Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting from a simple case with known applications, and then generalising it to a nonlinear (actually, quasilinear) case, and then finally explain the inverse problem.

I admit to not having studied physics at university level. Hence, the notation and language I use might be exotic, for which I apologise.

Conductivity equation

Suppose we have a conducting body, $\Omega$, with conductivity (or conductance) $\sigma$. I will be using terminology from electricity, but this might also be heat conductance. Conductivity is the inverse of resistance, which I will not write more about. We take the material to be isotropic, which corresponds to conductivity with scalar (real) values.

By (differential) Ohm's law the current J is proportional to conductivity and electric field, which we write as the gradient of potential v: $$J = -\sigma \nabla v$$

By Kirchhoff's law the current is solenoidal, or divergence-free: $\nabla \cdot J = 0$.

All in all, we have the conductivity equation $$-\nabla \cdot \sigma \nabla v = 0$$ If the conductivity is constant, say one, this is simply the Laplace equation $-\nabla^2 v = -\Delta v = 0$.

p-Laplace equation

We consider a nonlinear Ohm's law: $$J = -\sigma |\nabla v|^{p-2} \nabla v$$ with finite parameter p > 1. The case p = 2 is the standard Ohm's law, and also the standard conductivity equation.

Combining this with Kirchhoff's law we get the p-Laplace equation (with weight $\sigma$): $$-\nabla \cdot (\sigma |\nabla v|^{p-2} \nabla v) = 0$$

Calderón's problem

The inverse problem of Calderón asks one to reconstruct the conductivity within the body by making boundary measurements: One prescribes either the current or the voltage on the boundary and measures the other one. These correspond to Neumann and Dirichlet boundary values for the equation - one sets the other and measures the other. This is a steady-state measurement, so one should wait for the current pattern to stabilise before making measurements.

For the conductivity equation (p = 2 case) this problem has been studied, there are theoretical and computation results, and several applications, such as electrical impedance tomography which can be used to detect breast cancer, discover oil or image cracks in concrete.

Questions

Are there applications for the Calderón problem in the more general case where we need not have p = 2? That is, is there some medium on which the Ohm's law has the extra term $|\nabla v|^{m}$ for some non-zero m, and where finding out the conductivity by non-invasive boundary measurements would be useful?

Is there some other nonlinear Ohm's law that is more realistic in situations where electrical impedance tomography or similar imaging methods would be useful?

• Near equilibrium most physical systems are pretty linear, which is why perturbation analysis usually works. But as the system approaches a phase change nonlinearities may appear, so that is one place to look for applications. Ohm's law is, as pointed out by others, is material depenent. It – Peter Diehr Feb 6 '16 at 12:52
• I agree with Peter Diehr, if you're looking for applications of a non-linear Calderon's problem it might be more productive to look directly at current applications. For example, larger currents usually allow better signal-to-noise ratio, but can introduce heating. Since $sigma$ is actually a non-linear function of $T$, you could look into solving a time-dependent Calderon problem (instead of waiting for the temperature to settle). As far as I know intrinsic material non-linearities are more rare. – Real Aug 26 '16 at 4:30