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Truncated spectral regularization for an ill-posed non-linear parabolic problem

Ajoy Jana, M. Thamban Nair (2019)

Czechoslovak Mathematical Journal

It is known that the nonlinear nonhomogeneous backward Cauchy problem u t ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 t < τ with u ( τ ) = φ , where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f , that a solution of the above problem satisfies an integral equation involving the spectral representation of A , which is also ill-posed. Spectral truncation is used...

Two examples of fattening for the curvature flow with a driving force

Giovanni Bellettini, Maurizio Paolini (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We provide two examples of a regular curve evolving by curvature with a forcing term, which degenerates in a set having an interior part after a finite time.

Two Numerical Methods for the elliptic Monge-Ampère equation

Jean-David Benamou, Brittany D. Froese, Adam M. Oberman (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The numerical solution of the elliptic Monge-Ampère Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155–192; Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238; Dean and Glowinski, in Partial differential equations, Comput. Methods Appl. Sci. 16 (2008) 43–63; Glowinski et al., Japan...

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