X ℛ ( v , Λ v ) .

The conditions described in Theorem 4.1 are met, so the solution u to the two membranes problem (1) satisfies

(25) ℛ ( u , Λ u ) = inf v ∈ X ℛ ( v , Λ v ) = sup q * ∈ Q * − ℛ * ( Λ * q * , − q * ) .

Let v * ∈ X * and q * ∈ Q * , we have

ℛ * ( v * , q * ) = sup ( v , q ) ∈ X × Q { ⟨ v * , v ⟩ X + ⟨ q * , q ⟩ Q − G ( v ) − L ( q ) } = sup v ∈ X { ⟨ v * , v ⟩ X − G ( v ) } + sup q ∈ Q { ⟨ q * , q ⟩ Q − L ( q ) } = G * ( v * ) + L * ( q * ) .

Consequently, (25) implies that

ℛ ( u , Λ u ) = sup q * ∈ Q * − G * ( Λ * q * ) − L * ( − q * ) .

To obtain ℛ ( u , Λ u ) , we first need to find the conjugate functions of both G and L . We have

Proposition 4.3

Let q * ∈ Q * , we have

G * ( Λ * q * ) = ∫ Ω ∑ i = 1 2 ( q 1 , i * . ∇ g i + ( q 2 , i * − f i ) g i ) d x i f − div ( q 1 , i * ) + ( q 2 , i * − f i ) = 0 f o r i = 1 , 2 , + ∞ o t h e r w i s e .

Proof

Let q * ∈ Q * , we have

G * ( Λ * q * ) = sup v ∈ X { ⟨ Λ * q * , v ⟩ X * − G ( v ) } = sup v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) { ⟨ q * , Λ v ⟩ Q * − ( ( f , v ) ) } = sup v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) ∫ Ω ∑ i = 1 2 ( q 1 , i * . ∇ v i + ( q 2 , i * − f i ) v i ) d x = ∑ i = 1 2 sup v i ∈ H g i 1 ∫ Ω ( q 1 , i * . ∇ v i + ( q 2 , i * − f i ) v i ) d x = ∫ Ω ∑ i = 1 2 ( q 1 , i * . ∇ g i + ( q 2 , i * − f i ) g i ) d x + ∑ i = 1 2 sup ν ∈ H 0 1 ( Ω ) ∫ Ω ( q 1 , i * . ∇ ν + ( q 2 , i * − f i ) ν ) d x .

Since, in the distributional sense, for i = 1 , 2 we have

sup ν ∈ H 0 1 ( Ω ) ∫ Ω ( q 1 , i * . ∇ ν + ( q 2 , i * − f i ) ν ) d x = 0 if − div ( q 1 , i * ) + ( q 2 , i * − f i ) = 0 , + ∞ otherwise .

So,

□ G * ( Λ * q * ) = ∫ Ω ∑ i = 1 2 ( q 1 , i * . ∇ g i + ( q 2 , i * − f i ) g i ) d x if − div ( q 1 , i * ) + ( q 2 , i * − f i ) = 0 for i = 1 , 2 , + ∞ otherwise .

Proposition 4.4

Let q * ∈ Q * , we have

L * ( − q * ) = ∫ Ω 1 2 ( ∣ q 1 , 1 * ∣ 2 + ∣ q 1 , 2 * ∣ 2 ) d x i f q 2 , 1 * = − q 2 , 2 * a n d 0 ⩽ q 2 , 1 * ⩽ f ˜ a . e . i n Ω , + ∞ o t h e r w i s e .

Proof

To compute the conjugate function L , we have to compute the following conjugate functions:

L 1 : [ L 2 ( Ω ) ] n → R , L 1 ( v ) = ∫ Ω 1 2 ∣ v ( x ) ∣ 2 d x , v ∈ [ L 2 ( Ω ) ] n

and

L 2 : [ L 2 ( Ω ) ] 2 → R , L 2 ( l ) = ∫ Ω f ˜ ( l 2 − l 1 ) + d x , l = ( l 1 , l 2 ) ∈ [ L 2 ( Ω ) ] 2 .

Let v * ∈ [ L 2 ( Ω ) ] n . According to Proposition 4.2, we have

L 1 * ( v * ) = ∫ Ω h 1 * ( v * ( x ) ) d x ,

where

h 1 * ( z ) = sup s ∈ R n z ⋅ s − 1 2 ∣ s ∣ 2 .

Now, the quadratic function: s ↦ z ⋅ s − 1 2 ∣ s ∣ 2 has a maximum on s = z . Therefore,

h 1 * ( z ) = 1 2 ∣ z ∣ 2 .

Hence,

(26) L 1 * ( v * ) = ∫ Ω 1 2 ∣ v * ( x ) ∣ 2 d x .

Let l * ∈ [ L 2 ( Ω ) ] 2 . According to Proposition 4.2, we have

L 2 * ( l * ) = ∫ Ω h 2 * ( x , l * ( x ) ) d x ,

where

h 2 : Ω × R 2 → R , h 2 ( x , s ) = f ˜ ( x ) ( s 2 − s 1 ) + , with s = ( s 1 , s 2 ) ∈ R 2 .

Let x ∈ Ω and z ∈ R 2 , we have

h 2 * ( x , z ) = sup s ∈ R 2 { z ⋅ s − h 2 ( x , s ) } = sup ( s 1 , s 2 ) ∈ R 2 { z 1 s 1 + z 2 s 2 − f ˜ ( x ) ( s 2 − s 1 ) + } ,

with z = ( z 1 , z 2 ) . If we put μ 1 = s 1 and μ 2 = s 2 − s 1 , we obtain

h 2 * ( x , z ) = sup ( μ 1 , μ 2 ) ∈ R 2 { z 1 μ 1 + z 2 ( μ 2 + μ 1 ) − f ˜ ( x ) μ 2 + } = sup μ 1 ∈ R { μ 1 ( z 1 + z 2 ) } + sup μ 2 ∈ R { z 2 μ 2 − f ˜ ( x ) μ 2 + } . = sup μ 1 ∈ R { μ 1 ( z 1 + z 2 ) } + max { sup μ 2 ∈ R + { μ 2 ( z 2 − f ˜ ( x ) ) } , sup μ 2 ∈ R − { μ 2 z 2 } } .

We have

sup μ 1 ∈ R { μ 1 ( z 1 + z 2 ) } = 0 if z 1 = − z 2 + ∞ otherwise , sup μ 2 ∈ R + { μ 2 ( z 2 − f ˜ ( x ) ) } = 0 if z 2 ⩽ f ˜ ( x ) + ∞ otherwise , sup μ 2 ∈ R − { μ 2 z 2 } = 0 if z 2 ⩾ 0 + ∞ otherwise .

Thus,

h 2 * ( x , z ) = 0 if z 1 = − z 2 and 0 ⩽ z 2 ⩽ f ˜ ( x ) + ∞ otherwise .

Consequently,

L 2 * ( l * ) = 0 if l 1 * = − l 2 * and 0 ⩽ l 2 * ⩽ f ˜ a.e. in Ω , + ∞ otherwise ,

then

(27) L 2 * ( − l * ) = 0 if l 1 * = − l 2 * and 0 ⩽ l 1 * ⩽ f ˜ a.e. in Ω , + ∞ otherwise .

Finally, relations (26) and (27) imply that for any q * ∈ Q * , we have

L * ( − q * ) = ∫ Ω 1 2 ( ∣ q 1 , 1 * ∣ 2 + ∣ q 1 , 2 * ∣ 2 ) d x if q 2 , 1 * = − q 2 , 2 * and 0 ⩽ q 2 , 1 * ⩽ f ˜ a.e. in Ω , + ∞ otherwise ,

which completes the proof.□

Let D be the subset of Q * defined by

q ∈ D ⇔ q ∈ Q * and − div ( q 1 , i ) + ( q 2 , i − f i ) = 0 for i = 1 , 2 q 2 , 1 = − q 2 , 2 and 0 ⩽ q 2 , 1 ⩽ f ˜ a.e. in Ω ,

and consider the operator A : Q * → R defined by

A ( q ) ≔ ∫ Ω ∑ i = 1 2 1 2 ∣ q 1 , i ∣ 2 + q 1 , i . ∇ g i + ( q 2 , i − f i ) g i d x , q ∈ Q * .

Let u be the solution to problem (1). From Propositions 4.3 and 4.4, it follows that

(28) ℛ ( u , Λ u ) = sup q ∈ D { − A ( q ) } ,

which corresponds to the dual problem associated with the minimization problem (14).

5 A posteriori error estimation

This section provides an a posteriori error estimation between the solution u of the two membranes problem (13) and the solution u ξ of the regularized problem (16).

Proposition 5.1

Let u and u ξ be solutions of problems (13) and (16), respectively, we have

(29) ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ℛ ( u ξ , Λ u ξ ) + 2 A ( q ) , ∀ q ∈ D .

Proof

Let u and u ξ be solutions of problems (13) and (16), respectively. Taking v = u ξ in problem (13) yields

a ( u , u ξ − u ) + Ψ ( u ξ ) − Ψ ( u ) + ( ( f , u ξ − u ) ) ⩾ 0 .

Since

ℛ ( u ξ , Λ u ξ ) − ℛ ( u , Λ u ) = 1 2 a ( u ξ , u ξ ) − 1 2 a ( u , u ) + ( ( f , u ξ − u ) ) + Ψ ( u ξ ) − Ψ ( u ) ,

we have

ℛ ( u ξ , Λ u ξ ) − ℛ ( u , Λ u ) ⩾ 1 2 a ( u ξ , u ξ ) − 1 2 a ( u , u ) − a ( u , u ξ − u ) .

Hence,

1 2 ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ ℛ ( u ξ , Λ u ξ ) − ℛ ( u , Λ u ) .

Thus, from equality (28), we deduce that

□ ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ℛ ( u ξ , Λ u ξ ) + 2 A ( q ) , ∀ q ∈ D .

Proposition 5.2

Let u and u ξ be solutions of problems (13) and (16), respectively, and ψ ξ = ψ ξ i for i = 1 , 2 (choices (17) and (18)). The a posteriori error estimate is expressed as follows:

(30) ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ Ω f ˜ ( ( u ξ , 2 − u ξ , 1 ) + − ( u ξ , 2 − u ξ , 1 ) ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) d x .

Proof

Let u ξ be the solution of the regularized problem:

Find u ξ ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) such that a ( u ξ , v − u ξ ) + Ψ ξ ( v ) − Ψ ξ ( u ξ ) + ( ( f , v − u ξ ) ) ⩾ 0 ∀ v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) .

Then, u ξ is a solution of the following minimization problem:

Find u ξ ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) : J ξ ( u ξ ) = min v ∈ H g 1 1 ( Ω ) × H g 2 1 ( Ω ) J ξ ( v ) ,

where J ξ is the function defined on H g 1 1 ( Ω ) × H g 2 1 ( Ω ) by

J ξ ( v ) ≔ 1 2 a ( v , v ) + ( ( f , v ) ) + ∫ Ω f ˜ ψ ξ ( v 2 − v 1 ) d x .

J ξ is differentiable. Hence, the solution u ξ satisfies the following variational equation:

(31) a ( u ξ , v ) + ( ( f , v ) ) + ∫ Ω f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ( v 2 − v 1 ) d x = 0 , ∀ v = ( v 1 , v 2 ) ∈ [ H 0 1 ( Ω ) ] 2 .

By taking v 1 = 0 and v 2 = 0 in (31), we obtain the following system of equations:

(32) ∫ Ω ∇ u ξ , 2 ⋅ ∇ ν d x + ∫ Ω ( f 2 + f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) ν d x = 0 , ∀ ν ∈ H 0 1 ( Ω ) ∫ Ω ∇ u ξ , 1 ⋅ ∇ ν d x + ∫ Ω ( f 1 − f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) ν d x = 0 , ∀ ν ∈ H 0 1 ( Ω ) .

Thus, in the distributional sense, we obtain

(33) − div ( ∇ u ξ , 2 ) + ( f 2 + f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) = 0 , − div ( ∇ u ξ , 1 ) + ( f 1 − f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) = 0 .

We remark that for ψ ξ = ψ ξ i with i = 1 , 2 , we have

(34) 0 ⩽ ψ ξ ′ ( t ) ⩽ 1 , ∀ t ∈ R .

Since f ˜ ⩾ 0 , we have

(35) 0 ⩽ f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ⩽ f ˜ .

We set

q 1 , 1 = − ∇ u ξ , 1 , q 1 , 2 = − ∇ u ξ , 2 , q 2 , 1 = f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) , and q 2 , 2 = − f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) .

From relations (33), (34), and (35), we deduce that

− div ( q 1 , i ) + ( q 2 , i − f i ) = 0 for i = 1 , 2 , q 2 , 1 = − q 2 , 2 and 0 ⩽ q 2 , 1 ⩽ f ˜ a.e. in Ω .

As a result, q ∈ D . Thus, by Proposition 5.1, we obtain

‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 ⩽ 2 ℛ ( u ξ , Λ u ξ ) + 2 A ( q ) ,

where

A ( q ) = ∫ Ω 1 2 ∣ ∇ u ξ , 1 ∣ 2 + 1 2 ∣ ∇ u ξ , 2 ∣ 2 − ∇ u ξ , 1 . ∇ g 1 + ( f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) − f 1 ) g 1 − ∇ u ξ , 2 . ∇ g 2 − ( f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) + f 2 ) g 2 d x

and

ℛ ( u ξ , Λ u ξ ) = ∫ Ω 1 2 ∣ ∇ u ξ , 1 ∣ 2 + 1 2 ∣ ∇ u ξ , 2 ∣ 2 + f 1 u ξ , 1 + f 2 u ξ , 2 + f ˜ ( u ξ , 2 − u ξ , 1 ) + d x .

For i = 1 , 2 , we have u ξ , i − g i ∈ H 0 1 ( Ω ) . Therefore, from (32), we deduce that

∫ Ω ( f 2 + f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) g 2 d x = ∫ Ω ( f 2 + f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) u ξ , 2 + ∇ u ξ , 2 ⋅ ∇ ( u ξ , 2 − g 2 ) d x ∫ Ω ( f 1 − f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) g 1 d x = ∫ Ω ( f 1 − f ˜ ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) u ξ , 1 + ∇ u ξ , 1 ⋅ ∇ ( u ξ , 1 − g 1 ) d x .

By using these equalities in the expression of A ( q ) and simplifying, we obtain

ℛ ( u ξ , Λ u ξ ) + A ( q ) = ∫ Ω f ˜ ( ( u ξ , 2 − u ξ , 1 ) + − ( u ξ , 2 − u ξ , 1 ) ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) d x .

Since, by Proposition 5.1, we have

‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ℛ ( u ξ , Λ u ξ ) + 2 A ( q ) ,

hence

‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ Ω f ˜ ( ( u ξ , 2 − u ξ , 1 ) + − ( u ξ , 2 − u ξ , 1 ) ψ ξ ′ ( u ξ , 2 − u ξ , 1 ) ) d x .

Thus, the proof is complete.□

  • For ψ ξ = ψ ξ 1 (17), we have the a posteriori error estimate

    ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ [ 0 < u ξ , 2 − u ξ , 1 < ξ ] f ˜ ( u ξ , 2 − u ξ , 1 ) 1 − u ξ , 2 − u ξ , 1 ξ d x .

  • For ψ ξ = ψ ξ 2 (18), we have

    ‖ u − u ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ [ u ξ , 2 > u ξ , 1 ] f ˜ ( u ξ , 2 − u ξ , 1 ) 1 − u ξ , 2 − u ξ , 1 ( u ξ , 2 − u ξ , 1 ) 2 + ξ 2 d x .

6 Discrete problem

Let Ω be a bounded open domain of R n   ( n = 2 ) and ( T h ) h be a regular triangulation of Ω .

Consider ( V h ) h (resp. ( V 0 , h ) h ) the sequence of finite-dimension subspaces of H 1 ( Ω ) (resp. H 0 1 ( Ω ) ) given by

V h ≔ { v h ∈ C ( Ω ¯ ) ∩ H 1 ( Ω ) : ∀ T ∈ T h , v h ∣ T ∈ P 1 ( T ) } , V 0 , h ≔ { v h ∈ C ( Ω ¯ ) ∩ H 0 1 ( Ω ) : ∀ T ∈ T h , v h ∣ T ∈ P 1 ( T ) } ,

where P 1 ( T ) denotes the space of polynomial functions on T of degree less than or equal to one. For simplicity, we suppose that the boundary condition function g can be exactly represented by a function from V h × V h .

The discrete problem associated with the two membranes problem (13) is

(36) Find u h ∈ V h × V h , u h = g on ∂ Ω , such that a ( u h , v h − u h ) + Ψ ( v h ) − Ψ ( u h ) + ( ( f , v h − u h ) ) ⩾ 0 ∀ v h ∈ V h × V h , v h = g on ∂ Ω .

By setting u 0 , h ≔ u h − g , we see that u 0 , h satisfies

(37) Find u 0 , h ∈ V 0 , h × V 0 , h , such that a ( u 0 , h + g , v h − u 0 , h ) + Ψ ( v h + g ) − Ψ ( u 0 , h + g ) + ( ( f , v h − u 0 , h ) ) ⩾ 0 ∀ v h ∈ V 0 , h × V 0 , h .

We can adopt a similar approach to that outlined in [8] to establish the convergence of the finite element approximation and derive an a priori estimate.

The discrete regularized problem associated with problem (36) is

(38) Find u h , ξ ∈ V h × V h , u h , ξ = g on ∂ Ω , such that a ( u h , ξ , v h − u h , ξ ) + Ψ ξ ( v h ) − Ψ ξ ( u h , ξ ) + ( ( f , v h − u h , ξ ) ) ⩾ 0 ∀ v h ∈ V h × V h , v h = g on ∂ Ω .

Similar to the continuous problem, we derive the following a posteriori error estimates for the discrete problem:

Proposition 6.1

Let u h and u h , ξ be solutions of (36) and (38), respectively. Then, we have the following a posteriori error estimate:

(39) ‖ u h − u h , ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ Ω f ˜ ( ( u h , ξ , 2 − u h , ξ , 1 ) + − ( u h , ξ , 2 − u h , ξ , 1 ) ψ ξ ′ ( u h , ξ , 2 − u h , ξ , 1 ) ) d x .

  • For ψ ξ = ψ ξ 1 (17), the a posteriori error estimate is provided by

    (40) ‖ u h − u h , ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ [ 0 < u h , ξ , 2 − u h , ξ , 1 < ξ ] f ˜ ( u h , ξ , 2 − u h , ξ , 1 ) 1 − u h , ξ , 2 − u h , ξ , 1 ξ d x .

  • For ψ ξ = ψ ξ 2 (18), we have

    (41) ‖ u h − u h , ξ ‖ [ H 0 1 ( Ω ) ] 2 2 ⩽ 2 ∫ [ u h , ξ , 2 > u h , ξ , 1 ] f ˜ ( u h , ξ , 2 − u h , ξ , 1 ) 1 − u h , ξ , 2 − u h , ξ , 1 ( u h , ξ , 2 − u h , ξ , 1 ) 2 + ξ 2 d x .

7 Conclusion and remarks

In this study, we addressed a linear problem involving two membranes, formulated as a variational inequality of the first kind, and reformulated it into an unconstrained variational inequality problem. We then conducted a comprehensive analysis of the regularization approach for the two membranes problem, providing convergence results for the approximate solutions and a priori error estimates. In practice, however, this estimate cannot serve as a stopping criterion in the discrete case. Therefore, by applying the duality approach, we established a posteriori error estimates for both the continuous and discrete problems.

We note that for ψ ξ = ψ ξ 1 (refer (17)), we have

∫ [ 0 < u h , ξ , 2 − u h , ξ , 1 < ξ ] f ˜ ( u h , ξ , 2 − u h , ξ , 1 ) 1 − u h , ξ , 2 − u h , ξ , 1 ξ d x ⩽ ξ ‖ f ˜ ‖ L 1 ( Ω ) .

Hence, for the choice ψ ξ = ψ ξ 1 , the a posteriori error estimates (40) provide a more refined evaluation compared to the a priori error estimates (20). However, theoretically, we cannot confirm this remark for the choice ψ ξ = ψ ξ 2 .

Furthermore, let ( u h , 1 , u h , 2 ) be the solution of the discrete problem associated with the two membranes problem, we have

u h , 1 > u h , 2 a.e. in Ω \ ℐ and u h , 1 = u h , 2 a.e. in ℐ ,

where ℐ is the coincidence set of the two membranes. Let ( u h , ξ , 1 , u h , ξ , 2 ) be the approximate solution of the discrete regularized problem, we have

  • For the case where ψ ξ = ψ ξ 1 , the a posteriori error bounds will be calculated over the set [ 0 < u h , ξ , 2 − u h , ξ , 1 < ξ ] , this set represents the overlapping domain of the approximate solution with an amplitude less than ξ . Due to the limited extent of this domain, the a posteriori error bounds are expected to be small.

  • For the case where ψ ξ = ψ ξ 2 , the a posteriori error bounds will be calculated over the set [ u h , ξ , 2 > u h , ξ , 1 ] , which is the overlapping domain of the approximate solution, this overlap is not managed by any amplitude.

Based on the information provided, it appears that choosing ψ ξ = ψ ξ 1 yields better a posteriori error estimates for the two membranes problem.

Acknowledgements

We thank the two anonymous referees for carefully reading the first version of the study and for their helpful comments.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have collectively taken responsibility for the entirety of this manuscript’s content. They have approved the final version of the manuscript and consented to its submission to the journal.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2024-02-29
Revised: 2024-10-20
Accepted: 2024-12-04
Published Online: 2024-12-31

© 2024 the author(s), published by De Gruyter

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  88. One special kind of Kloosterman sum and its fourth-power mean
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  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0116
Keywords for this article
variational inequalities; two membranes problem; a posteriori error estimates; regularization method; duality theory
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  97. Generalized quadratic Gauss sums and their 2mth power mean
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  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
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  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
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  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
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  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
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  123. Solutions of linear control systems on Lie groups
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