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Concept# Cauchy boundary condition

Summary

In mathematics, a Cauchy (koʃi) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.
Second-order ordinary differential equations
Cauchy boundary conditions are simple and common in second-order ordinary differential equations,
:y''(s) = f\big(y(s), y'(s), s\big),
where, in order to ensure that a unique solution y(s) exists, one may specify the value of the function y and the value of the derivative y' at a given point s=a, i.e.,
:y(a) = \alpha,
and

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Miklos Lajko, Frédéric Mila, Pierre Marcel Nataf, Karlo Penc

We show that, in the presence of a π/2 artificial gauge field per plaquette, Mott insulating phases of ultracold fermions with SU(N) symmetry and one particle per site generically possess an extended chiral phase with intrinsic topological order characterized by an approximate ground space of N low-lying singlets for periodic boundary conditions, and by chiral edge states described by the SU(N)1 Wess-Zumino-Novikov-Witten conformal field theory for open boundary conditions. This has been achieved by extensive exact diagonalizations for N between 3 and 9, and by a parton construction based on a set of N Gutzwiller projected fermionic wave functions with flux π/N per triangular plaquette. Experimental implications are briefly discussed.

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Interface controls are unknown functions used as Dirichlet or Robin boundary data on the interfaces of an overlapping decomposition designed for solving second order elliptic boundary value problems. The controls are computed through an optimal control problem with either distributed or interface observation. Numerical results show that, when interface observation is considered, the resulting interface control domain decomposition method is robust with respect to coefficients variations; it can exploit nonconforming meshes and provides optimal convergence with respect to the discretization parameters; finally it can be easily used to face heterogeneous advectionadvection- diffusion couplings.

This paper is devoted to the problem of recovering a potential q in a domain in ℝd for d≥3 from the Dirichlet to Neumann map. This problem is related to the inverse Calder'on conductivity problem via the Liouville transformation. It is known from the work of Haberman and Tataru [11] and Nachman and Lavine [17] that uniqueness holds for the class of conductivities of one derivative and the class of W2,d/2 conductivities respectively. The proof of Haberman and Tataru is based on the construction of complex geometrical optics (CGO) solutions initially suggested by Sylvester and Uhlmann [22], in functional spaces introduced by Bourgain [2]. The proof of the second result, in the work of Ferreira et al. [10], is based on the construction of CGO solutions via Carleman estimates. The main goal of the paper is to understand whether or not an approach which is based on the construction of CGO solutions in the spirit of Sylvester and Uhlmann and involves only standard Sobolev spaces can be used to obtain these results. In fact, we are able to obtain a new proof of uniqueness for the Calder'on problem for 1) a slightly different class as the one in [11], and for 2) the class of W2,d/2 conductivities. The proof of statement 1) is based on a new estimate for CGO solutions and some averaging estimates in the same spirit as in [11]. The proof of statement 2) is on the one hand based on a generalized Sobolev inequality due to Kenig et al. [14] and on another hand, only involves standard estimates for CGO solutions [22]. We are also able to prove the uniqueness of a potential for 3) the class of Ws,3/s (⫌W2,3/2) conductivities with 3/2

2014