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Section: New Results

Boundary value problems, generalized Hardy classes

Participants : Laurent Baratchart, Slah Chaabi, Juliette Leblond, Dmitry Ponomarev.

This work has been performed in collaboration with Yannick Fischer from the Magique3D EPI (Inria Bordeaux, Pau).

In collaboration with the CMI-LATP (University Aix-Marseille I), the team considers 2-D diffusion processes with variable conductivity. In particular its complexified version, the so-called conjugate or real Beltrami equation, was investigated. In the case of a smooth domain, and for Lipschitz conductivity, we analyzed the Dirichlet problem for solutions in Sobolev and then in Hardy classes [5] .

Their traces merely lie in $L^p$ ($1<p<\infty$) of the boundary, a space which is suitable for identification from pointwise measurements. Again these traces turn out to be dense on strict subsets of the boundary. This allows us 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 .

We generalized the construction to finitely connected Dini-smooth domains and $W^{1,q}$-smooth conductivities, with $q>2$ [35] . The case of an annular geometry is the relevant one for the application to plasma shaping mentioned below [58] , [35] . The application that initially motivated this work came from free boundary problems in plasma confinement (in tokamaks) for thermonuclear fusion. This work was initiated in collaboration with the Laboratoire J. Dieudonné (University of Nice).

In the transversal section of a tokamak (which is a disk if the vessel is idealized into a torus), the so-called poloidal flux is subject to some conductivity equation outside the plasma volume for some simple explicit smooth conductivity function, while the boundary of the plasma (in the Tore Supra tokamak) is a level line of this flux [54] . Related magnetic measurements are available on the chamber, which furnish incomplete boundary data from which one wants to recover the inner (plasma) boundary. This free boundary problem (of Bernoulli type) can be handled through the solutions of a family of bounded extremal problems in generalized Hardy classes of solutions to real Beltrami equations, in the annular framework [35] .

In the particular case at hand, the conductivity is $1/x$ and the domain is an annulus embedded in the right half-plane. We obtained a basis of solutions (exponentials times Legendre functions) upon separating variables in toroidal coordinates. This provides a computational setting to solve the extremal problems mentioned before, and was the topic of the PhD thesis of Y. Fischer [58] , [27] . In the most recent tokamaks, like Jet or ITER, an interesting feature of the level curves of the poloidal flux is the occurrence of a cusp (a saddle point of the poloidal flux, called an X point), and it is desirable to shape the plasma according to a level line passing through this X point for physical reasons related to the efficiency of the energy transfer. We established well-posedness of the Dirichlet problem in weighted $L^p$ classes for harmonic measure on piecewise smooth domains without cusps, thereby laying ground for such a study. This issue is next in line, now that the present approach has been validated numerically on Tore Supra data, and the topic of the PhD thesis of D. Ponomarev.

The PhD work of S. Chaabi is devoted to further aspects of Dirichlet problems for the conjugate Beltrami equation. On the one hand, a method based on Foka's approach to boundary value problems, which uses Lax pairs and solves for a Riemann-Hilbert problem, has been devised to compute in semi explicit form solutions to Dirichlet and Neumann problems for the conductivity equation satisfied by the poloidal flux. Also, for more general conductivities, namely bounded below and lying in $W^{1,s}$ with $s\ge 2$, parameterization of solutions to Dirichlet problems on the disk by Hardy function was achieved through Bers-Nirenberg factorization. Note the conductivity may be unbounded when $s=2$, which is completely new. Two papers are being prepared reporting on these topics.

Finally, note that the conductivity equation can be expressed like a static Schrödinger equation, for smooth enough conductivity coefficients. This provides a link with the following results recently set up by D. Ponomarev, who recently join the team for his PhD. 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 such setting, Helmholtz equation is approximated by a linear Schrödinger equation with one of spatial coordinates being an evolutionary variable. Explicit comparison of the solutions in the whole half-space allows to establish global justification of the Schrodinger model for sufficiently smooth pulses [73] . This phenomenon can also be described by a nonstationary model that relies on the spatial nonlinear Schrödinger (NLS) equation with the time-dependent refractive index. A toy problem is considered in [71] , 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. The NLS approximation is derived from a 2-D quasi-linear wave equation, for small time intervals and smooth initial data. Numerical simulations illustrate the approximation result in the 1-D case.