We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary ...conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the q-Laplacian.
In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.
A multi-term two-dimensional time-fractional initial-boundary value problem, whose boundary conditions are Neumann type, is considered. A temporally graded and spatially uniform mesh is used to apply ...an L1-finite difference method. This method involves applying the L1 scheme to discretize each term of the multi-term time-fractional derivative and using the finite difference method for spatial discretization. By carefully handling the Neumann boundary discretization, a sharp error analysis is demonstrated at each time level. Numerical results supporting the sharp error estimates of our theoretical results are presented.
For each fixed integer
N
≥
2
let
Ω
⊂
R
N
be an open, bounded and convex set with smooth boundary. For each real number
p
∈
(
1
,
∞
)
define
M
(
p
;
Ω
)
=
inf
u
∈
W
C
2
,
∞
(
Ω
)
∖
{
0
}
∫
Ω
(
exp
(
...|
Δ
u
|
p
)
−
1
)
d
x
∫
Ω
(
exp
(
|
u
|
p
)
−
1
)
d
x
,
where
W
C
2
,
∞
(
Ω
)
:=
∩
1
<
p
<
∞
{
u
∈
W
0
2
,
p
(
Ω
)
:
Δ
u
∈
L
∞
(
Ω
)
}
. We show that if the radius of the largest ball which can be inscribed in
Ω
is strictly larger than a constant which depends on
N
then
M
(
p
;
Ω
)
vanishes while if the radius of the largest ball which can be inscribed in
Ω
is strictly less than
1
then
M
(
p
;
Ω
)
is a positive real number. Moreover, in the latter case when
p
is large enough we can identify the value of
M
(
p
;
Ω
)
as being the principal frequency of the
p
-Bilaplacian on
Ω
with coupled Dirichlet–Neumann boundary conditions.
Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, ...p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.
In this paper, we propose two approaches to apply boundary conditions for bond-based peridynamic models. There has been in recent years a renewed interest in the class of so-called non-local models, ...which include peridynamic models, for the simulation of structural mechanics problems as an alternative approach to classical local continuum models. However, a major issue, which is often disregarded when dealing with this class of models, is concerned with the manner by which boundary conditions should be prescribed. Our point of view here is that classical boundary conditions, since applied on surfaces of solid bodies, are naturally associated with local models. The paper describes two methods to incorporate classical Dirichlet and Neumann boundary conditions into bond-based peridynamics. The first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points. The second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the so-called horizon vanishes. The approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions. We present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a one-dimensional bar.
We consider a model for the regulation of growth and patterning in developing tissues by diffusing morphogens expressed in terms of a system of nonlinear PDEs. Transforming such a system in an ...equation with singular diffusion we analyze its long time behavior.
This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic ...convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H1-norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included.
Variational fracture has been viewed as incompatible with boundary loads, due to a straightforward non-existence argument. We introduce a different variational formulation that includes loads, and we ...illustrate a method for showing existence.
A nonlocal supercritical Neumann problem Cinti, Eleonora; Colasuonno, Francesca
Journal of Differential Equations,
02/2020, Letnik:
268, Številka:
5
Journal Article
Recenzirano
Odprti dostop
•Variational Neumann Fractional Laplacian problem in a radial set of Rn.•A priori estimates for solutions in a suitable cone.•Sobolev-supercritical nonlinearity. Truncation method. Mountain-pass-type ...solution.•No use of the expression of fractional Laplacian for radial functions, all the proofs are genuinely variational.•Strong maximum principle for nonlocal Neumann problems.
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the sense of Sobolev embedding). The consequent lack of compactness can be overcome, by working in the cone of non-negative and non-decreasing radial functions. Within this cone, we establish some a priori estimates which allow, via a truncation argument, to use variational methods for proving existence of solutions. As a side result, we prove a strong maximum principle for nonlocal Neumann problems, which is of independent interest.
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