On optimal control in a nonlinear interface problem described by hemivariational inequalities

2.1 Some preliminaries from Clarke’s generalized differential calculus

From Clarke’s generalized differential calculus [12], we need the concept of the generalized directional derivative of a locally Lipschitz function ϕ : X → R on a real Banach space X at x ∈ X in the direction z ∈ X defined by:

Note that the function z ∈ X ↦ ϕ 0 ( x ; z ) is finite, sublinear, hence convex, and continuous; furthermore, the function ( x , z ) ↦ ϕ 0 ( x ; z ) is upper semicontinuous. The generalized gradient of the function ϕ at x , denoted by (simply) ∂ ϕ ( x ) , is the unique nonempty weak * compact convex subset of the dual space X * , whose support function is ϕ 0 ( x ; . ) . Thus,

When X is finite dimensional, according to Rademacher’s theorem, ϕ is differentiable almost everywhere, and the generalized gradient of ϕ at a point x ∈ R n can be characterized by:

where “co” denotes the convex hull.

2.2 Interface problem

Let Ω ⊂ R d ( d ≥ 2 ) be a bounded domain with Lipschitz boundary Γ = cl Γ s ∪ cl Γ t with non-empty open disjoint boundary parts Γ s and Γ t . Let n denote the unit normal on Γ defined almost everywhere pointing from Ω into Ω c ≔ R d \ Ω ¯ . Let the data f ∈ L 2 ( Ω ) , u 0 ∈ H 1 ∕ 2 ( Γ ) , and q ∈ L 2 ( Γ ) be given.

In the interior part Ω , consider the nonlinear PDE

where p : [ 0 , ∞ ) → [ 0 , ∞ ) is a continuous function with t ⋅ p ( t ) being monotonously increasing with t .

In the exterior part Ω c , consider the Laplace equation

with the radiation condition at infinity for ∣ x ∣ → ∞ ,

where a is a real constant for any u , but may vary with u .

With u 1 ≔ u ∣ Ω and u 2 ≔ u ∣ Ω c , the tractions on the coupling boundary Γ are given by the traces of p ( ∣ ∇ u 1 ∣ ) ∂ u 1 ∂ n and − ∂ u 2 ∂ n , respectively. Prescribe classical transmission conditions on Γ t :

and on Γ s analogously for the tractions:

and the generally non-monotone, set-valued transmission condition:

Here, the function j : Γ s × R → R is such that j ( ⋅ , ξ ) : Γ s → R is measurable on Γ s for all ξ ∈ R and j ( s , ⋅ ) : R → R is locally Lipschitz for almost all (a.a.) s ∈ Γ s with ∂ j ( s , ξ ) ≔ ∂ j ( s , ⋅ ) ( ξ ) , the generalized gradient of j ( s , ⋅ ) at ξ .

Furthermore, require the following growth condition on the so-called superpotential j : there exist positive constants c j , 1 and c j , 2 such that for a.a. s ∈ Γ s , all ξ ∈ R and for all η ∈ ∂ j ( s , ξ ) , the following inequalities hold

Note that it follows from (2.7) ( i ) and (2.7) ( i i ) , respectively, that for a.a. s ∈ Γ s ,

and

Altogether, the interface problem consists in finding u 1 ∈ H 1 ( Ω ) and u 2 ∈ H loc 1 ( Ω c ) that satisfy (2.1)–(2.6) in a weak form.

To exhibit the relation of the above scalar interface problem to an elastic transmission problem with frictional contact, we insert the following remark.

To arrive at a first variational formulation of the interface problem in form of a HVI, introduce some function spaces. For the bounded Lipschitz domain Ω , use the standard Sobolev space H s ( Ω ) and the Sobolev spaces on the bounded Lipschitz boundary Γ (see [50, Sect 2.4.1]),

Furthermore, for the unbounded domain Ω c = R d \ Ω ¯ , introduce the Frechet space (see, e.g., [29, Section 4.1, (4.1.43)]):

By the trace theorem, u ∣ Γ ∈ H 1 ∕ 2 ( Γ ) for u ∈ H loc 1 ( Ω c ) . Next, define Φ : H 1 ( Ω ) × H loc 1 ( Ω c ) → R ∪ { ∞ } by:

Here the data f ∈ L 2 ( Ω ) , q ∈ L 2 ( Γ ) enter the linear functional:

Furthermore, in (2.10), the function g is given by p (see (2.1)) through

assume that p is C 1 , 0 ≤ p ( t ) ≤ p 0 < ∞ , and t ↦ t ⋅ p ( t ) is strictly monotonic increasing. Then, 0 ≤ g ( t ) ≤ 1 2 p 0 ⋅ t 2 and the real-valued functional

is strictly convex. The Frechet derivative of G ,

is Lipschitz continuous and strongly monotone in H 1 ( Ω ) with respect to the semi-norm:

i.e., there exists a constant c G > 0 such that

Analogously to [8,36], we first define

and then, the affine, hence convex set of admissible functions:

According to [8, Remark 4], C is closed in H 1 ( Ω ) × H loc 1 ( Ω c ) . Furthermore, it holds

Then, it can be proved [26, Theorem 1] that the interface problem (2.1)–(2.6) is equivalent in the sense of distributions to the HVI problem ( P Φ ) : find ( u ˆ 1 , u ˆ 2 ) ∈ C such that for all ( u 1 , u 2 ) ∈ C , there holds for δ u 1 ≔ u 1 − u ˆ 1 , δ u 2 ≔ u 2 − u ˆ 2 ,

However, since this HVI lives on the unbounded domain Ω × Ω c (as the original problem), this HVI cannot be treated in a reflexive Banach space setting and therefore provides only an intermediate step in the analysis. Therefore, employ boundary integral operator theory [27,29] to reformulate the interface problem (2.1)–(2.6) in the weak sense as a boundary-domain variational inequality on Γ × Ω . From now on, concentrate on the analysis to the case of dimension d = 3 , since as already the distinction in the radiation condition (2.3) indicates, in the case d = 2 , some peculiarities of boundary integral methods for exterior problems come up that need extra attention (see, e.g., [8], [27, Sec. 12.2]). As a result, arrive at an equivalent hemivariational formulation of the original interface problem (2.1)–(2.6) that lives on Ω × Γ and consists of a weak formulation of the nonlinear differential operator in the bounded domain Ω , the Poincaré–Steklov operator on the bounded boundary Γ , and a nonsmooth functional on the boundary part Γ s .

To this end, recall the Poincaré-Steklov operator for the exterior problem: S : H 1 ∕ 2 ( Γ ) → H − 1 ∕ 2 ( Γ ) is a selfadjoint operator with the defining property:

for solutions u 2 ∈ ℒ 0 of the Laplace equation on Ω c . The operator S enjoys the important property that it can be expressed as:

where I , V , K , K ′ , and W denote the identity, the single-layer boundary integral operator, the double-layer boundary integral operator, its formal adjoint, and the hypersingular integral operator, respectively (see [27, Section 12.2] for details).

Furthermore, S gives rise to the positive definite bilinear form ⟨ S ⋅ , ⋅ ⟩ , i.e., there exists a constant c S > 0 such that

where ⟨ ⋅ , ⋅ ⟩ = ⟨ ⋅ , ⋅ ⟩ H − 1 ∕ 2 ( Γ ) × H 1 ∕ 2 ( Γ ) extends the L 2 duality on Γ .

Let E ≔ H 1 ( Ω ) × H ˜ 1 ∕ 2 ( Γ s ) with H ˜ 1 ∕ 2 ( Γ s ) ≔ { w ∈ H 1 ∕ 2 ( Γ ) ∣ supp w ⊆ Γ ¯ s } . Next, define the linear functional λ ∈ E * by:

Using the representation formula of potential theory (see [17,36] for similar nonlinear interface problems), it can be proved [26, Theorem 2] that the intermediate HVI ( P Φ ) is equivalent to the following HVI problem ( P A ) : find ( u ˆ , v ˆ ) ∈ E such that for all ( u , v ) ∈ E ,

where A : E → E * is defined for all ( u , v ) , ( u ′ , v ′ ) ∈ E by:

2.3 An equilibrium approach to a class of HVIs – existence and uniqueness results

Next, describe the functional analytic setting for the interface problem and provide the existence and uniqueness results using an equilibrium approach. To this end, let X ≔ L 2 ( Γ s ) and introduce the real-valued locally Lipschitz functional:

Then, by Lebesgue’s theorem of majorized convergence,

where j 0 ( s , ⋅ ; ⋅ ) denotes the generalized directional derivative of j ( s , ⋅ ) .

As seen in the previous subsection, the weak formulation of problem (2.1)–(2.6) leads, in an abstract setting, to a HVI with a nonlinear operator A and the nonsmooth functional J , namely: find v ˆ ∈ C such that

Here, C ≠ ∅ is a closed convex subset of a real reflexive Banach space E , γ ≔ γ E → X is a linear continuous operator, the linear form λ belongs to the dual E * , and the nonlinear monotone operator A : E → E * is Lipschitz continuous and strongly monotone with some monotonicity constant c A > 0 , which results from the strong monotonicity of the nonlinear operator D G in H 1 ( Ω ) with respect to the semi-norm ∣ ⋅ ∣ H 1 ( Ω ) = ‖ ∇ ⋅ ‖ L 2 ( Ω ) and the positive definiteness of the Poincaré-Steklov operator S (see [8, Lemma 4.1]).

On the other hand, by (2.8) and the compactness of the operator γ , the real-valued upper semicontinuous bivariate function, shortly bifunction:

becomes pseudomonotone (see [42, Lemma 1], [25, Lemma 4.1]). The latter result also shows that (2.9) implies a linear growth of ψ ( ⋅ , 0 ) . This and the strong monotonicity of A imply coercivity. Therefore, by the theory of pseudomonotone VIs [20, Theorem 3], [58] (see [43] for the application to HVIs), we have solvability of (2.20).

Furthermore, suppose that the generalized directional derivative J 0 satisfies the one-sided Lipschitz condition: there exists c J > 0 such that

Then, the smallness condition

implies unique solvability of (2.20) (see, e.g., [41, Theorem 5.1] and [55, Theorem 83]).

It is noteworthy that under the smallness condition (2.22) together with (2.21), fixed point arguments [7] or the theory of set-valued pseudomonotone operators [55] are not needed, but simpler monotonicity arguments are sufficient to conclude unique solvability. Moreover, the compactness of the linear operator γ is not needed either. In fact, (2.20) can be framed as a monotone equilibrium problem in the sense of Blum-Oettli [6]:

Since strong monotonicity implies coercivity and uniqueness, the fundamental existence result [6, Theorem 1] applies to the HVI (2.20) to conclude the following.

Thus, under the smallness condition, unique solvability holds for ( P A ) .