Keywords

AMS (MOS) Subject Classification

1 Introduction

One of the most applied discretization methods for boundary value problems of different types, is FEM with its numerous variants. There are difficulties that may hinder the efficiency of this approach since the finite element grid may degenerate and certain regularity hypothesis have to be imposed in order to obtain good results, Ciarlet [7]. Starting with dimension three, the grid generation may involve a high degree of complexity (see for instance [14]).

In front tracking problems appearing in free boundary applications, such conditions may be difficult to preserve during the iterations of the algorithm. Alternative approaches are the recent virtual element method [3] or meshless methods [20]. In the second case, the used finite dimensional bases (for instance RBF) are quite complex, while the VEM has still to be developed to its full potential.

In this article, we review another approach (based on duality theory) that has already ensured powerful results [13, 21], in its variant known as the control variational method and can be applied to large classes of linear or nonlinear boundary value problems admitting a variational formulation. A recent development is due to Machalova and Netuka [15] for a beam model governed by a nonlinear fourth-order differential equation introduced by Gao [9].

Duality is an important principle in mathematics or physics. In the context of differential equations, dual problems may be obtained via the variational formulations (or otherwise) and are useful in optimal control, numerical approximation or theoretical advances [2, 4, 8, 11, 14]. We limit ourselves to the stationary case, characterized by the minimization of certain energies, but extensions to evolution equations are also possible [10, 23]. The dual optimization problem associated to it or to its approximation, is finite dimensional.

We discuss here three basic types of results associated to the minimization of energy, according to the constraints that are involved. In the next section, we describe problems governed by ordinary differential equations and the constraints are defined by some of the boundary conditions. Their number is finite and the dual problem is finite dimensional and provides the explicit solution to the model since no approximation is used here. Usual FEM techniques produce just approximate solutions and special techniques are needed to avoid critical situations, [5, 6].

In Sect. 3, we consider variational inequalities associated to the biharmonic equation. The constraints are given by the convex set characterizing the restriction (for instance, the obstacle problem). It is discretized by using a countable dense subset of points (no finite element) and the dual optimization problem is again finite dimensional and provides the desired approximation. This is an important advantage, removing any geometric regularity condition.

We also investigate the p-Laplacian problem and some fourth order problem by discretizing the constraints expressed via the boundary conditions. Here, we use recent papers [17, 18, 24], where numerical experiments are also reported. Partial results for the general linear elasticity system are discussed in [22].

In the last section, we briefly recall an application in shape optimization of the control variational method [21].

The variational problems that we discuss in this short review are in classical form or can be reformulated as an optimal control problem. The duality theory, based on the Fenchel theorem, has a wide range of applications to many classes of boundary value problems.

2 Ordinary Differential Equations

The classical Kirchhoff–Love model [7] in dimension two describes the deformation of a clamped cylindrical shell, under forcing acting in the normal plane, constantly along the shell. It consists in finding the displacement with components v 1 ∈ H 0 1(0, 1), v 2 ∈ H 0 2(0, 1) such that

$$\displaystyle{ \begin{array}{cc} \int _{0}^{1}\left [\frac{1} {\varepsilon } (v_{1}^{{\prime}}- cv_{ 2})(s)(u_{1}^{{\prime}}- cu_{ 2})(s) + (v_{2}^{{\prime}} + cv_{ 1})^{{\prime}}(s)(u_{ 2}^{{\prime}} + cu_{ 1})^{{\prime}}(s)\right ]ds \\ =\int _{ 0}^{1}(f_{ 1}u_{1} + f_{2}u_{2})(s)ds,\;\forall \;u_{1} \in H_{0}^{1}(0,1),\;\forall \;u_{ 2} \in H_{0}^{2}(0,1), \end{array} }$$
(2.1)

where ( f 1, f 2) are the normal, respectively the tangential component of the force acting on the arch, \(c: [0,1] \rightarrow \mathbb{R}\) denotes its curvature and \(\sqrt{\varepsilon }\) is its constant thickness. If \(\varphi = (\varphi _{1},\varphi _{2}): [0,1] \rightarrow \mathbb{R}^{2}\) is the parametrization of the arch, with respect to its arclength, then c(s) = φ 2 ″(s)φ 1 ′(s) −φ 1 ″(s)φ 2 ′(s) is its curvature and \(\theta (s) =\arctan \left (\frac{\varphi _{2}^{{\prime}}(s)} {\varphi _{1}^{{\prime}}(s)}\right )\) is the angle between the horizontal axis and the tangent vector φ ′ = (φ 1 ′, φ 2 ′). Consider the orthogonal matrix

$$\displaystyle{ W(s) = \left (\begin{array}{cc} \cos \theta (s) &\sin \theta (s)\\ -\sin \theta (s) &\cos \theta (s) \end{array} \right ) }$$
(2.2)

and define the functions

$$\displaystyle{ \left [\begin{array}{c} l\\ h \end{array} \right ](t) = -\int _{0}^{t}W(t)W^{-1}(s)\left [\begin{array}{c} f_{1}(s) \\ f_{2}(s) \end{array} \right ]ds, }$$
(2.3)
$$\displaystyle{ g_{1} =\varepsilon l,\;-g_{2}^{{\prime\prime}} = h,\;g_{ 2}(0) = g_{2}(1) = 0. }$$
(2.4)

The control variational method allows the reformulation of (2.1) as an optimal control problem. We use the notations introduced in (2.2)–(2.4):

$$\displaystyle{ \mathrm{Min}\left \{L(u,z) = \frac{1} {2\varepsilon }\int _{0}^{1}u^{2}(s)ds + \frac{1} {2}\int _{0}^{1}[z'(s)]^{2}ds\right \}, }$$
(2.5)

subject to the state equation

$$\displaystyle{ \left [\begin{array}{c} v_{1}(t) \\ v_{2}(t) \end{array} \right ] =\int _{ 0}^{t}W(t)W^{-1}(s)\left [\begin{array}{c} u(s) + g_{1}(s) \\ z(s) + g_{2}(s) \end{array} \right ]ds, }$$
(2.6)

with restriction

$$\displaystyle{ \int _{0}^{1}W^{-1}(s)\left [\begin{array}{c} u(s) + g_{1}(s) \\ z(s) + g_{2}(s) \end{array} \right ]ds = \left [\begin{array}{c} 0\\ 0 \end{array} \right ]. }$$
(2.7)

Relation (2.6) ensures the zero initial condition for v 1, v 2, while (2.7) expresses the zero final condition, i.e. the boundary conditions in (2.1). The formulation (2.5)–(2.7) (together with (2.2)–(2.4)) needs just θ ∈ L ∞(0, 1), i.e. φ ∈ W 1,∞(0, 1)2. Under more regularity conditions, we have (according to [13]):

Theorem 2.1

If φ ∈ W 3,∞(0, 1)2 and [v 1 ∗, v 2 ∗] is the optimal state of ( 2.5 )–( 2.7 ), then it satisfies ( 2.1 ).

The existence and uniqueness of [v 1 ∗, v 2 ∗] ∈ L ∞(0, 1)2 follows by the coercivity and strict convexity of (2.5). The above result shows that indeed (2.5)–(2.7) is a generalization of (2.1), under very weak regularity conditions on φ. Notice that the optimal control problem (2.5)–(2.7) has just two state (or equivalently, control) constraints given by (2.7). One can compute the dual problem [13], which is two dimensional (two Lagrange multipliers) and can be solved explicitly:

$$\displaystyle{ \begin{array}{cc} \mathop{\mathrm{Min}}\limits _{\lambda _{1},\lambda _{2}\in R}\{\frac{1} {2\varepsilon }\int _{0}^{1}[\lambda _{ 1}\varepsilon \cos \theta (s) +\lambda _{2}\varepsilon \sin \theta (s) +\varepsilon l(s)]^{2}ds+ \\ + \frac{1} {2}\int _{0}^{1}[(\lambda _{ 1}w_{1} +\lambda _{2}w_{2} + g_{2})'(s)]^{2}ds\}, \end{array} }$$
(2.8)

where we have denoted w 1, w 2 ∈ H 0 1(0, 1) ∩ H 2(0, 1):

$$\displaystyle{ w_{1}^{{\prime\prime}}(s) =\sin \theta (s),\;w_{ 2}^{{\prime\prime}}(s) = -\cos \theta (s). }$$

Theorem 2.2

If θ ∈ L ∞(0, 1), then the unique solution to ( 2.5 )–( 2.7 ) is given by ( 2.6 ) and

$$\displaystyle{ [u^{{\ast}},z^{{\ast}}] =\lambda _{ 1}^{{\ast}}[\varepsilon \cos \theta,w_{ 1}] +\lambda _{ 2}^{{\ast}}[\varepsilon \sin \theta,w_{ 2}], }$$

where [λ 1 ∗, λ 2 ∗] solve ( 2.8 ).

Note that the solution to (2.8) can be obtained by a 2 × 2 linear algebraic system that expresses the corresponding necessary optimality conditions. The results from Theorems 2.1, 2.2 show the efficiency of the duality arguments. In [5, 6] very special FEM schemes are used to approximate (2.1), due to its singular and stiff character (given by the small parameter ɛ > 0) and the so-called “locking problem” that affects standard FEM approaches, in such cases.

Applying a similar method, variational inequalities for arches, with unilateral conditions on the boundary, are discussed in [19]. In the recent paper [15], a nonlinear beam model is discussed by the control variational approach, but the obtained solution is no more explicit as in the case of the Kirchhoff-Love model for arches. The admissible control set has a complex structure and discretization procedures have to be used.

3 Partial Differential Equations

We start with an example related to the p-Laplacian, following [24]:

$$\displaystyle{ \mathop{\mathrm{Min}}\limits _{y\in W_{0}^{1,p}(\varOmega )}\left \{\frac{1} {p}\int _{\varOmega }\left [\;\vert \nabla y\vert ^{p} + \vert y\vert ^{p}\right ]dx -\int _{\varOmega }fydx\right \}, }$$
(3.1)

where \(\varOmega \subset \mathbb{R}^{d}\) is a bounded domain, p > d ≥ 2, f ∈ L q(Ω), p −1 + q −1 = 1.

The existence of a unique solution y ∈ W 0 1,p(Ω) is well known, due to the coercivity and strict convexity of (3.1). It may be interpreted as solving in a weak sense the p-Laplacian equation in Ω, with Dirichlet boundary conditions.

Let {x i } i ∈ N ⊂ ∂Ω be a dense subset. We approximate (3.1) by an optimization problem with a finite number of constraints (on the boundary ∂Ω):

$$\displaystyle{ \mathop{\mathrm{Min}}\limits _{\mathop{y\in W^{1,p}(\varOmega )}\limits _{ y(x_{i})=0,\;i=\overline{1,n}}}\left \{\frac{1} {p}\int _{\varOmega }\left [\;\vert \nabla y\vert ^{p} + \vert y\vert ^{p}\right ]dx -\int _{\varOmega }fydx\right \}. }$$
(3.2)

It makes sense due to the Sobolev embedding \(W^{1,p}(\varOmega ) \subset C(\overline{\varOmega })\) since p > d.

By the same argument as above, the minimization problem (3.2) has a unique solution y n ∈ W 1,p(Ω).

Formally, it may be interpreted as solving the p-Laplacian equation with mixed boundary conditions: Dirichlet conditions in \(\{x_{i}\}_{i=\overline{1,n}}\) and Neumann conditions in the remaining of ∂Ω.

One can prove via convex analysis techniques the following approximation result:

Theorem 3.1

We have y n → y, the solution to ( 3.1 ), strongly in W 1,p(Ω).

This allow to replace the study of (3.1), by (3.2).

We discuss the dual problem for (3.2) which is a finite dimensional optimization problem. We define g n : W 1,p(Ω) →  ] −∞, +∞] by

$$\displaystyle{ g_{n}(y) = \left \{\begin{array}{l} 0\quad \quad y(x_{i}) = 0,\;i = \overline{1,n}, \\ + \infty \quad \quad \mathrm{otherwise} \end{array} \right. }$$
(3.3)

and \(h: W^{1,p}(\varOmega ) \rightarrow \mathbb{R}\), given by

$$\displaystyle{ h(y) = -\frac{1} {p}\int _{\varOmega }\left [\vert \nabla y\vert ^{p} + \vert y\vert ^{p}\right ]dx +\int _{\varOmega }fydx. }$$
(3.4)

Clearly, the problem (3.2) can be reexpressed via (3.3), (3.4) as

$$\displaystyle{ \mathop{\mathrm{Min}}\limits _{y\in W^{1,p}(\varOmega )}\{g_{n}(y) - h(y)\}. }$$
(3.5)

Since g n is convex, proper, lower semicontinuous and h is concave, continuous on W 1,p(Ω), the Fenchel theorem [2] can be applied and the dual of (3.5) has the form

$$\displaystyle{ \mathop{\max }\limits _{z\in W^{1,p}(\varOmega )^{{\ast}}}\{h^{{\ast}}(z) - g_{ n}^{{\ast}}(z)\}, }$$
(3.6)

where h ∗, g n ∗ are the conjugate mappings. One can compute them and obtain

$$\displaystyle{ h^{{\ast}}(z) = -\frac{1} {p}\vert z - f\vert _{W^{1,p}(\varOmega )^{{\ast}}}^{q}, }$$
(3.7)
$$\displaystyle{ g_{n}^{{\ast}}(z) = \left \{\begin{array}{l} 0\quad \quad z =\mathop{ \sum }\limits _{i=1}^{n}\alpha _{ i}\delta _{x_{i}}, \\ + \infty,\quad \quad \mathrm{otherwise}, \end{array} \right. }$$
(3.8)

where \(\alpha _{i} \in \mathbb{R}\) and \(\delta _{x_{i}} \in W^{1,p}(\varOmega )^{{\ast}}\) is a Dirac-type functional concentrated in x i ∈ ∂Ω (but it is not a distribution).

By (3.6)–(3.8), we can state

Theorem 3.2

The dual problem is given by:

$$\displaystyle{ \mathrm{Min}\left \{\frac{1} {q}\vert \,f - z\vert _{W^{1,p}(\varOmega )^{{\ast}}}^{q};\;z =\mathop{ \sum }\limits _{ i=1}^{n}\alpha _{ i}\delta _{x_{i}},\;\alpha _{i} \in \mathbb{R}\right \} }$$

and it is a finite dimensional optimization problem.

It is known that, from the solution to the dual problem one can find the solution to the primal problem as well, [2, p. 188]. If p = 2, the involved equations become linear. The continuity may be obtained if f ∈ L s(Ω), with s sufficiently big, depending on dimension.

Similar ideas can be applied to fourth order elliptic variational inequalities. An example is given by

$$\displaystyle{ \mathop{\mathrm{Min}}\limits _{y\in K}\left \{\frac{1} {2}\int _{\varOmega }\vert \varDelta y\vert ^{2} -\int _{\varOmega }hydx\right \}, }$$
(3.9)
$$\displaystyle{ K = \left \{z \in H^{2}(\varOmega ) \cap H_{ 0}^{1}(\varOmega );\;\int _{\varOmega }hzdx \geq -1\right \}, }$$
(3.10)

which can be interpreted as a simply supported plate. The unilateral condition (3.10) is related to the mechanical work performed by the force h ∈ L 2(Ω).

The problem (3.9), (3.10) has a unique solution y ∈ K, due to the coercivity and strict convexity of the functional (3.9).

The approximating problem is

$$\displaystyle{ \mathop{\mathrm{Inf}}\limits _{y\in K_{n}}\left \{\frac{1} {2}\int _{\varOmega }\vert \varDelta y\vert ^{2}dx -\int _{\varOmega }hydx\right \}, }$$
(3.11)
$$\displaystyle{ K_{n} = \left \{z \in H^{2}(\varOmega );\;z(x_{ i}) = 0,\;i = \overline{1,n},\;\int _{\varOmega }hzdx \geq -1\right \}, }$$
(3.12)

where {x i } i ∈ N ⊂ ∂Ω is, as before, a dense subset.

Notice that (3.11), (3.12) may have no solution due to the possible lack of coercivity. One can use minimizing sequences in (3.11), (3.12). However, the dual problem has solutions and the Fenchel theorem can be applied.

Theorem 3.3

The dual problem is given by

$$\displaystyle{ \mathrm{Min}\frac{1} {2}\int _{\varOmega }\vert z\vert ^{2}dx }$$

subject to

$$\displaystyle{ D^{{\ast}}z - h \in \overline{\mathrm{conv}}\left \{\{0\} \cup A_{ n}\right \}, }$$
$$\displaystyle{ A_{n} = \left \{-h +\mathop{ \sum }\limits _{i=1}^{n}\alpha _{ i}\delta _{x_{i}},\;\alpha _{i} \in \mathbb{R}\right \}, }$$

where D ∗: L 2(Ω) → H 2(Ω)∗ is the adjoint of the linear continuous operator D: H 2(Ω) → L 2(Ω), Dy = Δy.

The finite dimensional character of A n is the key point in this approach. Other applications of the duality approach to second and fourth order variational inequalities of obstacle type are due to [16–18]. Different duality concepts, for high order nonlinear elliptic problems and for systems are discussed in [11, 12, 14].

4 An Application in Shape Optimization

The arguments from the previous sections have a variational character and are strongly related to optimization problems. We argue here via a shape optimization example [1], that they may have further consequences in optimization theory:

$$\displaystyle{ \mathrm{Min}\int _{\varOmega }u(x)dx, }$$
(4.1)
$$\displaystyle{ \varDelta (u^{3}\varDelta y) = f\quad \mathrm{in}\;\varOmega, }$$
(4.2)
$$\displaystyle{ y =\varDelta y = 0\quad \mathrm{in}\;\partial \varOmega, }$$
(4.3)
$$\displaystyle{ 0 <m \leq u(x) \leq M\;\mathrm{a.e.}\;\mathrm{in}\;\varOmega, }$$
(4.4)
$$\displaystyle{ y \in C, }$$
(4.5)

where Ω is a bounded domain in \(\mathbb{R}^{d}\), C ⊂ L 2(Ω) is a given nonempty closed subset, \(m,M \in \mathbb{R}\), f ∈ L 2(Ω).

In dimension two, relations (4.2), (4.3) model the equilibrium state of a simply supported plate with thickness u satisfying (4.4) and deflection y, under the vertical load f. The geometric optimization problem (4.1)–(4.5) consists in finding the plate of minimal volume, such that the deflection remains in the prescribed set C. For instance, we may take (\(\tau \in \mathbb{R}_{+}\) given):

$$\displaystyle{ C =\{ z \in L^{2}(\varOmega );\;z(x) \geq -\tau \;\mathrm{a.e.}\;\mathrm{in}\;\varOmega \}, }$$
(4.6)

which is a safety condition (the deflection should not overpass some limit). In this example (4.6), C is even convex. However, the optimization problem (4.1)–(4.6) remains strongly nonconvex, even for C convex, due to the nonlinear character of the dependence u → y defined by (4.2).

It enters the category of control by coefficients problems. Notice that the boundary value problem (4.2), (4.3) has a unique weak solution y ∈ H 2(Ω) ∩ H 0 1(Ω).

Denote by w ∈ H 2(Ω) ∩ H 0 1(Ω) the unique solution to the Dirichlet problem Δw = f in Ω. Then, (4.2), (4.3) is equivalent with

$$\displaystyle{ \varDelta y = wl\quad \mathrm{in}\;\varOmega, }$$
(4.7)
$$\displaystyle{ y = 0\quad \mathrm{on}\;\partial \varOmega, }$$
(4.8)

where l = u −3 ∈ L ∞(Ω). Equations (4.7), (4.8) together with the above definition of w may be interpreted as the optimality conditions for a linear-quadratic control problem and is one of the simplest examples of the application of the control variational method, [1].

The shape optimization problem (4.1)–(4.5) becomes

$$\displaystyle{ \mathrm{Min}\int _{\varOmega }l^{-\frac{1} {3} }(x)dx }$$
(4.9)

subject to (4.7), (4.8) and the constraints l ∈ [M −3, m −3] and (4.5). Due to the linearity of the dependence l → y defined by (4.7) and the strict convexity of the functional (4.9), we infer

Theorem 4.1

The problem ( 4.1 )–( 4.5 ) has at least one optimal pair [ y ∗, u ∗] ∈ H 2(Ω) × L 2(Ω). If C is convex, the optimal pair is unique.

The existence is a consequence of usual weak lower semicontinuity arguments and the boundedness of the set of admissible thicknesses. The problem (4.1)–(4.5) may have many local optimal pairs since it is nonconvex, but the global optimal pair is unique if C is convex. Uniqueness is a very unusual property in optimal design. Such results may be extended to clamped plates [1, 21]. If M is big enough, then one can prove that the set of admissible pairs is nonvoid. A general presentation of shape optimization problems can be found in [19].

5 Conclusion

We have performed a short review of the control variational approach and some of its applications. An important ingredient is the Fenchel duality theorem and the analysis of the corresponding dual problems. The literature on duality methods in differential equations is very rich and includes a large variety of arguments and results. Obtaining the exact solution in certain non autonomous boundary value problems, proving the uniqueness of the minimizer in some shape optimization examples or developing new numerical discretization procedures via dense subsets of points (in the considered domain) are useful properties that show the applicability of such ideas in many directions of interest.