2,722 research outputs found
Space-time adaptive finite elements for nonlocal parabolic variational inequalities
This article considers the error analysis of finite element discretizations
and adaptive mesh refinement procedures for nonlocal dynamic contact and
friction, both in the domain and on the boundary. For a large class of
parabolic variational inequalities associated to the fractional Laplacian we
obtain a priori and a posteriori error estimates and study the resulting
space-time adaptive mesh-refinement procedures. Particular emphasis is placed
on mixed formulations, which include the contact forces as a Lagrange
multiplier. Corresponding results are presented for elliptic problems. Our
numerical experiments for -dimensional model problems confirm the
theoretical results: They indicate the efficiency of the a posteriori error
estimates and illustrate the convergence properties of space-time adaptive, as
well as uniform and graded discretizations.Comment: 47 pages, 20 figure
Constructing solutions for a kinetic model of angiogenesis in annular domains
We prove existence and stability of solutions for a model of angiogenesis set
in an annular region. Branching, anastomosis and extension of blood vessel tips
are described by an integrodifferential kinetic equation of Fokker-Planck type
supplemented with nonlocal boundary conditions and coupled to a diffusion
problem with Neumann boundary conditions through the force field created by the
tumor induced angiogenic factor and the flux of vessel tips. Our technique
exploits balance equations, estimates of velocity decay and compactness results
for kinetic operators, combined with gradient estimates of heat kernels for
Neumann problems in non convex domains.Comment: to appear in Applied Mathematical Modellin
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
control, double obstacles, necessary optimality condition
Symmetrization for fractional Neumann problems
In this paper we complement the program concerning the application of
symmetrization methods to nonlocal PDEs by providing new estimates, in the
sense of mass concentration comparison, for solutions to linear fractional
elliptic and parabolic PDEs with Neumann boundary conditions. These results are
achieved by employing suitable symmetrization arguments to the Stinga-Torrea
local extension problems, corresponding to the fractional boundary value
problems considered. Sharp estimates are obtained first for elliptic equations
and a certain number of consequences in terms of regularity estimates is then
established. Finally, a parabolic symmetrization result is covered as an
application of the elliptic concentration estimates in the implicit time
discretization scheme.Comment: 34 page
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
Nonlinear Dirichlet problem for the nonlocal anisotropic operator
In this paper we study an equation driven by a nonlocal anisotropic operator
with homogeneous Dirichlet boundary conditions. We find at least three non
trivial solutions: one positive, one negative and one of unknown sign, using
variational methods and Morse theory. We present some results about regularity
of solutions as -bound and Hopf's lemma, for the latter we first
consider a non negative nonlinearity and then a strictly negative one.
Moreover, we prove that, for the corresponding functional, local minimizers
with respect to a -topology weighted with a suitable power of the distance
from the boundary are actually local minimizers in the -topology
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