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Section: New Results

Boundary value problems

Participants : Laurent Baratchart, Slah Chaabi, Sylvain Chevillard, Juliette Leblond, Dmitry Ponomarev, Elodie Pozzi.

This work was the occasion of collaborations with Alexander Borichev (Aix-Marseille University), Jonathan Partington (Univ. Leeds, UK), and 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 relations 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 reminiscent manner of what was done for analytic functions as discussed 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. Typical results here are that solution 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 which amounts to solvability of the Dirichlet problem for the initial conductivity equation with $L^p$ boundary data for all $p\in (1,\infty )$. Over the last two years, the case of $W^{1,q}$ conductivities over finitely connected domains, $q>2$, has been carried out in [13] [61] .

In 2013, completing a study begun last year in the framework of the PhD of S. Chaabi, we established similar results in the case where $log\sigma$ lies in $W^{1,2}$, which corresponds to the critical exponent in Vekua's theory of pseudo-holomorphic functions. This is completely new, and apparently the first example of a solvable Dirichlet problem with $L^p$ boundary data where the conductivity can be both unbounded an vanishing at some places. Accordingly, solutions may also be unbounded inside the domain of the equation, that is, the maximum principle no longer holds. The proof develops a refinement of the Bers factorization based on Muckenhoupt weights and on an original multiplier theorem for $log{W}^{1,2}$ functions. A paper on this topic has been submitted [28] .

The PhD work of S. Chaabi (defended December 2) contains further work on the Weinstein equation and certain generalizations thereof. This equation results from 2-D projection of Laplace's equation in the presence of rotation symmetry in 3-D. In particular, it is the equation governing the free boundary problem of plasma confinement in the plane section of a tokamak. A method dwelling on Fokas's approach to elliptic boundary value problems has been developed which uses Lax pairs and solves for a Riemann-Hilbert problem on a Riemann surface. It was used to devise semi-explicit forms of solutions to Dirichlet and Neumann problems for the conductivity equation satisfied by the poloidal flux.

In another connection, the conductivity equation can also be regarded as a static Schrödinger equation for smooth coefficients. In particular, a description of laser beam propagation in photopolymers can be crudely approximated by a stationary two-dimensional model of wave propagation in a medium with negligible change of refractive index. In this setting, Helmholtz equation is approximated by a linear Schrödinger equation with one spatial coordinate as evolutionary variable. This phenomenon can be described by a non-stationary model that relies on a spatial nonlinear Schrödinger (NLS) equation with time-dependent refractive index. A model problem has been considered in [20] , when the rate of change of refractive index is proportional to the squared amplitude of the electric field and the spatial domain is a plane.

We have also studied composition operators on generalized Hardy spaces in the framework of [13] . In the work [32] submitted for publication, we provide necessary and/or sufficient conditions on the composition map, depending on the geometry of the domains, ensuring that these operators are bounded, invertible, isometric or compact.

Best constrained analytic approximation

Several questions about the behavior of solutions to the bounded extremal problem $\left(P\right)$ of Section  3.3.1 have been considered. For instance, truncated Toeplitz operators have been studied in [17] , that can be used to quantify robustness properties of our resolution schemes in $H^2$ and to establish error estimates. Moreover we considered additional interpolation constraints on the disk in Problem $\left(P\right)$, and derived new stability estimates for the solution [46] . Such interpolation constraints arise naturally in inverse boundary problems like plasma shaping in last generation tokamaks, where some measurements are performed inside the chamber 4.4 . Of course the version studied so far is much simplified, as it must be carried over to non-constant conductivities and annular geometries.