Section: New Results

Boundary value problems

Participants : Laurent Baratchart, Sylvain Chevillard, Juliette Leblond, Dmitry Ponomarev.

Collaboration with Laurent Bourgeois (ENSTA ParisTech, Lab. Poems), Elodie Pozzi (Univ. Bordeaux, IMB), Emmanuel Russ (Univ. Grenoble, IJF).

Generalized Hardy classes

As we mentioned in section 4.4 , 2-D diffusion equations of the form $\mathrm{div}\left(\sigma \nabla u\right)=0$ with real non-negative valued conductivity $\sigma$ can be viewed as compatibility conditions for the so-called conjugate Beltrami equation: $\overline{\partial }f=\nu \overline{\partial f}$ with $\nu =(1-\sigma )/(1+\sigma )$ [4] . Thus, the conjugate Beltrami equation is a means to replace the initial second order diffusion equation by a first order system of two real equations, merged into a single complex one. Hardy spaces under study here are those of this conjugate Beltrami equation: they are comprised of solutions to that equation in the considered domain whose $L^p$ means over curves tending to the boundary of the domain remain bounded. They will for example replace holomorphic Hardy spaces in problem $\left(P\right)$ when dealing with non-constant (isotropic) conductivity. Their traces merely lie in $L^p$ ($1<p<\infty$), which is suitable for identification from point-wise measurements, and turn out to be dense on strict subsets of the boundary. This allows one to state Cauchy problems as bounded extremal issues in $L^p$ classes of generalized analytic functions, in a manner which is reminiscent of what we discussed for analytic functions in section  3.3.1 .

The study of such Hardy spaces for Lipschitz $\sigma$ was reduced in [4] to that of spaces of pseudo-holomorphic functions with bounded coefficients, which were apparently first considered on the disk by S. Klimentov. Solutions factorize as $e^{s}F$, where $F$ is a holomorphic Hardy function while $s$ is in the Sobolev space $W^{1,r}$ for all $r<\infty$ (Bers factorization), and the analog to the M. Riesz theorem holds which amounts to solvability of the Dirichlet problem with $L^p$ boundary data. The case of finitely connected domains was carried out in [14] .

This year, we addressed in [25] the uniqueness issue for the classical Robin inverse problem on a Lipschitz-smooth domain $\Omega \subset {ℝ}^n$, with $L^{\infty }$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}\left(\Omega \right)$, $r>n$. The Robin inverse problem consists in recovering the ratio of the normal derivative and the solution (the so-called Robin coefficient) on a subset of the boundary, knowing them on the complementary subset. We showed that uniqueness of the Robin coefficient on a subset of the boundary, given Cauchy data on the complementary subset, does hold when $n=2$ whenever the boundary subsets are of positive Lebesgue measure. We also showed that this no longer holds in higher dimension, and we gave counterexamples when $n=3$. The subsets in these counterexamples look very bad, and it is natural to ask whether uniqueness prevails if they have interior points. This raises an interesting open issue on harmonic gradients, namely: can a nonzero harmonic function vanish together with its normal derivative on a subset of the boundary of positive measure, and still the Robin coefficient is bounded in a neighborhood of that set? This question is worth investigating

Best constrained analytic approximation

Several questions about the behavior of solutions to the bounded extremal problem $\left(P\right)$ in section 3.3.1 , and of some generalizations thereof, are still under study by Apics.. We considered additional interpolation constraints on the disk in problem $\left(P\right)$, and derived new stability estimates for the solution [24] . An article is being written on the subject. Ongoing work is geared towards applications of [24] . New insight leads us to relate these results to overdetermined boundary value problems for 2D Laplace equations on irregular boundaries. This has applications in set-ups where measurements are obtained from oddly distributed sensors. Treating some of the measurements as pointwise interpolation constraints seems a reasonable strategy in comparison with interpolation of the data along a geometrically complicated boundary. Such interpolation constraints arise naturally in inverse boundary problems like plasma shaping, when some of the measurements are performed inside the chamber of the tokamak, see section 4.4 .