We study a source identification problem for prototypical elliptic PDE from Dirichlet boundary data. This is ill-posed, and the involved forward operator has significant nullspace. Standard Tikhonov regularization yields solutions which approach minimum \begin{document}$ L^2 $\end{document}-norm least-squ...

This paper presents an a posteriori error estimator for the preconditioned iteration scheme of Bramble, Knyazev and Pasciak to compute the smallest eigenvalues together with its invariant subspace of an elliptic diierential operator. The error estimator is applied both to the nite element space in which the preconditioned iteration is performed (iteration error estimation) as well as to its hie...

This paper deals with the Dirichlet problem for convex differential (PC) inclusions of elliptic type. On the basis of conjugacy correspondence the dual problems are constructed. Using the new concepts of locally adjoint mappings in the form of Euler-Lagrange type inclusion is established extremal relations for primary and dual problems. Then duality problems are formulated for convex problems a...

From the dynamical twisting of the classical r-matrix, we obtain a new Lax operator for the elliptic Ruijsenaars-Schneider model (cf. Ruisenaars’). The corresponding r-matrix is shown to be the classical Zn-symmetric elliptic r-matrix, which is the same as that obtained in the study of the nonrelativistic version—the An−1 Calogero-Moser model givened by us. Mathematics Subject Classification : ...

We solve the Kato problem for divergence form elliptic operators whose heat kernels satisfy a pointwise Gaussian upper bound. More precisely, given the Gaussian hypothesis, we establish that the domain of the square root of a complex uniformly elliptic operator L = −div(A∇) with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate ‖ √ Lf‖2 ∼ ‖∇f‖2...

The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for areapreserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integrati...

We construct a realization of the L-operator satisfying the RLL-relation of the face type elliptic quantum group Bq,λ(A (2) 2 ). The construction is based on the elliptic analogue of the Drinfeld currents of Uq(A (2) 2 ), which forms the elliptic algebra Uq,p(A (2) 2 ). We give a realization of the elliptic currents E(z), F (z) and K(z) as a tensor product of the Drinfeld currents of Uq(A (2) 2...

Using the framework of operator or Calderón preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The as composition an opposite order operator, on same ansatz space, and two identical diagonal scaling operators, whose matrix representation is lumped mass matrix.

In this paper, we discuss the following conjecture raised by BaumDouglas: For any first-order elliptic differential operator D on smooth manifold M with boundary ∂M , D possesses an elliptic boundary condition if and only if ∂[D] = 0 in K1(∂M), where [D] is the relative K-cycle in K0(M, ∂M) corresponding to D. We prove the “if” part of this conjecture for dim(M) 6= 4, 5, 6, 7 and the “only if” ...