Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

Abstract

Many physical equations have the form $\mathbf{J }(\mathbf{x })=\mathbf{L }(\mathbf{x })\mathbf{E }(\mathbf{x })-\mathbf{h }(\mathbf{x })$ with source $\mathbf{h }(\mathbf{x })$ and fields $\mathbf{E }$ and $\mathbf{J }$ satisfying differential constraints, symbolized by $\mathbf{E }\in \mathcal E$, $\mathbf{J }\in \mathcal J$ where $\mathcal E$, $\mathcal J$ are orthogonal spaces. We show that if $\mathbf{L }(\mathbf{x })$ takes values in certain nonlinear manifolds $\mathcal M$, and coercivity and boundedness conditions hold, then the infinite body Green’s function (fundamental solution) satisfies exact identities. The theory also links Green’s functions of different problems. The analysis is based on the theory of exact relations for composites, but without assumptions about the length scales of variations in $\mathbf{L }(\mathbf{x })$, and more general equations, such as for waves in lossy media, are allowed. For bodies $\Omega$, inside which $\mathbf{L }(\mathbf{x })\in \mathcal{M}$, the “Dirichlet-to-Neumann map” giving the response also satisfies exact relations. These boundary field equalities generalize the notion of conservation laws: the field inside $\Omega$ satisfies certain constraints that leave a wide choice in these fields, but which give identities satisfied by the boundary fields, and moreover provide constraints on the fields inside the body. A consequence is the following: if a matrix-valued field $\mathbf{Q }(\mathbf{x })$ with divergence-free columns takes values within $\Omega$ in a set $\mathcal B$ (independent of $\mathbf{x }$) that lies on a nonlinear manifold, we find conditions on the manifold, and on $\mathcal B$, that with appropriate conditions on the boundary fluxes $\mathbf{q }(\mathbf{x })=\mathbf{n }(\mathbf{x })\cdot \mathbf{Q }(\mathbf{x })$ (where $\mathbf{n }(\mathbf{x })$ is the outward normal to $\partial \Omega$) force $\mathbf{Q }(\mathbf{x })$ within $\Omega$ to take values in a subspace $\mathcal D$. This forces $\mathbf{q }(\mathbf{x })$ to take values in $\mathbf{n }(\mathbf{x })\cdot \mathcal D$. We find there are additional divergence-free fields inside $\Omega$ that in turn generate additional boundary field equalities. Consequently, there exist partial null Lagrangians, functionals $F(\mathbf{w },\nabla \mathbf{w })$ of a vector potential $\mathbf{w }$ and its gradient that act as null Lagrangians when $\nabla \mathbf{w }$ is constrained for $\mathbf{x }\in \Omega$ to take values in certain sets $\mathcal A$, of appropriate nonlinear manifolds, and when $\mathbf{w }$ satisfies appropriate boundary conditions. The extension to certain nonlinear minimization problems is also sketched.

Appendix: Some physical equations that can be expressed in the required canonical form

Here we give examples of some physical equations that can be expressed in the required canonical form. The examples are by no means comprehensive: for further examples, see [26] Chap. 2, [35] Chap. 1, [27, 28] and the appendix of [29]. In the equations that follow we omit the source terms. We emphasize that if one is interested in exact relations satisfied by the DtN map and the associated boundary field equalities, then it is not necessary that the source terms have a physical significance provided they are localized outside the body $\Omega$.

The simplest canonical equations are those of electrical conductivity

$$\begin{aligned} \underbrace{\mathbf{j }(\mathbf{x })}_{\mathbf{J }(\mathbf{x })}=\underbrace{{\varvec{\sigma }}(\mathbf{x })}_{\mathbf{L }(\mathbf{x })}\underbrace{\mathbf{e }(\mathbf{x })}_{\mathbf{E }(\mathbf{x })},\quad \nabla \cdot \mathbf{j }=0, \quad \mathbf{e }=-\nabla V=0, \end{aligned}$$

(10.4)

where $\mathbf{j }(\mathbf{x })$ and $\mathbf{e }(\mathbf{x })$ are the electrical current and electric field and $V(\mathbf{x })$ is the electrical potential. The boundary fields $\partial \mathbf{J }$ and $\partial \mathbf{E }$ are the flux $\mathbf{n }\cdot \mathbf{j }(\mathbf{x })$ and boundary voltage $V(\mathbf{x })$, respectively, with $\mathbf{x }\in \partial \Omega$ and $\mathbf{n }$ being the outward normal to $\Omega$. As displayed in the table at the beginning of Sect. 2.1 in [26] (adapted from one of Batchelor [4]), the equations for dielectrics, magnetostatics, heat conduction, particle diffusion, flow in porous media, and antiplane elasticity all take the same form as (10.4) and so any analysis applicable to (10.4) applies to them as well.

Another important example is that of linear elasticity,

$$\begin{aligned} \underbrace{{\varvec{\sigma }}(\mathbf{x })}_{\mathbf{J }(\mathbf{x })}=\underbrace{{\varvec{\mathcal {C}}}(\mathbf{x })}_{\mathbf{L }(\mathbf{x })}\underbrace{{\varvec{\epsilon }}(\mathbf{x })}_{\mathbf{E }(\mathbf{x })},\quad \nabla \cdot {\varvec{\sigma }}=0,\quad {\varvec{\epsilon }}=[\nabla \mathbf{u }+(\nabla \mathbf{u })^T]/2, \end{aligned}$$

(10.5)

where ${\varvec{\sigma }}(\mathbf{x })$ (not to be confused for the conductivity tensor field) is the stress, ${\varvec{\epsilon }}(\mathbf{x })$ is the strain, $\mathbf{u }(\mathbf{x })$ is the displacement, and ${\varvec{\mathcal {C}}}(\mathbf{x })$ is the elasticity tensor field. The boundary fields $\partial \mathbf{J }(\mathbf{x })$ and $\partial \mathbf{E }(\mathbf{x })$ are the traction $\mathbf{n }\cdot {\varvec{\sigma }}(\mathbf{x })$ and boundary displacement field $\mathbf{u }(\mathbf{x })$, respectively.

It is also possible to have equations that couple fields together, such as the magnetoelectric equations,

$$\begin{aligned} \underbrace{\begin{pmatrix}\mathbf{d }\\ \mathbf{b }\end{pmatrix}}_{\mathbf{J }(\mathbf{x })}=\underbrace{\begin{pmatrix}{\varvec{\varepsilon }}&{}\quad {\varvec{\beta }}\\ {\varvec{\beta }}^T &{}\quad {\varvec{\mu }}\end{pmatrix}}_{\mathbf{L }(\mathbf{x })}\underbrace{\begin{pmatrix}\mathbf{e }\\ \mathbf{h }\end{pmatrix}}_{\mathbf{E }(\mathbf{x })},\quad \nabla \cdot \mathbf{d }=\nabla \cdot \mathbf{b }=0,\quad \mathbf{e }=-\nabla V,\quad \mathbf{h }=-\nabla \psi , \end{aligned}$$

(10.6)

where $\mathbf{d }$ and $\mathbf{b }$ are the electric displacement field and magnetic induction, $\mathbf{e }$ and $\mathbf{h }$ are the electric and magnetic fields, V and $\psi$ are the electric potential and magnetic scalar potential (assuming there are no free currents), ${\varvec{\varepsilon }}$(x) is the free-body electrical permittivity (with $\mathbf{h }=0$), ${\varvec{\beta }}(\mathbf{x })$ is the second-order magnetoelectric coupling tensor, ${\varvec{\mu }}(\mathbf{x })$ is the free-body magnetic permeability (with $\mathbf{e }=0$). The boundary fields $\partial \mathbf{J }(\mathbf{x })$ and $\partial \mathbf{E }(\mathbf{x })$ are then the flux pair $(\mathbf{n }\cdot \mathbf{d }(\mathbf{x }),\mathbf{n }\cdot \mathbf{b }(\mathbf{x }))$ and the potential pair $(V(\mathbf{x }),\psi (\mathbf{x }))$, respectively, with $\mathbf{x }\in \partial \Omega$.

Thermoelectricity also takes this form, but one has to be careful in defining the fields to ensure that the associated tensor $\mathbf{L }(\mathbf{x })$ is symmetric (see, e.g., Sect. 2.4 in [26]).

Fields that are coupled together need not have the same tensorial rank, an example being the equations of piezoelectricity,

$$\begin{aligned} \underbrace{\begin{pmatrix}{\varvec{\epsilon }}\\ \mathbf{d }\end{pmatrix}}_{\mathbf{J }(\mathbf{x })}=\underbrace{\begin{pmatrix}{\varvec{\mathcal {S}}}&{}\quad {\varvec{\mathcal {D}}}\\ {\varvec{\mathcal {D}}}^T &{}\quad {\varvec{\varepsilon }}\end{pmatrix}}_{\mathbf{L }(\mathbf{x })} \underbrace{\begin{pmatrix}{\varvec{\sigma }}\\ \mathbf{e }\end{pmatrix}}_{\mathbf{E }(\mathbf{x })}, \end{aligned}$$

(10.7)

where ${\varvec{\mathcal {S}}}(\mathbf{x })$ is the compliance tensor under short-circuit boundary conditions (i.e., with $\mathbf{e }=0$), ${\varvec{\mathcal {D}}}(\mathbf{x })$ is the piezoelectric stress coupling tensor, and ${\varvec{\varepsilon }}(\mathbf{x })$ is the free-body dielectric tensor (i.e., with ${\varvec{\tau }}=0$). The strain field ${\varvec{\epsilon }}$, electric displacement field $\mathbf{d }$, stress field ${\varvec{\sigma }}$, and electric field $\mathbf{e }$ satisfy the usual differential constraints:

$$\begin{aligned} {\varvec{\epsilon }}= [\nabla \mathbf{u }+(\nabla \mathbf{u })^T]/2,\quad \nabla \cdot \mathbf{d }= 0,\quad \nabla \cdot {\varvec{\sigma }}= 0,\quad \mathbf{e }= -\nabla V. \end{aligned}$$

(10.8)

Since the stresses and strains are symmetric matrices, ${\varvec{\mathcal {D}}}$ is a third-order tensor that maps vectors to symmetric matrices. The boundary fields $\partial \mathbf{J }(\mathbf{x })$ and $\partial \mathbf{E }(\mathbf{x })$ are the displacement, flux pair $(\mathbf{u }(\mathbf{x }),\mathbf{n }\cdot \mathbf{d }(\mathbf{x }))$ and the traction, voltage pair $(\mathbf{n }\cdot {\varvec{\sigma }},V)$, respectively.

Of course, more than two fields can be coupled together. Thus, by combining a piezoelectric material and a magnetostrictive material in a composite, we can obtain a material where there is coupling between electric fields, elastic fields, and magnetic fields.

For fields varying in time at constant frequency $\omega$, with wavelengths and attenuation lengths much bigger than the size of the body under consideration, the quasistatic equations are applicable. For dielectrics these take the same form as (10.4):

$$\begin{aligned} \mathbf{d }(\mathbf{x })={\varvec{\varepsilon }}(\mathbf{x })\mathbf{e }(\mathbf{x }),\quad \nabla \cdot \mathbf{j }=0, \quad \mathbf{e }=-\nabla V=0, \end{aligned}$$

(10.9)

where everything is now complex-valued: $\mathbf{d }(\mathbf{x })$ and $\mathbf{e }(\mathbf{x })$ are the complex-valued electrical displacement field and electric field, $V(\mathbf{x })$ is the complex-valued electrical potential, and ${\varvec{\varepsilon }}(\mathbf{x })$ is the complex-valued electrical permittivity. (The physical displacement field, electric field, and potential are the real parts of $\mathbf{d }(\mathbf{x })e^{-i\omega t}$, $\mathbf{e }(\mathbf{x })e^{-i\omega t}$, and $V e^{-i\omega t}$, respectively). Let us set

$$\begin{aligned} \mathbf{d }=\mathbf{d }'+i\mathbf{d }'',\quad \mathbf{e }=\mathbf{e }'+i\mathbf{e }'',\quad V=V'+iV'',\quad {\varvec{\varepsilon }}={\varvec{\varepsilon }}'+{\varvec{\varepsilon }}'', \end{aligned}$$

(10.10)

where the primed fields denote the real parts, while the doubled primed fields denote the imaginary parts. Physically, ${\varvec{\varepsilon }}''(\mathbf{x })$ is associated with electrical energy loss into heat and is positive semidefinite. Assuming it is positive definite and that an inverse $[{\varvec{\varepsilon }}'']^{-1}$ exists, substitution of (10.10) in (10.9), followed by suitable manipulation, gives the equivalent coupled field equations of Gibiansky and Cherkaev [7]:

$$\begin{aligned} \underbrace{\begin{pmatrix}\mathbf{e }'' \\ \mathbf{d }'' \\ \end{pmatrix}}_{\mathbf{J }(\mathbf{x })}= \underbrace{\begin{pmatrix}[{\varvec{\varepsilon }}'']^{-1} &{}\quad [{\varvec{\varepsilon }}'']^{-1} {\varvec{\varepsilon }}' \\ {\varvec{\varepsilon }}'[{\varvec{\varepsilon }}'']^{-1} &{}\quad {\varvec{\varepsilon }}'' + {\varvec{\varepsilon }}'[{\varvec{\varepsilon }}'']^{-1} {\varvec{\varepsilon }}' \end{pmatrix}}_{\mathbf{L }(\mathbf{x })}\underbrace{\begin{pmatrix}-\mathbf{d }' \\ \mathbf{e }' \\ \end{pmatrix}}_{\mathbf{E }(\mathbf{x })}, \end{aligned}$$

(10.11)

Clearly $\mathbf{L }$ is real and symmetric and by inspection of the quadratic form associated with $\mathbf{L }$ one sees that it is positive definite. Now $\partial \mathbf{J }$ consists of the voltage, flux pair $(V'',\mathbf{n }\cdot \mathbf{d }'')$ while $\partial \mathbf{E }$ consists of the flux, voltage pair $(-\mathbf{n }\cdot \mathbf{d }',V')$. As Gibiansky and Cherkaev show, similar manipulations can be done for viscoelasticity in the quasistatic limit where the equations have the form (10.5), but with all fields being complex. More generally, the Gibiansky–Cherkaev approach can be applied to equations where the tensor entering the constitutive law is not self-adjoint, but its self-adjoint part is positive definite, to an equivalent form where the tensor $\mathbf{L }(\mathbf{x })$ entering the constitutive law is self-adjoint and positive definite [25] (see also Sect. 13.4 of [26]): such manipulations can be applied, for example, to electrical conduction in the presence of a magnetic field where, due to the Hall effect, the conductivity tensor ${\varvec{\sigma }}(\mathbf{x })$ entering (10.4) is not symmetric.

Wave equations, can be expressed in the form (4.12) with an identity like (4.15) holding. For example, at fixed frequency $\omega$ with a $\mathrm{{e}}^{-i\omega t}$ time dependence, as recognized in [32] the acoustic equations, with $P(\mathbf{x })$ the (complex) pressure, $\mathbf{v }(\mathbf{x })$ the (complex) velocity, ${\varvec{\rho }}(\mathbf{x },\omega )$ the effective mass density matrix, and $\kappa (\mathbf{x },\omega )$ the bulk modulus, take the form

$$\begin{aligned} \underbrace{\begin{pmatrix}-i\mathbf{v }\\ -i\nabla \cdot \mathbf{v }\end{pmatrix}}_{\mathbf{J }(\mathbf{x })} =\underbrace{\begin{pmatrix}-(\omega {\varvec{\rho }})^{-1} &{}\quad 0 \\ 0 &{}\quad \omega /\kappa \end{pmatrix}}_{\mathbf{L }(\mathbf{x })}\underbrace{\begin{pmatrix}\nabla P \\ P\end{pmatrix}}_{\mathbf{E }(\mathbf{x })}, \end{aligned}$$

(10.12)

(and $\partial \mathbf{E }$ and $\partial \mathbf{J }$ can be identified with the boundary values of $P(\mathbf{x })$ and $\mathbf{n }\cdot \mathbf{v }(\mathbf{x })$ at $\partial \Omega$, respectively). Here we allow for effective mass density matrices that, at a given frequency, can be anisotropic and complex-valued as may be the case in metamaterials [30, 33, 44, 50]. Maxwell’s equations, with $\mathbf{e }(\mathbf{x })$ the (complex) electric field, $\mathbf{h }(\mathbf{x })$ the (complex) magnetizing field, ${\varvec{\mu }}(\mathbf{x },\omega )$ the magnetic permeability, ${\varvec{\varepsilon }}(\mathbf{x })$ the electric permittivity, take the form

$$\begin{aligned} \underbrace{\begin{pmatrix}-i\mathbf{h }\\ i\nabla \times \mathbf{h }\end{pmatrix}}_{\mathbf{J }(\mathbf{x })} =\underbrace{\begin{pmatrix}-{[\omega {\varvec{\mu }}]}^{-1} &{}\quad 0 \\ 0 &{}\quad \omega {\varvec{\varepsilon }}\end{pmatrix}}_{\mathbf{L }(\mathbf{x })} \underbrace{\begin{pmatrix}\nabla \times \mathbf{e }\\ \mathbf{e }\end{pmatrix}}_{\mathbf{E }(\mathbf{x })}, \end{aligned}$$

(10.13)

(and $\partial \mathbf{E }$ and $\partial \mathbf{J }$ can be identified with the tangential values of $\mathbf{e }(\mathbf{x })$ and $\mathbf{h }(\mathbf{x })$ at $\partial \Omega$, respectively). The linear elastodynamic equations, with $\mathbf{u }(\mathbf{x })$ the (complex) displacement, ${\varvec{\sigma }}(\mathbf{x })$ the (complex) stress, ${\varvec{\mathcal {C}}}(\mathbf{x },\omega )$ the elasticity tensor, ${\varvec{\rho }}(\mathbf{x },\omega )$ the effective mass density matrix, take the form

$$\begin{aligned} \underbrace{\begin{pmatrix} -{\varvec{\sigma }}/\omega \\ -\nabla \cdot {\varvec{\sigma }}/\omega \end{pmatrix}}_{\mathbf{J }(\mathbf{x })} =\underbrace{\begin{pmatrix}-{\varvec{\mathcal {C}}}/\omega &{}\quad 0 \\ 0 &{}\quad \omega {\varvec{\rho }}\end{pmatrix}}_{\mathbf{L }(\mathbf{x })}\underbrace{\begin{pmatrix}/2 \\ \mathbf{u }\end{pmatrix}}_{\mathbf{E }(\mathbf{x })}, \end{aligned}$$

(10.14)

(and $\partial \mathbf{E }$ and $\partial \mathbf{J }$ can be identified with the values of $\mathbf{u }(\mathbf{x })$ and the traction $\mathbf{n }\cdot {\varvec{\sigma }}(\mathbf{x })$ at $\partial \Omega$, respectively). The preceeding three equations have been written in this form so ${\mathrm{Im}}\mathbf{L }(\mathbf{x })\ge 0$ when ${\mathrm{Im}}\omega \ge 0$, where complex frequencies have the physical meaning of the solution increasing exponentially in time. Under assumptions that the material moduli are lossy, or that the frequency $\omega$ is complex with positive imaginary part, one can easily manipulate them into equivalent forms similar to the Gibiansky–Cherkaev form in (10.11) with a positive semidefinite tensor entering the constitutive law [32, 34]. Of course, the boundary fields $\partial \mathbf{E }$ and $\partial \mathbf{J }$ then need to be appropriately redefined.

For thin plates, the dynamic plate equations at constant frequency can be written in the form

$$\begin{aligned} \underbrace{\begin{pmatrix} i\mathbf{M }\\ \nabla \cdot (\nabla \cdot \mathbf{M }) \end{pmatrix}}_{\mathbf{J }(\mathbf{x })} = \underbrace{\begin{pmatrix} -{\varvec{\mathcal {D}}}(\mathbf{x })/\omega &{}\quad 0 \\ 0 &{}\quad h(\mathbf{x })\omega \rho (\mathbf{x }) \end{pmatrix}}_{\mathbf{L }(\mathbf{x })} \underbrace{\begin{pmatrix} \nabla \nabla v \\ i v \end{pmatrix}}_{\mathbf{E }(\mathbf{x })}. \end{aligned}$$

(10.15)

Here $\mathbf{M }(\mathbf{x },t)$ is the (complex) bending moment tensor, ${\varvec{\mathcal {D}}}(\mathbf{x })$ is the fourth-order tensor of plate rigidity coefficients, $h(\mathbf{x })$ is the plate thickness, $\rho (\mathbf{x })$ is the density, and $v=\partial w/\partial t$ is the velocity of the (complex) vertical deflection $w(\mathbf{x },t)$ of the plate. Note that the matrix $\mathbf{L }(\mathbf{x })$ has positive definite imaginary part when $\omega$ has positive imaginary part. $\partial \mathbf{E }$ can be identified with the boundary values of the pair $(\nabla v,v)$ while $\partial \mathbf{J }$ can be identified with the boundary values of the pair $(\mathbf{M }\mathbf{n },(\nabla \cdot \mathbf{M })\cdot \mathbf{n })$, in which $\mathbf{n }$ is the outward normal to $\partial \Omega$. Again, when the material moduli are lossy, or the frequency $\omega$ is complex with positive imaginary part, this can be manipulated into the Gibiansky–Cherkaev form in (10.11) with a positive semidefinite tensor entering the constitutive law, and with appropriately redefined boundary fields.

Further examples of wave equations at constant frequency that can be represented in the required form are given in the appendix of [29].