575 research outputs found
Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation
This paper is concerned with the existence and uniqueness of global weak
solutions to a generalized Camassa-Holm equation on real line. By introducing
some new variables, the equation is transformed into two different semi-linear
systems. Then the existence and uniqueness of global weak solutions to the
original equation are obtained from that of the two semi-linear systems,
respectively.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1601.03889,
arXiv:1509.08569, arXiv:1401.0312 by other author
Pohozaev identity for the anisotropic -Laplacian and estimates of torsion function
In this paper we prove the Pohozaev identity for the weighted anisotropic
-Laplace operator. As an application of our identity, we deduce the
nonexistence of nontrivial solutions of the Dirichlet problem for the weighted
anisotropic -Laplacian in star-shaped domains of . We also
provide an upper bound estimate for the first Dirichet eigenvalue of the
anisotropic -Laplacian on bounded domains of , some sharp
estimates for the torsion function of compact manifolds with boundary and a
nonexistence result for the solutions of the Laplace equation on closed
Riemannian manifolds.Comment: 21 page
On Payne-Schaefer's Conjecture about an Overdetermined Boundary Problem of Sixth Order
This paper considers overdetermined boundary problems. Firstly, we give a
proof to the Payne-Schaefer conjecture about an overdetermined problem of sixth
order in the two dimensional case and under an additional condition for the
case of dimension no less than three. Secondly, we prove an integral identity
for an overdetermined problem of fourth order which can be used to deduce
Bennett's symmetry theorem. Finally, we prove a symmetry result for an
overdetermined problem of second order by integral identities.Comment: 17 page
On Ashbaugh-Benguria's Conjecture about Lower Order Dirichlet Eigenvalues of the Laplacian
In this paper, we prove an isoperimetric inequality for lower order
eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space
which strengthens the well-known Ashbaugh-Beguria inequality conjectured by
Payne-P\'olya-Weinberger on the ratio of the first two Dirichlet eigenvalues
and makes an important step toward the proof of a conjecture by
Ashbaugh-Benguria.Comment: 12 pages, comments are welcom
On a Conjecture of Ashbaugh and Benguria about Lower Eigenvalues of the Neumann Laplacian
In this paper, we prove an isoperimetric inequality for lower order
eigenvalues of the free membrane problem on bounded domains of a Euclidean
space or a hyperbolic space which strengthens the well-known Szeg\"o-Weinberger
inequality and supports strongly an important conjecture of Ashbaugh-Benguria.Comment: 16 page
Inequalities for the Steklov Eigenvalues
This paper studies eigenvalues of some Steklov problems. Among other things,
we show the following sharp estimtes. Let be a bounded smooth domain
in an -dimensional Hadamard manifold an let denote the eigenvalues of the Steklov
problem: in and
on . Then with equality holding if and only if
is isometric to an -dimensional Euclidean ball. Let be an
-dimensional compact connected Riemannian manifold with boundary and
non-negative Ricci curvature. Assume that the mean curvature of \pa M is
bounded below by a positive constant and let be the first eigenvalue
of the Steklov problem: in and on . Then
with equality holding if and only if is isometric to a ball of
radius in .Comment: 17 page
Eigenvalues of the Wentzell-Laplace Operator and of the Fourth Order Steklov Problems
We prove a sharp upper bound and a lower bound for the first nonzero
eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary
and an isoperimetric inequality for the same eigenvalue in the case where the
manifold is a bounded domain in a Euclidean space. We study some fourth order
Stekolv problems and obtain isoperimetric upper bound for the first eigenvalue
of them. We also find all the eigenvalues and eigenfunctions for two kind of
fourth order Stekolv problems on a Euclidean ball.Comment: 20 page
Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator
We obtain sharp lower bounds for the first eigenvalue of four types of
eigenvalue problem defined by the bi-Laplace operator on compact manifolds with
boundary and determine all the eigenvalues and the corresponding eigenfunctions
of a Wentzell-type bi-Laplace problem on Euclidean balls.Comment: In this version, we corrected some typo
A reaction-diffusion-advection competition model with two free boundaries in heterogeneous time-periodic environment
In this paper, we study the dynamics of a two-species competition model with
two different free boundaries in heterogeneous time-periodic environment, where
the two species adopt a combination of random movement and advection upward or
downward along the resource gradient. We show that the dynamics of this model
can be classified into four cases, which forms a spreading-vanishing
quartering. The notion of the minimal habitat size for spreading is introduced
to determine if species can always spread. Rough estimates of the asymptotic
spreading speed of free boundaries and the long time behavior of solutions are
also established when spreading occurs. Furthermore, some sufficient conditions
for spreading and vanishing are provided.Comment: 26 page
A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat
We study the diffusive logistic equation with a free boundary in timeperiodic
environment. To understand the effect of the dispersal rate , the original
habitat radius , the spreading capability , and the initial density
on the dynamics of the problem, we divide the time-periodic habitat into
two cases: favorable habitat and unfavorable habitat. By choosing , ,
and as variable parameters, we obtain a spreading-vanishing
dichotomy and sharp criteria for the spreading and vanishing in time-periodic
environment. We introduce the principal eigenvalue to determine the spreading and vanishing of the invasive
species. We prove that if , the
spreading must happen; while if , the
spreading is also possible. Our results show that the species in the favorable
habitat can establish itself if the dispersal rate is slow or the occupying
habitat is large. In an unfavorable habitat, the species vanishes if the
initial density of the species is small, while survive successfully if the
initial value is big. Moreover, when spreading occurs, the asymptotic spreading
speed of the free boundary is determined.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1311.7254 by
other author
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