On some non-linear systems of PDEs related to inverse problems in 3-d conductivity

Abstract

We focus on certain non-linear, non-convex, non-coercive systems of PDEs in three dimensions that are directly motivated by inverse problems in conductivity for the three-dimensional case. It turns out that such systems are variational, as they formally are the Euler–Lagrange systems associated with an explicit first-order functional, and thus we exploit both its variational structure as well as its connection to inverse problems. In particular, boundary conditions play a central role.

1 Introduction

Let $\Omega \subset \mathbb {R}^3$ be a bounded domain as regular as necessary. We may think of $\Omega$ as a ball ${\mathbf {B}}$ from the outset. We would like to examine the system of three non-linear PDEs for three unknowns $\varvec{u}=(u, v, w)$ in three independent variables $\varvec{x}=(x_1, x_2, x_3)$ given in the form

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega , \end{aligned}$$

(1.1)

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\right] =0\hbox { in }\Omega ,\end{aligned}$$

(1.2)

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla v|^2\mathbf {1}-\nabla v\otimes \nabla v)\nabla w\right] =0\hbox { in }\Omega . \end{aligned}$$

(1.3)

$\mathbf {1}$ is the $3\times 3$-identity matrix, and for two three-dimensional vectors $\mathbf {x}$, $\mathbf {y}$, $\mathbf {x}\otimes \mathbf {y}$ is the $3\times 3$-, rank-one matrix

$$\begin{aligned} (\mathbf {x}\otimes \mathbf {y})_{ij}=x_iy_j, \quad 1\le i, j\le 3. \end{aligned}$$

$\mathbf {x}\wedge \mathbf {y}$ stands for the usual cross product in $\mathbb {R}^3$. In addition to the impressive form of the above system, the kind of boundary conditions that we would like to consider for the triplet (u, v, w) make things even more involved. We can certainly impose typical boundary conditions, either Dirichlet or Neumann, for each component. But the nature of such a system coming from inverse problems in conductivity, as will be explained below, forces us to explore boundary conditions of the form

$$\begin{aligned} u=u^\circ ,\quad (\nabla v\wedge \nabla w)\cdot \varvec{n}=v^\circ , \end{aligned}$$

(1.4)

on $\partial \Omega$, where $u^\circ$ and $v^\circ$ are suitable functions. $\varvec{n}$ is the outer, unit normal to the boundary $\partial \Omega$. There are, at least, two special issues concerning the second boundary condition in (1.4):

• (1) it is definitely a non-linear boundary condition for v and w;

• (2) there is some degeneracy involved as we have just one boundary condition determined through $v^\circ$ for two unknowns v and w.

These difficulties push us to regard the same system (1.1)–(1.3), thinking of the function w as given a priori, and discard the third equation (1.3). Namely, for a given a priori function $w(\varvec{x})\in H^1(\Omega )$ as smooth and regular as we may need it to be, we would like to examine the system of two non-linear PDEs for two unknowns (u, v) in three independent variables $\varvec{x}=(x_1, x_2, x_3)$ given in the form

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega , \end{aligned}$$

(1.5)

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\right] =0\hbox { in }\Omega . \end{aligned}$$

(1.6)

This time there is no mismatch in the number of boundary conditions and unknowns, though the boundary condition for the second component v, which is now linear in v, is definitely of a special nature

$$\begin{aligned} \nabla v\cdot (\nabla w\wedge \varvec{n})=v^\circ \hbox { on }\partial \Omega . \end{aligned}$$

This kind of boundary condition for v has been analyzed recently in [12]. We will recall its meaning below.

This contribution is like a second part to [10], in which we deal with the similar situation in 2-D. As we will see, the extension to 3-D is more than just a straightforward generalization. Though we have chosen to stay here as close as possible to the ideas in that paper, some other arguments are explored in [3].

As remarked in [10], “for vector problems, like the one we are considering here, there are not many ways to show existence of solutions. In fact, as far as we can tell, there are two classic methods to prove such existence of solutions. Both come directly from the treatment of equilibrium solutions in finite elasticity [2]. The first is to make use of the implicit function theorem [4, 15] in a suitable framework. By its very perturbation nature, this method can only deliver solutions which are sufficiently close to solutions of approximating linear problems. It demands quite restrictive hypotheses. The second alternative is to show that the system to be studied is, at least formally, the Euler–Lagrange system of a functional which admits minimizers in appropriate function spaces. This too asks for important structural assumptions on the underlying functional which cannot always be taken for granted [2, 4]”.

It turns out that our systems of PDEs correspond exactly to the Euler–Lagrange equations for the functional

$$\begin{aligned} E=\int _\Omega |\nabla u(\varvec{x})|\,|\nabla v(\varvec{x})\wedge \nabla w(\varvec{x})|\,d\varvec{x}\end{aligned}$$

(1.7)

depending on whether we regard it as a funcional $E=E(u, v, w)$ of the three functions (u, v, w), or just of the first two $E=E(u, v)$ for given w. As indicated, boundary conditions play a special role. The only possibility we envision to show existence of solutions is to deal with this functional and prove that, possibly under circumstances related to choosing appropriate boundary data as well as the auxiliary function w for the two-dimensional case, there are minimizers. Note that problem (1.7) is a vector, non-(quasi)convex, non-coercive problem in either situation. In addition, as anticipated above, the suitable boundary condition for the second component v coming from inverse problems in conductivity is of a new type [12]. It is therefore an explicit problem under the most adverse of circumstances. Yet some interesting things can be tried out, as in the 2-D [10], and even the 3-D case [3] from a different perspective.

Our results concerning solutions of this system of PDEs either under typical Dirichlet boundary conditions or under the ones coming from inverse problems in conductivity are far from being complete. Three are our main conclusions:

• (1) a criterium to ensure when a solution (u, v) of our system (1.5)–(1.6) generates a solution of the inverse conductivity problem (Proposition 3.4);

• (2) a result (Theorem 3.5) pointing in the direction that existence of solutions may depend on the nature of boundary data;

• (3) an explicit method to produce boundary data inherited from inverse problems in conductivity for which system (1.5)–(1.6) admits solutions (Theorem 4.1).

Because of its special nature, we briefly discuss this new boundary condition for the second component v in connection with Theorem 4.1 below. It was rigorously introduced and studied in [12] in the context of a typical scalar, quadratic or strictly convex functional. We just describe it here informally for the sake of readers. The function $w(\varvec{x})\in H^1(\Omega )$ is selected a priori and in such a way that ${\text {image}}(w)$ is an interval of $\mathbb {R}$. Consider the subspaces

$$\begin{aligned} \mathbb {L}_w=\{\psi \circ w: \psi \circ w\in H^1(\Omega )\}, \quad \mathbb {H}_w=\mathbb {L}_w+H^1_0(\Omega ). \end{aligned}$$

Then the boundary condition for v becomes

$$\begin{aligned} v\in v_0+\mathbb {H}_w, \end{aligned}$$

for some $v_0\in H^1(\Omega )$. The intuitive interpretation of this condition is

$$\begin{aligned} (\nabla v-\nabla v_0)\cdot (\nabla w\wedge \varvec{n})=0\hbox { on }\partial \Omega . \end{aligned}$$

When boundary conditions

$$\begin{aligned} u-u_0\in H^1_0(\Omega ),\quad v-v_0\in \mathbb {H}_w, \end{aligned}$$

permit, we can find a solution of the problem

$$\begin{aligned}&{\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega ,\\&{\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\right] =0\hbox { in }\Omega , \end{aligned}$$

under those boundary conditions together with the funny, additional constraint (coming from optimality)

$$\begin{aligned} \int _{\{w=\lambda \}\cap \partial \Omega } \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\cdot \varvec{n}\,dS(\varvec{x})=0 \end{aligned}$$

for every $\lambda \in {\text {image}}(w)$. $\varvec{n}$ is the outer, unit normal to $\partial \Omega$. As announced, we will provide a certificate that ensures under what circumstances this solution (u, v) determines a valid conductivity coefficient through the natural formula

$$\begin{aligned} \gamma =\frac{|\nabla v\wedge \nabla w|}{|\nabla u|} \end{aligned}$$

for the boundary measurement $(u^\circ , v^\circ )$ where

$$\begin{aligned} u^\circ =u_0,\quad v^\circ =(\nabla v_0\wedge \nabla w)\cdot \varvec{n}\hbox { on }\partial \Omega . \end{aligned}$$

The paper is divided in another three sections. We first motivate the systems of PDEs that we deal with in this work through an informal discussion of 3-D inverse conductivity problems, in which the special boundary condition for v is discovered too. We then examine the difficulties with the problem of existence of minimizers for the underlying vector variational problem associated with the functional (1.7), and prove Proposition 3.4 and Theorem 3.5. Finally, we focus on building synthetic boundary data in a suitable way to ensure existence of solutions of our PDE-system under that special boundary condition for the second component v (Theorem 4.1).

2 Inverse problems in conductivity

We explain here where system (1.5)–(1.6) comes from, as it is fundamental to understand how to proceed to find solutions for it. Simultaneously, it will serve us to explain the nature of the boundary condition for the second component v. The discussion is informal and follows closely that in [12].

A typical Calderón problem in dimension 3 (for one measurement) reads as follows:

For a pair of boundary data $(u^\circ , v^\circ )$ taken from a suitable class on $\partial \Omega$, find a conductivity coefficient

$$\begin{aligned} \gamma (\varvec{x}):\Omega \rightarrow \mathbb {R}^+ \end{aligned}$$

such that the unique solution u of the problem

$$\begin{aligned} {\text {div}}(\gamma \nabla u)=0\hbox { in }\Omega , \quad u=u^\circ \hbox { on }\partial \Omega , \end{aligned}$$

(2.1)

complies with

$$\begin{aligned} \gamma \nabla u\cdot \varvec{n}=v^\circ \hbox { on }\partial \Omega \end{aligned}$$

(2.2)

as well.

The literature on this problem is quite abundant (check, for instance, the two recent general sources [1, 7]). The so-called Dirichlet-to-Neumann operator is one main tool in this analysis.

A variational perspective on this problem, as developed in [10] for the 2-D case, aims at determining the conductivity coefficient $\gamma$ through the solution of a non-linear, non-convex vector variational problem. In the 3-D case, problem (2.1) can be formally interpreted as

$$\begin{aligned} \gamma \nabla u=\nabla v\wedge \nabla w\hbox { in }\Omega ,\quad u=u^\circ \hbox { on }\partial \Omega , \end{aligned}$$

(2.3)

for suitable functions v and w, while the Neumann condition (2.2) becomes

$$\begin{aligned} (\nabla v\wedge \nabla w)\cdot \varvec{n}=v^\circ \hbox { on }\partial \Omega . \end{aligned}$$

Some times the functions v and w in (2.3) are referred to as Clebsch potentials (check [8, 11]).

If we multiply (2.3) by $\nabla w$, and divide through by $\gamma$, we find

$$\begin{aligned} \nabla u\wedge \nabla w=\frac{1}{\gamma }(\nabla v\wedge \nabla w)\wedge \nabla w. \end{aligned}$$

This identity is informing us that

$$\begin{aligned} {\text {div}}\left[ \frac{1}{\gamma }(\nabla v\wedge \nabla w)\wedge \nabla w\right] =0\hbox { in }\Omega . \end{aligned}$$

(2.4)

If the coefficient $\gamma$ is unknown, we can always try to recover it through the quotient

$$\begin{aligned} \gamma =\frac{|\nabla v\wedge \nabla w|}{|\nabla u|}. \end{aligned}$$

If we replace this formula in the two equations (2.1) and (2.4), we arrive at the two coupled equations

$$\begin{aligned} {\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega , \end{aligned}$$

(2.5)

$$\begin{aligned} {\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|} (\nabla v\wedge \nabla w)\wedge \nabla w\right] =0\hbox { in }\Omega , \end{aligned}$$

(2.6)

that must be completed with boundary conditions

$$\begin{aligned} u=u^\circ , (\nabla v\wedge \nabla w)\cdot \varvec{n}=v^\circ \quad \hbox { on }\partial \Omega . \end{aligned}$$

Note that similar computations lead to the third equation (1.3) for w. It turns out that the two previous PDEs correspond exactly to the Euler–Lagrange equations for the functional

$$\begin{aligned} E(u, v)=\int _\Omega |\nabla u(\varvec{x})|\,|\nabla v(\varvec{x})\wedge \nabla w(\varvec{x})|\,d\varvec{x}\end{aligned}$$

(2.7)

where the function $w(\varvec{x})$ is assumed to be given. The boundary condition for u is clear as it is a standard Dirichlet condition. The one for v as

$$\begin{aligned} (\nabla v\wedge \nabla w)\cdot \varvec{n}=v^\circ \quad \hbox { on }\partial \Omega \end{aligned}$$

(2.8)

is not so. The strategy to determine or approximate the unknown conductivity coefficient $\gamma$ from the measurement $(u^\circ , v^\circ )$ is then to solve the system (2.5)–(2.6) as the optimality system of the functional E in (2.7) under the given boundary condition $u=u^\circ$ for u, and (2.8) for v.

As already indicated in the Introduction, this special boundary condition for w was introduced in [12]. If we informally introduce the spaces

$$\begin{aligned} \mathbb {L}_w=\{\psi \circ w: \psi \circ w\in H^1(\Omega )\}, \quad \mathbb {H}_w=\mathbb {L}_w+H^1_0(\Omega ), \end{aligned}$$

and suppose that $v_0$ is a particular function such that (2.8) holds

$$\begin{aligned} (\nabla v_0\wedge \nabla w)\cdot \varvec{n}=v^\circ \quad \hbox { on }\partial \Omega , \end{aligned}$$

then the boundary condition (2.8) for v becomes

$$\begin{aligned} v\in v_0+\mathbb {H}_w. \end{aligned}$$

A main useful, heuristic interpretation of the condition $v\in \mathbb {H}_w$ [12] is

$$\begin{aligned} (\nabla v\wedge \nabla w)\cdot \varvec{n}=0\hbox { on }\partial \Omega , \end{aligned}$$

and so $v\in v_0+\mathbb {H}_w$ becomes

$$\begin{aligned} (\nabla v\wedge \nabla w)\cdot \varvec{n}=v^\circ \quad \hbox { on }\partial \Omega . \end{aligned}$$

In this way, we become interested in finding minimizers of the vector variational problem

$$\begin{aligned} \hbox {Minimize in }(u, v)\in H^1(\Omega ; \mathbb {R}^2):\quad \int _\Omega |\nabla u|\,|\nabla v\wedge \nabla w|\,d\varvec{x}\end{aligned}$$

(2.9)

under

$$\begin{aligned} u-u^\circ \in H^1_0(\Omega ), \quad v-v_0\in \mathbb {H}_w. \end{aligned}$$

The 3-D inverse-conductivity problem for a single measurement $(u^\circ , v^\circ )$ can be treated in three steps as follows.

• (1) Select a non-constant function $w\in H^1(\Omega )$, and find $v_0\in H^1(\Omega )$ such that

  $$\begin{aligned} v^\circ =\nabla v_0\cdot (\nabla w\wedge \varvec{n}) \hbox { on }\partial \Omega . \end{aligned}$$

• (2) Find a solution (u, v) of the coupled, non-linear system of PDEs

  $$\begin{aligned} {\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega , \end{aligned}$$

  (2.10)
  $$\begin{aligned} {\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\right] =0\hbox { in }\Omega , \end{aligned}$$

  (2.11)

  under the boundary conditions

  $$\begin{aligned} u-u^\circ \in H^1_0(\Omega ), \quad v-v_0\in \mathbb {H}_w \end{aligned}$$

  (2.12)

  on $\partial \Omega$ by searching for a minimizer (u, v) for our vector variational problem (2.9). In particular, one may find some additional constraint on $\partial \Omega$ coming from optimality much in the same way as with Neumann boundary conditions. Indeed, it is so as Proposition 4.1 below states.

• (3) Put

  $$\begin{aligned} \gamma =\frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\hbox { in }\Omega . \end{aligned}$$

There is however a fundamental step to be examined to validate this procedure. As explained earlier, it is always true that

$$\begin{aligned} \gamma \nabla u=\nabla v\wedge \nabla w \end{aligned}$$

implies

$$\begin{aligned} \nabla u\wedge \nabla w=\frac{1}{\gamma }(\nabla v\wedge \nabla w)\wedge \nabla w. \end{aligned}$$

However, the reverse is not always correct unless $\nabla v\cdot \nabla w=0$.

3 Search for solutions

Our system (2.5)–(2.6) is very special, basically because it is the Euler–Lagrange system for the functional in (2.7). We put

$$\begin{aligned} W({\mathbf {F}})=|F_1|\,|F_2\wedge F_3|,\quad {\mathbf {F}}=\begin{pmatrix} F_1\\ F_2\\ F_3\end{pmatrix}\in \mathbb {R}^{3\times 3}, \end{aligned}$$

(3.1)

for the integrand corresponding to the integral functional in (2.7), and

$$\begin{aligned} L({\mathbf {F}})=\det {\mathbf {F}}. \end{aligned}$$

Lemma 3.1

Under the notation just introduced, for every matrix ${\mathbf {F}}$, it is always true that $W({\mathbf {F}})\ge L({\mathbf {F}})$, and $W({\mathbf {F}})=L({\mathbf {F}})$ if and only if

$$\begin{aligned} \gamma F_1=F_2\wedge F_3, \hbox { for some } \gamma \ge 0. \end{aligned}$$

This is an elementary lemma, but it is the clue to understanding our system because condition (2.3) is equivalent to equality in this lemma. This statement also explains why it may be advantageous to change our functional in (2.7) to

$$\begin{aligned} E^*=\int _\Omega [W(\nabla \varvec{u}(\varvec{x}))-L(\nabla \varvec{u}(\varvec{x}))]\,d\varvec{x},\quad \varvec{u}=(u, v, w). \end{aligned}$$

Again, we can put

$$\begin{aligned} E^*=E^*(u, v, w),\quad E^*=E^*(u, v)\hbox { for given }w. \end{aligned}$$

Proposition 3.2

Suppose $\varvec{u}=(u, v, w)$, with (u, v) complying with the appropriate boundary conditions, is such that $E^*(u, v)=0$. Then (u, v) is a weak solution of (2.5)–(2.6), and the coefficient

$$\begin{aligned} \gamma (\varvec{x})=\frac{|\nabla v(\varvec{x})\wedge \nabla w(\varvec{x})|}{|\nabla u(\varvec{x})|} \end{aligned}$$

is a solution of the associated inverse conductivity problem.

Proof

Note that the Euler–Lagrange system for this new functional is exactly our initial non-linear system of PDEs because the added term

$$\begin{aligned} \det (\nabla u, \nabla v, \nabla w)=\nabla u\cdot (\nabla v\wedge \nabla w) \end{aligned}$$

is a null-lagrangian

$$\begin{aligned} {\text {div}}D_{\nabla u}\det (\nabla u, \nabla v, \nabla w)={\text {div}}D_{\nabla v}\det (\nabla u, \nabla v, \nabla w)=0 \end{aligned}$$

in $\Omega$. The proof, then, reduces to noticing that, by Lemma 3.1, the condition $E^*(u, v)=0$ implies

$$\begin{aligned} \gamma (\varvec{x})\nabla u(\varvec{x})=\nabla v(\varvec{x})\wedge \nabla w(\varvec{x}) \end{aligned}$$

for a.e. $\varvec{x}$ and some non-negative coefficient $\gamma$. This identity in $\Omega$, by all of our previous discussion, implies the conclusion. $\square$

We thus realize that our only hope to find a solution of such a system which is compatible with a solution of the inverse conductivity problem is to ensure that the infimum value m of this last variational problem vanishes, and it is attained. We therefore change our problem (2.9) to minimizing the functional

$$\begin{aligned} \int _\Omega [|\nabla u|\,|\nabla v\wedge \nabla w|-\det (\nabla u, \nabla v, \nabla w)]\,d\varvec{x}\end{aligned}$$

(3.2)

over $(u, v)\in H^1(\Omega ; \mathbb {R}^2)$, for a given w, under

$$\begin{aligned} u-u^\circ \in H^1_0(\Omega ), \quad v-v_0\in \mathbb {H}_w. \end{aligned}$$

(3.3)

As in [10] we introduce the following definition.

Definition 3.3

Three functions $u, v, w\in H^1(\Omega )$ form a feasible solution for a 3-D inverse problem in conductivity if

$$\begin{aligned} |\nabla v\wedge \nabla w|\nabla u=|\nabla u|(\nabla v\wedge \nabla w)\hbox { in }\Omega . \end{aligned}$$

The corresponding conductivity coefficient is the non-negative measurable function

$$\begin{aligned} \gamma =\frac{|\nabla v\wedge \nabla w|}{|\nabla u|}. \end{aligned}$$

Our discussion above can be summarized in the following statement.

Proposition 3.4

If $w\in H^1(\Omega )$ is given, feasible solutions for inverse problems in conductivity in the 3-dimensional case according to the previous definition correspond exactly to triplets (u, v, w) with minimizers (u, v) of (3.2) such that the value m of the infimum vanishes.

One main result is a sufficient condition to ensure that $m=0$ for the same functional in (3.2) regarded as a functional on the three unknowns (u, v, w). Namely, we focus on

$$\begin{aligned} m=\inf _{(u, v, w)}\int _\Omega [|\nabla u|\,|\nabla v\wedge \nabla w|-\det (\nabla u, \nabla v, \nabla w)]\,d\varvec{x}, \end{aligned}$$

(3.4)

though we cannot impose (3.3) as these are non-linear boundary conditions on the triplet (u, v, w). For our next result to be valid, we would need to restrict (u, v, w) through typical Dirichlet boundary conditions.

Theorem 3.5

If there is a feasible triplet (u, v, w) such that

$$\begin{aligned} \det (\nabla u, \nabla v, \nabla w)>0\hbox { in }\Omega , \end{aligned}$$

then the infimum m in (3.4) under usual Dirichlet boundary conditions (determined by (u, v, w) themselves) for (u, v, w) vanishes.

The proof of the result is a typical relaxation fact. This is a well-established process in the vectorial Calculus of Variations [5]. For the integrand $W({\mathbf {F}})$ given in (3.1), one needs to calculate its quasiconvexification. This is, in general, an impossible task except for a few distinguished examples. Surprisingly enough, our density $W({\mathbf {F}})$ is one such example, as it was the 2-D version in [10]. These and some other similar examples and calculations can be found in [13].

Proof

The main step of the proof focuses on the computation of the quasiconvexification of the integrand $W({\mathbf {F}})$ in (3.1). The strategy is the same as in every example where such calculation is possible: one computes the first step $R_1W({\mathbf {F}})$ towards the rank-one convex hull of W, and realizes that the outcome is a polyconvex function. If this is so, then $QW=R_1W$. We refer to [5, 13] for more information on the various convex hulls involved in vector, variational problems. The function $R_1W$ is defined through the formula

$$\begin{aligned}&R_1W({\mathbf {F}})=\min \{tW({\mathbf {F}}_1)+(1-t)W({\mathbf {F}}_0): t\in [0, 1],\\&{\mathbf {F}}=t{\mathbf {F}}_1+(1-t){\mathbf {F}}_0, {\mathbf {F}}_1-{\mathbf {F}}_0\hbox { is a rank-one matrix}\}. \end{aligned}$$

It can be checked that

$$\begin{aligned} R_1W({\mathbf {F}})=|\det {\mathbf {F}}|, \end{aligned}$$

which is well-known to be a polyconvex function, and, hence,

$$\begin{aligned} QW({\mathbf {F}})=|\det {\mathbf {F}}|. \end{aligned}$$

As a matter of fact, this example is just a particular case of a more general calculation performed in [13]. There are some algebraic manipulations involved that are irrelevant to our discussion, and so we will not transcribe them here, but refer interested readers to [13].

A typical relaxation result [5] ensures that

$$\begin{aligned} m&=\inf _{(u, v, w)\in {\mathcal {A}}}\int _\Omega [W(\nabla u, \nabla v, \nabla w)-\det (\nabla u, \nabla v, \nabla w)]\,d\varvec{x}\nonumber \\&= \inf _{(u, v, w)\in {\mathcal {A}}}\int _\Omega [QW(\nabla u, \nabla v, \nabla w)-\det (\nabla u, \nabla v, \nabla w)]\,d\varvec{x} \end{aligned}$$

(3.5)

if ${\mathcal {A}}$ stands for the class of triplets under a standard Dirichlet boundary conditions. Note, on the other hand, that $-\det {\mathbf {F}}$ is a quasiaffine function (a null-lagrangian) and so it does not affect the quasiconvexification in (3.5). If our claim above holds and we have that

$$\begin{aligned} QW({\mathbf {F}})=|\det {\mathbf {F}}| \end{aligned}$$

then

$$\begin{aligned} QW({\mathbf {F}})-\det {\mathbf {F}}=|\det {\mathbf {F}}|-\det {\mathbf {F}}=2{\det }^-{\mathbf {F}}, \end{aligned}$$

and

$$\begin{aligned} m=2\inf _{(u, v, w)\in {\mathcal {A}}}\int _\Omega {\det }^-(\nabla u, \nabla v, \nabla w)\,d\varvec{x}. \end{aligned}$$

Under our hypothesis in the statement of the theorem, clearly $m=0$. $\square$

If such infimum value m is attained, there is a solution for our initial PDE system, and for the inverse conductivity problem; if it is not, there is a homogenized solution in which $\gamma \mathbf {1}$ is replaced by a suitable homogenized matrix ${\mathbf {A}}(\varvec{x})$.

Unfortunately, a parallel result for problem (3.2) under (3.3), when the function w is regarded as given is, in general, not possible. The integrand for this problem is

$$\begin{aligned} W(\varvec{x}, {\mathbf {F}})=W({\mathbf {F}})=|F_1|\,|F_2\wedge \nabla w(\varvec{x})|,\quad {\mathbf {F}}=\begin{pmatrix} F_1\\ F_2\end{pmatrix}\in \mathbb {R}^{2\times 3}. \end{aligned}$$

This time the quasiconvexification of this integrand for a.e. $\varvec{x}$ cannot seem to be computable. The above observations for the same problem regarding w as an unknown do not apply. All we can say is that

$$\begin{aligned} QW(\varvec{x}, {\mathbf {F}})\ge \left| \det \begin{pmatrix}F_1\\ F_2\\ \nabla w(\varvec{x})\end{pmatrix}\right| \end{aligned}$$

but equality looks far from being correct.

4 Synthetic data

Suppose we face a real inverse conductivity problem in 3 dimensions for a smooth, regular domain $\Omega \subset \mathbb {R}^3$. We know that there is indeed an unknown conductivity coefficient $\gamma (\varvec{x})$ that we would like to know or approximate through boundary measurements $(u^\circ , v^\circ )$. Assume then that we give a priori such conductivity coefficient $\gamma (\varvec{x})$, and by means of boundary synthetic data, we would like to test our ability to recover it. The situation is much more sophisticated that in the 2-D case. We establish the following method to furnish boundary measurements in a coherent way:

• (1) Take $u_0(\varvec{x})=u^\circ (\varvec{x})\in H^1(\Omega )$.

• (2) Solve the standard conductivity problem

  $$\begin{aligned} {\text {div}}(\gamma \nabla u)=0\hbox { in }\Omega ,\quad u=u_0\hbox { on }\partial \Omega , \end{aligned}$$

  (4.1)

  and put

  $$\begin{aligned} v^\circ =\gamma \nabla u\cdot \varvec{n}\hbox { on }\partial \Omega . \end{aligned}$$

• (3) Find two Clebsch potential $v, w\in H^1(\Omega )$ so that

  $$\begin{aligned} \gamma \nabla u=\nabla v\wedge \nabla w, \end{aligned}$$

  (4.2)

  and take w as the auxiliary function to setup functional (1.7) and system (1.1)–(1.2).

• (4) Look for a function $v_0(\varvec{x})\in H^1(\Omega )$ with

  $$\begin{aligned} v^\circ =(\nabla v_0\wedge \nabla w)\cdot \varvec{n}\hbox { on }\partial \Omega . \end{aligned}$$

  (4.3)

Note how the pair $(u^\circ , v^\circ )$ is a boundary measurement compatible with the initial conductivity coefficient $\gamma$, by construction.

Let us pretend not to know $\gamma$, and starting from $(u_0, v_0)$, let us try to apply our method to recover it. We would like to find a solution of the problem

$$\begin{aligned} \hbox {Minimize in }&(u, v)\in H^1(\Omega ; \mathbb {R}^2):\nonumber \\&E^*(u, v)=\int _\Omega [|\nabla u|\,|\nabla v\wedge \nabla w|-\det (\nabla u, \nabla v, \nabla w)]\,d\varvec{x} \end{aligned}$$

(4.4)

under

$$\begin{aligned} u-u_0\in H^1_0(\Omega ), \quad v-v_0\in \mathbb {H}_w. \end{aligned}$$

Recall that the Euler–Lagrange system for this functional still is our initial system of PDEs (1.5)–(1.6) because the added term involving the determinant is a null-lagrangian as explained earlier.

Theorem 4.1

Let

$$\begin{aligned} \gamma (\varvec{x})\ge \gamma _0>0 \end{aligned}$$

(4.5)

and boundary data $u_0$ be given. Suppose $w\in H^1(\Omega )$ and $v_0$ are determined through (4.1) and (4.2).

• (1) There is a solution of the problem

  $$\begin{aligned} {\text {div}}\left[ \frac{|\nabla v\wedge \nabla w|}{|\nabla u|}\nabla u\right] =0\hbox { in }\Omega ,\nonumber \\ {\text {div}}\left[ \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\right] =0\hbox { in }\Omega , \end{aligned}$$

  (4.6)

  under the boundary conditions

  $$\begin{aligned} u-u_0\in H^1_0(\Omega ),\quad v-v_0\in \mathbb {H}_w, \end{aligned}$$

  and

  $$\begin{aligned} \int _{\{w=\lambda \}\cap \partial \Omega } \frac{|\nabla u|}{|\nabla v\wedge \nabla w|}(|\nabla w|^2\mathbf {1}-\nabla w\otimes \nabla w)\nabla v\cdot \varvec{n}\,dS(\varvec{x})=0 \end{aligned}$$

  for every $\lambda \in {\text {image}}(w)$.

• (2) The conductivity coefficient ${\overline{\gamma }}$, recovered through the formula

  $$\begin{aligned} {\overline{\gamma }}=\frac{|\nabla v\wedge \nabla w|}{|\nabla u|}, \end{aligned}$$

  (4.7)

  is compatible with the measurement $(u^\circ , v^\circ )$, in the sense of Definition 3.3.

At this stage, the proof is elementary since all ingredients have been set up to favor the existence of a solution. Even so, the existence result is appealing because of the non-linear, non-convex, non-coercive character of the underlying functional, as already stressed earlier.

Proof

Simply notice that the pair (u, v) where u is the solution of (4.1) and v is the additional function in (4.2) is a solution of the system. This is so because the identity in (4.2) implies, on the one hand, (4.7), but, on the other, $E^*(u, v)=0$. Since $E^*\ge 0$ always, the pair (u, v) so determined is truly a global minimizer for $E^*$ and, hence, a weak solution of our system. It would be hard to conclude this without the use of functional $E^*$ in (4.4). This is basically the conclusions of Lemma 3.1 and Proposition 3.2. $\square$

Condition (4.2) is all that is required for a solution of the inverse conductivity problem. However, it is not clear how to find both Clebsch potentials v and w from u and $\gamma$. In fact, once we can provide such v and w, the pair (u, v) automatically becomes a solution of our system (1.5)–(1.6) compatible with the solution of an inverse conductivity problem. More interesting is to learn how to determine the auxiliary function w with those important properties. Namely, the more appealing mechanism would be:

• (1) Take $u_0(\varvec{x})=u^\circ (\varvec{x})\in H^1(\Omega )$.

• (2) Solve the standard conductivity problem

  $$\begin{aligned} {\text {div}}(\gamma \nabla u)=0\hbox { in }\Omega ,\quad u=u_0\hbox { on }\partial \Omega , \end{aligned}$$

  (4.8)

  and put

  $$\begin{aligned} v^\circ =\gamma \nabla u\cdot \varvec{n}\hbox { on }\partial \Omega . \end{aligned}$$

• (3) Take two functions $w, v_0\in H^1(\Omega )$ such that

  $$\begin{aligned} v^\circ =(\nabla v_0\wedge \nabla w)\cdot \varvec{n}\hbox { on }\partial \Omega . \end{aligned}$$

  (4.9)

For the triplet $(u_0, v_0, w)$, consider variational problem (4.4), and check if there is a minimizer (u, v) with $E^*(u, v)=0$. If it is so, the pair (u, v) becomes a solution of (1.5)–(1.6) compatible with an inverse conductivity problem for the coefficient ${\overline{\gamma }}$ given in (4.7), which might or might not be equal to the initial $\gamma$. If $\inf E^*>0$, then one would need to change w and $v_0$ in the third stage above. It is not clear, however, on what grounds such selection should be made.

Nonetheless, it would be quite interesting to tackle the numerical treatment of an inverse problem in conductivity in the 3-D case through this procedure. The idea is to produce data $(u^\circ , v^\circ )$ from a selected $\gamma$ according to the mechanism just described, and try to recover it by approximating a minimizer of the variational problem (4.4) through a typical Newton-Raphson method, or other standard algorithm. If numerical computations show a regular and steady path to the minimum value $m=0$, one might reasonably infer that we are getting close to a solution (u, v) of our system of PDEs which is compatible with a conductivity coefficient. If not, then one would have to select a different pair $(w, v_0)$ compatible with (4.9), and go again through the approximation procedure. This enterprise still requires further insight related to the following issues:

• (1) investigate how to deal with the special boundary condition $v-v_0\in \mathbb {H}_w$ for a variational problem like (4.4), or a differential equation like (1.2).

• (2) describe how to find w and $v_0$ in practice complying with (4.9) in such a way that if m is the minimum value of problem (4.4) then $m=0$.

We plan to explore this line of research in the future. It looks challenging.