# Electrostatic Approximation of Vector Fields

Giles Auchmuty

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Giles Auchmuty. Electrostatic Approximation of Vector Fields. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. pp.89-94, 10.1007/978-3-319-55795-3_7. hal-01626889

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# Electrostatic Approximation of Vector Fields.

Giles Auchmuty

Department of Mathematics, University of Houston, Houston, Tx 77204-3008 USA,
auchmuty@uh.edu **

**Abstract.** This paper provides expressions for the boundary potential that provides the best electrostatic potential approximation of a given $L^2$ vector field on a nice bounded region in $\mathbb{R}^N$. The permittivity of the region is assumed to be known and the potential is required to be zero on the conducting part of the boundary. The boundary potential is found by solving the minimization conditions and using a special basis of the trace space for the space of allowable potentials. The trace space is identified by its representation with respect to a basis of $\Sigma$-Steklov eigenfunctions.

## 1 Introduction

Quite often in physical applications one wishes to produce electrostatic fields in a region of known permittivity $\epsilon(x)$ that approximate a prescribed (given) field $F$. Such fields are determined by their boundary values, so a natural question is what imposed potentials on the boundary provide good approximations, in an energy norm, to $F$? Very often the boundary includes surface patches that are conductors as well as patches where nonzero potentials may be imposed. This may be regarded as a problem of stationary control or approximation.

The difficulty with such problems has been how to work with the control space of allowable boundary conditions as it will be a trace space of allowable $H^1$ functions. The standard Lions-Magenes description of trace spaces is not amenable to nice constructions of solutions for problems of this type. This problem is treated here using methods based on the spectral characterization of trace spaces as described in Auchmuty [3] and [4] which provides constructive methods and explicit bases for the traces. The allowable trace space is characterized as being isomorphic to a class of weak solutions of a linear elliptic equation and a basis of $\Sigma$-Steklov eigenfunctions of this space is identified.

Here an explicit expression for the best approximation in terms of the data is found. Moreover boundary data that is close in a boundary norm to this optimal solution will provide good approximations to the field $F$ in an energy norm on $\Omega$. The analysis described here is described for quite general N-dimensional regions since the results are essentially independent of the dimension $N \ge 2$ and the methods may be of interest for other approximation questions.

<footnote>** This research has been supported in part by NSF award DMS 11008754.

*2010 Mathematics Subject classification.* Primary 35Q60, Secondary 35J25, 78A30.

*Key words and phrases.* Boundary control, Trace Spaces, Best approximation, Steklov eigenproblems.</footnote>
## 2 Definitions and Requirements.

To analyze this problem, standard definitions, terminology and assumptions will be used as in Attouch, Buttazzo and Michaille [1]. All functions in this paper will take values in $\mathbb{R} := [-\infty, \infty]$, derivatives should be taken in a weak sense and $N \ge 2$ throughout.

A region is a non-empty, connected, open subset of $\mathbb{R}^N$. Its closure is denoted $\bar{\Omega}$ and its boundary is $\partial\Omega := \bar{\Omega}\setminus\Omega$. Let $L^p(\Omega)$, $H^1(\Omega)$ be the usual real Lebesgue and Sobolev spaces of functions on $\Omega$. The norm on $L^p(\Omega)$ is denoted $\|\cdot\|_p$ and the inner product on $L^2(\Omega)$ by $\langle\cdot, \cdot\rangle$. The basic requirement on $\Omega$ is

(B1): $\Omega$ is a bounded region in $\mathbb{R}^N$ whose boundary $\partial\Omega$ is the union of a finite number of disjoint closed Lipschitz surfaces; each surface having finite surface area.

The region $\Omega$ is said to satisfy Rellich's theorem provided the imbedding of $H^1(\Omega)$ into $L^p(\Omega)$ is compact for $1 \le p < p_S$ where $p_S(N) := 2N/(N-2)$ when $n \ge 3$, or $p_S(2) = \infty$ when $N=2$.

The trace map is the linear extension of the map restricting Lipschitz continuous functions on $\bar{\Omega}$ to $\partial\Omega$. When (B1) holds, this map has an extension to $W^{1,1}(\Omega)$ and then the trace of $u$ on $\partial\Omega$ will be Lebesgue integrable with respect to $\sigma$, see [5], Section 4.2 for details. The region $\Omega$ is said to satisfy the compact trace theorem provided the trace mapping $\gamma: H^1(\Omega) \to L^2(\partial\Omega, d\sigma)$ is compact. We will use the inner product

$$[u, v]_{\partial} := \int_{\Omega} \nabla u(x) \cdot \nabla v(x) \, dx + \int_{\partial\Omega} \gamma(u) \gamma(v) \, d\sigma \quad (2.1)$$

on $H^1(\Omega)$ and the associated norm is denoted $\|u\|_{\partial}$. This is an equivalent inner product to the usual inner product when $\Omega$ obeys (B1) - see [2] for a proof. Here $\nabla u := (D_1 u, \dots, D_n u)$ is the gradient of the function $u$

Our interest is in a problem that arises in electrostatics where part of the boundary $\Sigma$ is a conductor and a potential can be imposed on the complementary part of the boundary $\tilde{\Sigma} := \partial\Omega \setminus \bar{\Sigma}$. Mathematically our requirements are (B2): $\Sigma$ is an nonempty open subset of $\partial\Omega$, $\Sigma$ and $\tilde{\Sigma}$ have strictly positive surface measure and $\sigma(\partial\Sigma) = 0$.

A function $u \in H^1(\Omega)$ is said to be in $H_{\Sigma_0}^1(\Omega)$ provided $\gamma(u) = 0$, $\sigma$ a.e. on $\Sigma$. This is equivalent to requiring that

$$\int_{\partial\Omega} \gamma(u) \gamma(v) \, d\sigma = 0 \quad \text{for all} \quad v \in X_{\Sigma} \quad (2.2)$$

Let $X$ be the space $H^1(\Omega) \cap C(\bar{\Omega})$ and $X_{\Sigma}$ be the subspace of functions in $X$ with $\text{supp } v \cap \partial\Omega \subset \Sigma$. The space $H_{\Sigma_0}^1(\Omega)$ is a closed subspace of $H^1(\Omega)$ that contains $H_0^1(\Omega)$.
### 3 The Boundary Control Problem

The problem to be studied here is given a vector field $\mathbf{F}$ on $\Omega$ to find the potential $\varphi$ that provides the best $L^2$-approximation when the region $\Omega$ has known permittivity tensor $\epsilon(\cdot)$ and part of the boundary $\Sigma$ is a conductor held at zero potential. That is we want to find the function $\tilde{\varphi}$ that minimizes

$$ \| \epsilon \nabla \varphi - \mathbf{F} \|_{2}^{2} := \int_{\Omega} |\epsilon \nabla \varphi - \mathbf{F}|^{2} dx \quad \text{over all } \varphi \in H_{\Sigma 0}^{1}(\Omega). \quad (3.3) $$

with $|\cdot|$ is the Euclidean norm on $\mathbb{R}^N$. Since $\mathbf{F}$ is known this reduces to minimizing the functional

$$ \mathcal{E}(\varphi) := \int_{\Omega} [ (A(x)\nabla\varphi) \cdot \nabla\varphi - 2 \mathbf{G} \cdot \nabla\varphi ] dx \quad \text{over all } \varphi \in H_{\Sigma 0}^{1}(\Omega). \quad (3.4) $$

where $A(x) := \epsilon(x)^T \epsilon(x)$ is real symmetric, $\mathbf{G}(x) := \epsilon(x)^T \mathbf{F}(x)$ on $\Omega$ and the superscript T denotes the vector transpose. The following will be assumed. (A1): $A(x) := (a_{jk}(x))$ is a real symmetric matrix whose components are bounded Lebesgue-measurable functions on $\Omega$ and there exist constants $c_0, c_1$ such that

$$ c_0 |\xi|^2 \le \xi^T A(x) \xi \le c_1 |\xi|^2 \quad \text{for all } \xi \in \mathbb{R}^N, x \in \Omega. \quad (3.5) $$

Existence uniqueness and extremality conditions for this problem may be obtained using standard methods. The problem is a convex quadratic minimization problem on a Hilbert space so the existence may be stated as follows.

**Theorem 31** Assume that (A1), (B1), (B2) hold and $\mathbf{F} \in L^2(\Omega : \mathbb{R}^N)$ is given, then there is a unique minimizer $\tilde{\varphi}$ of $\mathcal{E}$ on $H_{\Sigma 0}^{1}(\Omega)$.

The functional $\mathcal{E}$ also is $\mathbf{G}$-differentiable so the minimizers satisfy the following.

**Theorem 32** Assume that (A1), (B1), (B2) hold and $\mathbf{F} \in L^2(\Omega : \mathbb{R}^N)$ is given, then the minimizer $\tilde{\varphi}$ of $\mathcal{E}$ on $H_{\Sigma 0}^{1}(\Omega)$ satisfies the equation

$$ \int_{\Omega} (A(x)\nabla\varphi - \mathbf{G}) \cdot \nabla\psi \, dx = 0 \quad \text{for all } \psi \in H_{\Sigma 0}^{1}(\Omega). \quad (3.6) $$

To obtain further results about this problem a decomposition of the space $H_{\Sigma 0}^{1}(\Omega)$ will be used. Consider the bilinear form $a : H_{\Sigma 0}^{1}(\Omega) \to \mathbb{R}$ defined by

$$ a(u, v) := [u, v]_a := \int_{\Omega} (A \nabla u) \cdot \nabla v \, d^N x + \int_{\tilde{\Sigma}} \gamma(u) \gamma(v) \, d\sigma \quad (3.7) $$

This bilinear form defines the a-inner product on $H_{\Sigma 0}^{1}(\Omega)$ and is equivalent to the $\partial$-norm (2.1) when $A$ satisfies (A1).
Observe that a function $w \in H_{\Sigma_0}^1(\Omega)$ is a-orthogonal to $H_0^1(\Omega)$ if and only if

$$ \int_{\Omega} (A\nabla w) \cdot \nabla v \, dx = 0 \quad \text{for all} \quad v \in H_0^1(\Omega). \qquad (3.8) $$

That is $\gamma(w)$ is zero on $\Sigma$ and $\mathcal{L}_A w(x) := \text{div}(A\nabla w)(x) = 0$ in a weak sense on $\Omega$. The space of all such functions will be denoted $N(\mathcal{L}_A, \Sigma)$ so we have the orthogonal decomposition

$$ H_{\Sigma_0}^1(\Omega) = H_0^1(\Omega) \oplus_a N(\mathcal{L}_A, \Sigma) \qquad (3.9) $$

where $\oplus_a$ indicates that the a-inner product is used.

In light of this result, the minimizer $\tilde{\varphi}$ has a decomposition of the form $\tilde{\varphi} = \phi_0 + \phi_b$ where $\phi_0$ is the minimizer of $\mathcal{E}$ on $H_0^1(\Omega)$ and $\phi_b \in N(\mathcal{L}_A, \Sigma)$ is the solution of

$$ \int_{\Omega} (A\nabla\varphi) \cdot \nabla\psi \, dx = \int_{\Omega} (G - A\nabla\varphi_0) \cdot \nabla\psi \, dx \quad \text{for all} \quad \psi \in \mathbf{N}(\mathcal{L}_A, \Sigma). \qquad (3.10) $$

The fact that $\phi_0$ is a solution of the extremality condition on $H_0^1(\Omega)$ implies that $\text{div}(A\nabla\varphi_0 - G) = 0$ on $\Omega$ in a weak sense so this last equation may be written

$$ \int_{\Omega} (A\nabla\varphi) \cdot \nabla\psi \, dx = \int_{\tilde{\Sigma}} \psi (G - A\nabla\varphi_0) \cdot \nu \, d\sigma \quad \text{for all} \quad \psi \in \mathbf{N}(\mathcal{L}_A, \Sigma). \qquad (3.11) $$

This is the weak form of the equation $\mathcal{L}_A\varphi = 0$ on $\Omega$ subject to the boundary conditions

$$ \varphi(z) = 0 \quad \text{on} \quad \Sigma \quad \text{and} \quad A\nabla\varphi \cdot \nu = (G - A\nabla\varphi_0) \cdot \nu \quad \text{on} \quad \tilde{\Sigma}. \qquad (3.12) $$

The solution of the problem for $\phi_0$ is a standard Dirichlet boundary value problem and it is worth noting that the value of $\mathcal{E}(\phi_0) = 0$ if and only if $\text{div} G = 0$ on $\Omega$ in a weak sense. In this case the general problem reduces to that of solving (3.10) or (3.11) alone.

Our interest is in the problem of finding an expression for the boundary trace of $\tilde{\varphi}$ or $\phi_b$ on $\tilde{\Sigma}$. That is what boundary data gives the best approximating potential for the given field $F$ on $\Omega$?

# 4 Bases and Representations of $N(\mathcal{L}_A, \Sigma)$

To find the boundary data that provides the best $L^2$-approximation to the field $F$, an orthogonal basis of the space consisting of certain Steklov-type eigenfunctions of $\mathcal{L}_A$ is constructed and used. This will yield a spectral representation of $\phi_b$ as described in theorem 5.xx below.
An a-orthonormal basis of $N(\mathcal{L}_A, \Sigma)$ may be found using the algorithm described in Auchmuty [4]. In the notation of that paper take $V = N(\mathcal{L}_A, \Sigma)$, $a$ as above and

$$m(u, v) := \int_{\tilde{\Sigma}} \gamma(u) \gamma(v) \, d\sigma. \qquad (4.13)$$

A function $\chi \in N(\mathcal{L}_A, \Sigma)$ is said to be a $\Sigma$-Steklov eigenfunction of $\mathcal{L}_A$ on $\Omega$ provided it is a solution of

$$\int_{\Omega} (A \nabla \chi) \cdot \nabla \psi \, dx = \lambda \int_{\tilde{\Sigma}} \gamma(\chi) \gamma(\psi) \, d\sigma \quad \text{for all } \psi \in H_{\Sigma 0}^{1}(\Omega). \quad (4.14)$$

This problem has the form of equation 2.1 of [4] and the bilinear forms $a, m$ satisfy conditions (A1)-(A4) of that paper. Moreover condition (A5) there holds with $H = L^2(\tilde{\Sigma}, d\sigma)$.

Define $\mathcal{A}(\chi) = a(\chi, \chi)$, $\mathcal{M}(\chi) := m(\chi, \chi)$ and $C_1$ to be the closed unit ball in $H_{\Sigma 0}^1(\Omega)$ with respect to the a-norm. Consider the variational problem of maximizing $\mathcal{M}$ on $C_1$. This problem has maximizers $\pm \chi_1$ that have a-norm 1 and are solutions of (4.14) associated with an eigenvalue $\lambda_1 > 0$. Moreover one has the coercivity inequality

$$\mathcal{A}(\chi) \geq (\lambda_1 + 1) \int_{\tilde{\Sigma}} \gamma(\chi)^2 \, d\sigma \quad \text{for all } \chi \in H_{\Sigma 0}^1(\Omega). \quad (4.15)$$

Using the construction of section 4 of [4], a countably infinite a-orthonormal basis $\mathcal{B} := \{\chi_j : j \geq 1\}$ of $N(\mathcal{L}_A, \Sigma)$ may be constructed using a sequence of constrained maximization problems for $\mathcal{M}$.

These eigenfunctions also are m-orthogonal so that $m(\chi_j, \chi_k) = 0$ when $j \neq k$. Define $\tilde{\chi}_j := \chi_j / \sqrt{\lambda_j + 1}$ then $\tilde{\mathcal{B}} := \{\gamma(\tilde{\chi}_j) : j \geq 1\}$ will be an m-orthonormal basis of $L^2(\tilde{\Sigma}, d\sigma)$ from theorem 4.6. Since these functions constitute bases of the various Hilbert spaces, there is a spectral representation of functions $\varphi \in N(\mathcal{L}_A, \Sigma)$ in terms of their boundary values. Namely when $\varphi \in N(\mathcal{L}_A, \Sigma)$ then,

$$\varphi(x) = \sum_{j=1}^{\infty} c_j \tilde{\chi}_j(x) \quad \text{on } \Omega \quad \text{with } c_j := m(\varphi, \tilde{\chi}_j) \quad (4.16)$$

and this series converges strongly in $H_{\Sigma 0}^1(\Omega)$. Thus these constructions yield the following result.

**Theorem 41** Assume that (A1), (B1), (B2) hold and $\mathcal{B}, \tilde{\mathcal{B}}$ are defined as above. If $\varphi \in N(\mathcal{L}_A, \Sigma)$ then (4.16) holds and the series converges in a-norm. Moreover $a(\varphi, \varphi) = \sum_{j=1}^{\infty} (1 + \lambda_j) c_j^2$.

Let $H^{1/2}(\tilde{\Sigma})$ be the subspace of $L^2(\tilde{\Sigma}, d\sigma)$ of all functions with $\sum_{j=1}^{\infty} (1 + \lambda_j) c_j^2 < \infty$. It will be a Hilbert space with respect to the inner product

$$\langle \varphi, \psi \rangle_{1/2, \tilde{\Sigma}} := \sum_{j=1}^{\infty} (1 + \lambda_j) m(\phi, \chi_j) m(\psi, \chi_j) \quad (4.17)$$
In particular this yields an isomorphism between functions in $N(\mathcal{L}_A, \Sigma)$ and functions in $H^{1/2}(\tilde{\Sigma})$. Thus $H^{1/2}(\tilde{\Sigma})$ may be regarded as the boundary trace subspace (on $\tilde{\Sigma}$) of functions in $N(\mathcal{L}_A, \Sigma)$ and this is an isometry with $a(\varphi, \varphi) = \langle\varphi, \varphi\rangle_{1/2, \tilde{\Sigma}}^2$.

# 5 The best approximating potential

We are now in a position to specify the boundary data for the potential that minimizes (3.3). The preceding analysis enables the derivation of an explicit representation of the solution $\phi_b$ of (3.10) or (3.11). Equation (3.10) implies that $\phi_b$ satisfies

$$a(\phi_b, \tilde{\chi}_j) = g_j := \int_{\Omega} (\mathbf{G} - \Lambda \nabla \varphi_0) \cdot \nabla \tilde{\chi}_j \, \mathrm{d}\mathbf{x} \quad \text{for } \mathbf{j} \ge \mathbf{1}.$$

Note that the $g_j$ depend only on the data, the eigenfunction $\tilde{\chi}_j$ and the solution $\phi_0$ of the zero-Dirichlet variational problem. Then the eigenfunction equation (4.14) yields that the solution is

$$\phi_b(x) = \sum_{j=1}^{\infty} \frac{g_j}{(1 + \lambda_j)} \tilde{\chi}_j(x). \qquad (5.18)$$

Thus the boundary trace on $\tilde{\Sigma}$ of the best approximation is given by the boundary trace of this right hand side. That is imposing Dirichlet boundary data $\gamma(\phi_b)$ on $\tilde{\Sigma}$ given by (5.18) yields the minimizing potential $\phi_b$ of $\mathcal{E}$.

**Theorem 51** Assume that (A1), (B1), (B2) hold and $\tilde{B}, g_j$ are defined as above. Then the potential $\tilde{\varphi}$ that minimizes the norm in (3.3) or $\mathcal{E}$ on $H_{\Sigma_0}^1(\Omega)$ is given by $\tilde{\varphi} = \phi_0 + \phi_b$ where $\phi_0$ minimizes $\mathcal{E}$ on $H_0^1(\Omega)$ and $\phi_b$ is given by (5.18). When $\mathrm{div} \mathbf{G} = \mathbf{0}$ on $\Omega$, then $\mathbf{F} = \epsilon \nabla \tilde{\varphi}$ on $\Omega$.

*Proof.* This is a restatement of the preceding results. Note that when $\mathrm{div} \mathbf{G} = \mathbf{0}$ on $\Omega$, then $\phi_0 = 0$ so the last sentence holds.

Moreover when the boundary potential $\varphi$ is a good approximation of this $\phi_b$ in the norm of $H^{1/2}(\tilde{\Sigma})$ then the fact that the a-norm and the norm on $H^{1/2}(\tilde{\Sigma})$ are isometric implies that such potentials $\varphi$ will provide a good approximation of $\mathbf{F}$ on $\Omega$ in any equivalent norm on $H_{\Sigma_0}^1(\Omega)$.
References

[1] H. Attouch, G. Buttazzo and G. Michaille, *Variational Analysis in Sobolev and BV Spaces*, SIAM Publications, Philadelphia (2006).

[2] G. Auchmuty, "Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems", Numerical Functional Analysis and Optimization, **25** (2004) 321-348.

[3] G. Auchmuty, "Spectral Characterizations of the Trace Spaces $H^s(\partial\Omega)$", SIAM J Math Analysis **38** (2006), 894-905.

[4] G. Auchmuty, "Bases and Comparison Results for Linear Elliptic Eigenproblems," J. Math. Anal & Appns, **390** (2012), 394-406.

[5] L. C. Evans and R. F. Gariepy, *Measure Theory and Fine Properties of Functions*, CRC Press, Boca Raton (1992).