Discussion Board Lecture 08

Dear Damstadt Team,

we did not speak (too much) about the physical interpretation of the DtN operator. Let me start by quoting from the Wikipedia page on the DtN Operator:

"Consider a steady-state distribution of temperature in a body for given temperature values on the body surface. Then the resulting heat flux through the boundary (that is, the heat flux that would be required to maintain the given surface temperature) is determined uniquely. The mapping of the surface temperature to the surface heat flux is a Poincaré–Steklov operator. This particular Poincaré–Steklov operator is called the Dirichlet to Neumann (DtN) operator. The values of the temperature on the surface is the Dirichlet boundary condition of the Laplace equation, which describes the distribution of the temperature inside the body. The heat flux through the surface is the Neumann boundary condition (proportional to the normal derivative of the temperature)."

As is treated in the lecture, the DtN operator results from solving the Dirichlet problem to a function $ f $ on $ \partial\Omega $, and then attributing to $ f $ the normal derivative $ g $ of the solution. So, where $ f $ is `large', heat will flow out, i.e., the normal derivative is positive, and conversely. Thus, the semigroup generated by $ -D_0 $ has as the driving term $ -D_0u(t) $ a function that is negative where $ u(t) $ is large (compared to other values of $ u(t) $) and positive where $ u(t) $ is small (compared to other values of $ u(t) $). This means that the semigroup should have a compensatory effect on the solutions.

It was shown in the lecture that $ D_0 $ has compact resolvent. This implies that the generated semigroup can be written down explicitly in terms of the eigenfunctions and eigenvalues. Assume that $ \Omega $ is connected. Then $ 0 $ is the smallest eigenvalue, and $ 1 $ is the unique eigenfunction. In consequence the solution $ u(t) $ will converge to a constant function, the mean value of the initial function.

Concerning your precise question on Exercise 8.4, I think you should rather look at Exercise 8.5, where the solution is indicated, and Exercise 8.4 is a special case of 8.5. In the solutions of Exercise 8.4 as well of 8.5 you can observe the described behaviour.

Concluding, let me ask whether this was helpful. I have to emphasise that I am not a specialist in the DtN operator, and I am only interpreting what I can see from the mathematics.

Best wishes, Jürgen

Dear Darmstadt Team, dear Jürgen,

in Ulm we discussed the same topic today and we respectfully disagree with Jürgens interpretation, for the following reasons:

- If we prescribe the temperature of a body at its boundary, then it takes some time until the diffusion arrives (approximately) at its equilibrium. Hence, the interpretation by means of a steady-state heat distribution is fine for the Dirichlet-to-Neumann operator, but not for the generated semigroup. Indeed, the associated evolution equation $ \dot{u} = -Au $ would mean that the change of temperature at the boundary at time t is driven by the gradient of some future heat distrubtion.

- The asymptotic behaviour described by Jürgen does not seem to be consistent with what happens physically: Assume that the initial boundary temperature is constant, but higher then the temperature inside the body. Then we should expect the temperature on the boundary to converge to some lower value because some of the energy from the boundary diffuses into the body; hence, the boundary temperature does not converge to the mean values of the intial function. If we want the temperature on the boundary not to decrease, then we have to install a heating, which means mathematically to fix the boundary values; but then it does not make sense to describe their evolution by an abstract Cauchy problem.

After some discussion we found a different physical model whose time evolution can be described by the semigroup associated to the one-dimensional Dirichlet-to-Neumann operator:

Consider a capacitor which consists of two small plates located at the points $ -1 $ and $ 1 $ (we consider only a one-dimensional model, so let us assume that both plates have only the size of a point). The electrical charges on both plates are given by a vector $ q=(q_{-1},q_1) $. These charges generate an electrical potential at the points $ -1 $ and $ 1 $ which is proportional to the charges. Assuming all physical constants to be $ 1 $, we may suppose that the potential at $ -1 $ and $ 1 $ is also given by the vector $ q=(q_{-1},q_1) $. Now, assume that both plates of the capacitor are connected by a one-dimensional conductor which we identify with the interval $ [-1,1] $. Of course, an electrical current will flow between both plates.

The genesis of this current can be modelled in the following way: The charges on the plates generate an electrical field $ E $ on the interval $ [-1,1] $, and $ E $ is the negative gradient $ E = -u' $ of an electrical potential $ u $ on $ [-1,1] $. Since there are (almost) no charges in the conductor, the potential $ u $ is harmonic, i.e. $ u'' = 0 $. Since we already know that the boundary values of $ u $ are given by the vector $ q $, the potential $ u $ is the solution of the Dirichlet problem with boundary values $ q_{-1} $ and $ q_{1} $.

The inner normal derivatives of $ u $ in $ -1 $ and $ 1 $, given by the vector $ -D_0 q $, describe the electrical field pointing from the interior of the conductor to the plates. The electrical field thus induces a current between the conductor and the plates, and this current is proportional to the electrical field (and since we assume all physical constants to be $ 1 $, the current is in fact equal to the field). Hence, the vector $ -D_0q $ describes the electrical current towards the plates and thus coincides with the rate of change of the charges $ q = (q_{-1},q_1) $ on the plates. Therefore, those charges fulfil the evolution equation $\cdot{q} = -D_0 q$.

From Exercise 8.4 we know the matrix $ D_0 $ and the semigroup $ e^{-tD_0} $. We can use this knowledge to observe the behaviour of the system for different initial charges. If we charge the left plate with $ q_{-1}(0)=-1 $ and the right plate with $ q_1(0)=1 $, then we obtain $ q(t)=(-e^{-t},e^{-t}) $ for the evolution of the charges, i.e. the capacator discharges exponentially fast. However, if we use $ q(0)=(1,1) $ as initial values, we see that the semigroup leaves these values fixed. This is consistent with the physical interpretation since both positive charges on the left and on the right plate repel each other. More generally, for every initial vector $ q(0) = (q_{-1}(0),q_1(0)) $, the system converges to $ \frac{q_{-1}(0) + q_1(0)}{2}(1,1) $ which means that both charges compensate each other until both plates of the capacitor are equally charged.

A similar interpretation can be used in three dimensions.

Best regards, Jochen --~~~~

Dear Jochen,

I agree with you concerning your interpratation concerning the two capacitors.

Concerning your first two dashes, you have to distinguish between the physical reality and the model the equation decsribes. Your assumption that you have a body with high temperature at the boundary and low temperature inside simply is not representable in the model. Note that the Hilbert space where you put in initial values is a function space on the boundary. You simply would not know where to put the temerature inside the body in the description of the initial values. The model the equations describe is a model where intantaneous equilibrium is assumed to be possible. You agree to accept this for electricity because the speed of light is `near infinity', but not for heat diffusion. If you want to describe this you have to make up another model, where the temperature distribution at the inside is taken into account. But this probably is just the Robin Laplacian.

Best wishes, Jürgen

Dear Jürgen,

thanks a lot for your reply.

I'm yet not fully convinced that heat diffusion provides an appropriate physical interpretation for to DtN operator. Of course, you always have to distinguish between physical reality and the properties of a mathematical model. Yet, if we want to find a physical interpretation of a mathematical object, I think we should look for a physcial situation (more technically: the description of a physical experiment) for which the real physical behaviour is approximately described by the mathematical object that we want to interpret.

I don't really understand what kind of physical experiment involving heat diffusion could (approximately) be described by the semigroup generated by the DtN operator. I also think that this is not only a problem of the speed of heat diffusion: Assume we have a material with very high heat conductivity at hand such that the heat diffusion converges almost immediately to an equilibrium; what kind of experiment do we set up to observe a physical quantity whose behaviour is approximately described by the semigroup generated by the DtN operator?

Best wishes, Jochen

Dear Johannes,

thanks!

Best wishes, Jürgen