Regularization for a class of ill-posed evolution problems in Banach space

We study evolution equations in Banach space, and provide a general framework for regularizing a wide class of ill-posed Cauchy problems by proving continuous dependence on modeling for nonautonomous equations. We approximate the ill-posed problem by a well-posed one, and obtain H?lder-continuous dependence results that provide estimates of the error for a class of solutions under certain stabilizing conditions. For examples that include the linearized Korteweg-de Vries equation and the Schr?dinger equation in L ^p ,p??2, we obtain a family of regularizing operators for the ill-posed problem. This work extends to the nonautonomous case several recent results for ill-posed problems with constant coefficients.     

WEAK REGULARIZATION FOR A CLASS OF ILL-POSED CAUCHY PROBLEMS

This article is concerned with the ill-posed Cauchy problem associated with a densely defined linear operator A in a Banach space. A family of weak regularizing operators is introduced. If the spectrum of A is contained in a sector of right-half complex plane and its resolvent is polynomially bounded, the weak regularization for such ill-posed Cauchy problem can be shown by using the quasi-reversibility method and regularized semigroups. Finally, an example is given.     

A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

     

On k-summability of formal solutions for a class of partial differential operators with time dependent coefficients

We study the k-summability of divergent formal solutions for the Cauchy problem of a certain class of linear partial differential operators with time dependent coefficients. The problem is reduced to a k-summability property of formal solutions for a linear similar ordinary differential equation associated with the Cauchy problem.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

A stability estimate for an elliptic problem

We investigate the Caucy problem for linear elliptic operators withC ^∞-coefficients at a regular domain ℝ ⊂ ℝ, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ⊂∂Ω and our goal is to obtain a stability estimate inH ₄(Ω).     

Boundary problems for systems of partial differential operators of mixed order

On a compact n-dimensional C^∞ manifold $\text{Ω}$ with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere.     

Missing boundary data reconstruction by the factorization method

We consider the data completion problem for the Laplace equation in a cylindrical domain. The Neumann and Dirichlet boundary conditions are given on one face of the cylinder while there is no condition on the other face. This Cauchy problem has been known since Hadamard (1953) to be ill-posed. Here it is set as an optimal control problem with a regularized cost function. We use the factorization method for elliptic boundary value problems. For each set of Cauchy data, to obtain the estimate of the missing data one has to solve a parabolic Cauchy problem in the cylinder and a linear equation. The operator appearing in these problems satisfy a Riccati equation that does not depend on the data. To cite this article: A. Ben Abda et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Convergence estimates of a projection-difference method for an operator-differential equation

This article investigates the projection-difference method for a Cauchy problem for a linear operator-differential equation with a leading self-adjoint operator A(t) and a subordinate linear operator K(t) in Hilbert space. This method leads to the solution of a system of linear algebraic equations on each time level; moreover, the projection subspaces are linear spans of eigenvectors of an operator similar to A(t). The convergence estimates are obtained. The application of the developed method for solving the initial boundary value problem is given.     

On a complete discretization scheme for an ill-posed Cauchy problem in a Banach space

A complete discretization scheme for an ill-posed Cauchy problem for abstract firstorder linear differential equations with sectorial operators in a Banach space is validated. The scheme combines a time semidiscretization of the equations and a finite-dimensional approximation of the spaces and operators. Regularization properties of the scheme are established. Error estimates are obtained in the case of approximate initial data under various a priori assumptions concerning the solution.