Three-Field Modelling of Nonlinear Nonsmooth Boundary Value Problems and Stability of Differential Mixed Variational Inequalities

and Applied Analysis 3 By the boundary conditions, the previous second term vanishes. The second inequality in the Signorini boundary condition (2) 3 tells us that we have to require that σ belongs to the convex cone as follows: H + := H + (div, Ω, Γ S ) := {τ ∈ H (div, Ω) : γ^τ | ΓS ≥ 0} , (7) where “≥ 0” means that ⟨γ 0 󰜚, γ^τ⟩ ≥ 0 for any smooth function 󰜚 on Ω with 󰜚 = 0 on Γ D and 󰜚 ≥ 0 on Γ S . Thus we obtain for any τ ∈ H + (div, Ω, Γ S ), (p, τ − σ) L 2 (Ω,R) + (u, div (τ − σ))L2(Ω) ≥ ⟨g, γ^ (τ − )L2(Γ S ) . (8) Altogether we arrive at the following variational inequality ofmixed form: find [p,σ, u] ∈ L(Ω,R)×H + (div, Ω, Γ S )× L 2 (Ω), such that for all [q, τ, V] ∈ L(Ω,R)×H + (div, Ω, Γ S )×


Introduction
The classical Babuška-Brezzi theory for mixed variational problems has been extended by Gatica [1,2] to some classes of variational problems and nonlinear operator equations. This extension leads to three-field variational models that can be understood as dual-dual mixed variational models or as twofold saddle point formulations. Such augmented variational models are well adapted for multiphysics problems with different coupled unknown quantities and in particular for engineering problems, where speaking in terms of solid mechanics, strains and stresses are often of more interest then the displacements.
In a series of papers Gatica with coauthors applied his duality approach to the numerical treatment of various linear/nonlinear interior/exterior elliptic boundary value problems by the finite element method (FEM), the boundary element method (BEM), or by the coupling of these discretization methods. Here we refer to the paper of Gatica et al. [3] that presents a numerical analysis of nonlinear two-fold saddle point problems involving a nonlinear operator equation with a uniformly monotone operator.
This novel duality approach to nonlinear nonsmooth boundary value problems has to be distinguished from the standard duality approach which hinges on the Lagrange duality theory of convex analysis in calculus of variations (see [4] for a systematic study) and which is employed in the numerical FEM analysis of various unilateral boundary and obstacle problems as pioneered by Haslinger and Lovíšek [5,6]; see also the monograph [7].
In this paper, we address a simplified scalar model of steady-state unilateral contact problems with Tresca friction and nonlinear transient heat conduction problems with unilateral boundary conditions. We extend the duality approach of Gatica to such problems. This approach leads to variational inequalities of mixed form for three coupled fields as unknowns and to related differential mixed variational inequalities (DMVI) in the time-dependent case.
Differential variational inequalities have recently been introduced and studied in depth by Pang and Stewart [8] in finite dimensions as a new modeling paradigm of variational analysis. In their seminal paper the authors have already shown that this new class of differential inclusions contains ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems, dynamic complementarity systems, and evolutionary variational systems. More recently, some results of [8] have been extended to DMVI by Li et al. [9] in finite dimensions.
Furthermore in this paper, we are concerned with stability of the solution set to DMVI. In this connection, let us refer to [10], where a Lyapunov approach is developed for strong solutions of evolution variational inequalities and to [11], where first several sensitivity results are established for initial value problems of ordinary differential equations with nonsmooth right hand sides and then applied to treat differential variational inequalities. Related stability results for more general evolution inclusions by Papageorgiou [12,13] and by Hu and Papageorgiou in the memoir [14] are not applicable here since these results are limited to finite dimensions, respectively, and need more stringent compactness assumptions.
Here we present a novel upper set convergence result for DMVI with respect to perturbations in the data. In particular, we admit perturbations of the nonlinear maps, of the nonsmooth convex functionals, and the convex constraint set that describe the DMVI. We employ epiconvergence for the convergence of the functionals and Mosco convergence for set convergence. Since our analysis of the underlying mixed nonlinear variational inequality relies on the monotonicity method of Browder and Minty (see e.g., [15,16]), we need to impose comparably weak convergence assumptions on the perturbations in the nonlinear maps. Thus we extend the stability results of [17] (without considering slow solutions here) and of [18] to this new more general class of mixed differential variational inequalities.
The outline of this paper is as follows. In Sections 2 and 3, we show how the duality approach of Gatica can be extended to a nonlinear transient heat problem with unilateral boundary conditions and to a scalar nonsmooth boundary value problem modelling the steady-state unilateral contact of an elastic body with Tresca friction. Thus we obtain variational inequalities of mixed form involving three unknown fields and related differential mixed variational inequalities (DMVI) in the time-dependent case. Then in Section 4, we turn to the stability analysis of a general class of DMVI. After a discussion of some preliminaries including Mosco set convergence and epiconvergence (Section 4.2), we establish our novel stability result in Section 4.3 based on the Browder-Minty monotonicity method. Section 5 gives an outlook to some open directions of research.

A Nonlinear Nonsmooth Boundary Value Problem from Heat Conduction
In this section we consider a nonlinear boundary value problem with Signorini boundary conditions that arises from nonlinear heat conduction [19] with semipermeable walls [20]. We first show how the steady-state problem can be variationally formulated as a variational inequality in mixed form. Then we turn to the transient problem and derive the associated differential mixed variational inequality.

A Simplified Scalar Nonsmooth Boundary Value Problem from Frictional Contact
In this section we treat a non-smooth boundary value problem that can be considered as a simplified scalar model of a nonsmooth contact mechanics problem involving Tresca friction and a unilateral constraint of an elastic body with a rigid foundation. Instead of the vector Navier-Lamé pde system or a nonlinear extension of it to model nonlinear elastic material, we are concerned with a nonlinear Helmholtzlike pde. This pde will be complemented by nonclassical boundary conditions involving the non-smooth modulus function. We show how a three-field modelling transforms this nonsmooth boundary value problem to a variational inequality of mixed type. Similar to the preceding section, let Ω ⊂ R 2 be a bounded plane domain with the Lipschitz boundary. Here instead of the Navier-Lamé-system and instead of (2) we treat the elliptic Helmholz-like partial differential equation Again we use the gradient p := ∇ in Ω and the flux := a(⋅, p) in Ω as additional unknowns. In this way, the elliptic pde (14) writes as the following three equations: that should hold in the distributional sense in Ω. By this reformation we can again relax the regularity of the unknown . We require that ∈ 2 (Ω), p ∈ 2 (Ω, R 2 ), and ∈ (div, Ω). Thus testing (15)

Abstract and Applied Analysis
Now with the given function ∈ ∞ (Γ) with ( ) ≥ 0 > 0 a.e. on Γ, we impose the nonclassical boundary conditions Note that (17) is equivalent to These implications reflect Tresca's law of friction (given friction model) and the more general Coulomb's law of friction [20]. Namely, with ⋅^denoting (the tangential component of) the traction in the general elasticity problem, the first implication means that when the body is not in contact with the obstacle, there is no tangential stress due to friction. If on the other hand, the body is in contact with the rigid obstacle, then-this is the meaning of the second implication-there arises a tangential stress proportional and opposite to the tangential displacement .
To obtain a variational formulation of the boundary condition (17), we consider the boundary integral and claim that this boundary integral is nonnegative for any ∈ (div, Ω). Indeed, if | | < holds on some part of Γ, then^= ⋅^= 0, and the integrand reduces to If otherwise | | = holds, then insert^= − 0 , and the integrand becomes Thus we are led to the convex (actually sublinear) nonsmooth functional Remark 1. The obtained convex functional and the constraint | | ≤ are related by Fenchel duality of convex analysis, see for example [4]. Namely, introduce the convex, which is the indicator function of the previous constraint.
Moreover it is known that the Fenchel bidual function ( * ) * coincides with . Finally in virtue of Green's formula (5), the proven claim provides the variational inequality of the second kind as follows: for any ∈ (div, Ω), and altogether we arrive at the variational inequality of mixed form: find

Differential Mixed Variational Inequalities and Their Stability
Motivated by the non-smooth boundary value problems and their variational formulation in the previous sections, we deal in this section with a general class of differential mixed variational inequalities. As we will see that, with some changes of notation, all the concrete variational inequalities of mixed form of the previous sections can be subsumed in this class, when introducing some appropriate product spaces. Since in our stability analysis, we permit perturbations in the non-smooth convex functionals and in the convex constraint set, we provide auxiliary results on epiconvergence and Mosco convergence. Using the monotonicity method of Browder and Minty, we can establish a general stability result under weak convergence assumptions.
To give a precise meaning to a DMVI we have to introduce appropriate function spaces and impose some hypotheses on the data. The fixed finite time interval [0, ] gives rise to the Hilbert space 2 (0, ; ) endowed with the scalar product As in [8,9] we consider weak solutions of the differential equation in a DMVI in the sense of Caratheodory. In particular, the -valued function has to be absolutely continuous with derivative( ) defined almost everywhere. Moreover to define the initial condition, the "trace" (0) is needed. Therefore (see [23, Theorem 1, page 473]) we are led to the function space We assume that the map satisfies the following growth condition: there exist 0 ∈ ∞ (0, ), and 0 ∈ 2 (0, ) such that for all ∈ (0, ), for all ( , ) ∈ × there holds ( , , ) ≤ 0 ( ) (‖ ‖ + ‖ ‖ ) + 0 ( ) .
Then it makes sense to introduce the closed convex subset and replace the previous pointwise formulation of the mixed variational inequality in a DMVI by its integrated counterpart, Clearly, it is sufficient to test the variational inequality with any dense subset of K, for example, the -valued continuous functions on [0, ], as in [8, Section 2.1].

Preliminaries; Mosco
Here the prefix and → mean sequentially weak convergence in contrast to strong convergence denoted by the prefix and by →. Further, lim sup, lim inf respectively are in the sense of Kuratowski upper, lower limits respectively of sequences of sets (see [25] for more information on Mosco convergence). Here we note that for the nonempty set the second inclusion provides ∈ , such that → for some ∈ . Clearly, → , if and only if := − → := − , this simple translation argument shows that there is no loss of generality to assume when needed that 0 ∈ , .
As a preliminary result we next show that Mosco convergence of convex closed sets inherits to Mosco convergence of the polars 0 and to Mosco convergence of the associated sets K = 2 (0, ; ), derived from similar to (34).
It is enough to verify the claim for a dense subset of K; this follows from a diagonal sequence argument; see also [26,Lemma 2.6] for a similar reasoning. Here we use the wellknown fact from Bochner-Lebesgue integration theory that the set S(0, ; ) of simple -valued functions on (0, ) is dense in 2 (0, ; ). This extends to density of S(0, ; )∩K, the -valued simple functions on (0, ), in K. This can be seen by taking averages or mean value approximations; see [26] for approximations on a multidimensional integration domain instead of the interval (0, ).
Let us note that part (b) of the preceding lemma is of intrinsic interest for time-dependent variational inequalities. An analogous implication (b) (1). was already shown in [26] in the more general context of probability spaces instead of the interval (0, ), however for the restricted class of translated convex closed cones.
Since the epigraph epi := {( , ) : ≥ ( )} of a convex, lower semicontinuous (lsc) proper function : → R ∪ +∞ is nonempty, closed, and convex in R × , Mosco convergence applies to such sets. This is known as epiconvergence (see e.g., [25]). Thus a sequence { } of convex lsc proper functions on the Hilbert space is called epiconvergent to a convex lsc proper function on , written epi → , if and only if for all with ( ) < +∞ ∃{ } ∈N , As a further preliminary result we next show that epiconvergence of convex lsc functions inherits to epiconvergence of the associated functionals Φ , derived from similar to (33).
The details of the proof are omitted. As a further tool in our stability analysis we recall from [17] the following technical result.

The Stability Result.
Before stating the result, some remarks are in order. In view of the existence theory of variational inequalities in infinite dimensional spaces (see, e.g., [15]) the best one can hope for is weak convergence of the perturbations in the general case of nonunique solutions of the underlying variational inequalities in (DMVI). Weak convergence can, namely, be readily derived from a posteriori estimates. However, continuity of a nonlinear map (here , ) with respect to weak convergence is a hard requirement. To circumvent these weak convergence difficulties we apply the monotonicity method of Browder and Minty. Then as we shall see below, a stability condition on the maps with respect to the basic Hilbert space norm suffices.
These weak convergence difficulties also affect . Therefore we have to impose a generally strong stability condition on the nonlinear maps . In the situation of linear operators this condition can be drastically simplified to a stability condition with respect to convergence in the operator norm, see [18,Theorem 4.1] in the case = = 0. On the other hand, stronger assumptions on , like uniform monotonicity, imply that the solution sets Σ( , , ( , ⋅)) are single valued. Uniform monotonicity with respect to moreover entails that the sequence strongly converges. Then the stability assumption for can be relaxed.
Since our stability assumptions pertain the given maps , , not the derived maps , , we have a delicate interplay between the pointwise almost everywhere formulation and the integrated formulation of the variational inequality in the perturbed DMVI and in the limit DMVI.
We need the following hypotheses on the convergence of ( , ) to ( , ).
When the DMVI has a separable structure, the hypotheses (H1) and (H2) can be expressed more explicitly. Similar to [9], we have with fixed 0 , 0 , and an appropriate linear operator with its adjoint * . Then 2 inherits monotonicity from the nonlinear operator A.
We refrain from deriving a stability result for stationary problems from Theorem 6. Instead we can refer to [27, Theorem 3] for a much stronger result.

Some Concluding Remarks
Let us first shortly remark on possible extensions and limitations of the previous stability result. The monotonicity method extends easily to set-valued operators. Also, monotonicity can be replaced by the more general, however more abstract notion of (order-) pseudomonotonicity.
Here motivated by the considered non-smooth boundary value problems we discussed differential variational inequalities in a Hilbert space framework. Let us point out that differential variational inequalities and their stability using Mosco convergence can be investigated in more general reflexive Banach spaces, but not beyond [28].
Finally let us give an outlook of some open directions of research. In this paper we confined the three-field modelling to a simplified scalar model of nonsmooth contact in continuum mechanics. An extension of the three-field duality approach to such non-smooth boundary value problems will involve the tensor fields of continuum mechanics. In particular, the basic Green's formula (5) has then to be replaced by the more general "integration by parts" formula [29, (3.46)].
In this paper, we did not touch the issue of existence of solutions to DMVI. As shown by Pang and Stewart [8] and Li et al. respectively, [9], existence of solutions to DMVI can be obtained from the theory of multivalued differential equations [30] and the theory of the more general differential inclusions [31]. However, compactness assumptions as used in [8,9] are too strong in infinite dimensions; moreover the feasible set defined by the unilateral boundary conditions is a convex cone, but not compact. For an existence result for linear differential variational inequalities based on maximal monotone operator theory, we can refer to [18].
When it comes to numerical approximation, note that Galerkin approximation with respect to space discretization leads in the case of higher than piecewise linear approximation; to a nonconforming approximation, that is, the approximating convex set ℎ is not a subset of the given convex . In this situation Mosco convergence (and more refined Glowinski convergence) is instrumental to arrive at convergence of Galerkin approximation; see [7,32]. Thus our stability result can be seen as an important step towards convergence of semidiscretization methods that provide finite dimensional differential variational inequalities. Clearly a full space time discretization needs an additional analysis (see, e.g., [33]) and is beyond the scope of the present paper.