16,963 research outputs found
Qualitative properties and existence of sign changing solutions with compact support for an equation with a p-Laplace operator
We consider radial solutions of an elliptic equation involving the p-Laplace
operator and prove by a shooting method the existence of compactly supported
solutions with any prescribed number of nodes. The method is based on a change
of variables in the phase plane corresponding to an asymptotic Hamiltonian
system and provides qualitative properties of the solutions
A Conformally Invariant Holographic Two-Point Function on the Berger Sphere
We apply our previous work on Green's functions for the four-dimensional
quaternionic Taub-NUT manifold to obtain a scalar two-point function on the
homogeneously squashed three-sphere (otherwise known as the Berger sphere),
which lies at its conformal infinity. Using basic notions from conformal
geometry and the theory of boundary value problems, in particular the
Dirichlet-to-Robin operator, we establish that our two-point correlation
function is conformally invariant and corresponds to a boundary operator of
conformal dimension one. It is plausible that the methods we use could have
more general applications in an AdS/CFT context.Comment: 1+49 pages, no figures. v2: Several typos correcte
Existence results to a nonlinear p(k)-Laplacian difference equation
In the present paper, by using variational method, the existence of
non-trivial solutions to an anisotropic discrete non-linear problem involving
p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The
main technical tools applied here are the two local minimum theorems for
differentiable functionals given by Bonanno.Comment: The final version of this paper will be published in Journal of
Difference Equations and Applications in 201
On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
We review applications of theory of classical and quantum integrable systems
to the free-boundary problems of fluid mechanics as well as to corresponding
problems of statistical mechanics. We also review important exact results
obtained in the theory of multi-fractal spectra of the stochastic models
related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions
Normal random matrix ensemble as a growth problem
In general or normal random matrix ensembles, the support of eigenvalues of
large size matrices is a planar domain (or several domains) with a sharp
boundary. This domain evolves under a change of parameters of the potential and
of the size of matrices. The boundary of the support of eigenvalues is a real
section of a complex curve. Algebro-geometrical properties of this curve encode
physical properties of random matrix ensembles. This curve can be treated as a
limit of a spectral curve which is canonically defined for models of finite
matrices. We interpret the evolution of the eigenvalue distribution as a growth
problem, and describe the growth in terms of evolution of the spectral curve.
We discuss algebro-geometrical properties of the spectral curve and describe
the wave functions (normalized characteristic polynomials) in terms of
differentials on the curve. General formulae and emergence of the spectral
curve are illustrated by three meaningful examples.Comment: 44 pages, 14 figures; contains the first part of the original file.
The second part will be submitted separatel
Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties
We investigate the equation where corresponds to the
fractional Laplacian on hyperbolic space for and is a
smooth nonlinearity that typically comes from a double well potential. We prove
the existence of heteroclinic connections in the following sense; a so-called
layer solution is a smooth solution of the previous equation converging to at any point of the two hemispheres and which is strictly increasing with respect to the signed distance to a
totally geodesic hyperplane We prove that under additional conditions on
the nonlinearity uniqueness holds up to isometry. Then we provide several
symmetry results and qualitative properties of the layer solutions. Finally, we
consider the multilayer case, at least when is close to one
Existence and multiplicity results for resonant fractional boundary value problems
We study a Dirichlet-type boundary value problem for a pseudo-differential
equation driven by the fractional Laplacian, with a non-linear reaction term
which is resonant at infinity between two non-principal eigenvalues: for such
equation we prove existence of a non-trivial solution. Under further
assumptions on the behavior of the reaction at zero, we detect at least three
non-trivial solutions (one positive, one negative, and one of undetermined
sign). All results are based on the properties of weighted fractional
eigenvalues, and on Morse theory
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