139 research outputs found
Wellposedness of a nonlocal nonlinear diffusion equation of image processing
Existence and uniqueness are established for a degenerate regularization of
the well-known Perona-Malik equation proposed by the first author for
non-smooth initial data. The results heavily rely on the choice of appropriate
functional setting inspired by a recent approach to degenerate parabolic
equations via so-called singular Riemannian manifolds introduced by Herbert
Amann
A Class of Free Boundary Problems with Onset of a new Phase
A class of diffusion driven Free Boundary Problems is considered which is
characterized by the initial onset of a phase and by an explicit kinematic
condition for the evolution of the free boundary. By a domain fixing change of
variables it naturally leads to coupled systems comprised of a singular
parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even
though the one dimensional case has been thoroughly investigated, results as
basic as well-posedness and regularity have so far not been obtained for its
higher dimensional counterpart. In this paper a recently developed regularity
theory for abstract singular parabolic Cauchy problems is utilized to obtain
the first well-posedness results for the Free Boundary Problems under
consideration. The derivation of elliptic regularity results for the underlying
static singular problems will play an important role
The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature
We revisit the well-known Curve Shortening Flow for immersed curves in the
-dimensional Euclidean space. We exploit a fundamental structure of the
problem to derive a new global construction of a solution, that is, a
construction that is valid for all times and is insensitive to singularities.
The construction is characterized by discretization in time and the
approximant, while still exhibiting the possibile formation of finitely many
singularities at a finite set of singular times, exists globally and is well
behaved and simpler to analyze than a solution of the CSF. A solution of the
latter is obtained in the limit. Estimates for a natural (geometric) norm
involving length and total absolute curvature allow passage to the limit. Many
classical qualitative results about the flow can be recovered by exploiting the
simplicity of the approximant and new ones can be proved. The construction also
suggests a numerical procedure for the computation of the flow which proves
very effective as demonstrated by a series of numerical experiments scattered
throughout the paper.Comment: 32 pages, 9 figure
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