4,410 research outputs found
Integral equations PS-3 and moduli of pants
More than a hundred years ago H.Poincare and V.A.Steklov considered a problem
for the Laplace equation with spectral parameter in the boundary conditions.
Today similar problems for two adjacent domains with the spectral parameter in
the conditions on the common boundary of the domains arises in a variety of
situations: in justification and optimization of domain decomposition method,
simple 2D models of oil extraction, (thermo)conductivity of composite
materials. Singular 1D integral Poincare-Steklov equation with spectral
parameter naturally emerges after reducing this 2D problem to the common
boundary of the domains. We present a constructive representation for the
eigenvalues and eigenfunctions of this integral equation in terms of moduli of
explicitly constructed pants, one of the simplest Riemann surfaces with
boundary. Essentially the solution of integral equation is reduced to the
solution of three transcendent equations with three unknown numbers, moduli of
pants. The discreet spectrum of the equation is related to certain surgery
procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and
D.Sullivan- W.Thurston (1983).Comment: 27 pages, 13 figure
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity
This paper deals with the existence and the asymptotic behavior of
non-negative solutions for a class of stationary Kirchhoff problems driven by a
fractional integro-differential operator and involving a
critical nonlinearity. The main feature, as well as the main difficulty, of the
analysis is the fact that the Kirchhoff function can be zero at zero, that
is the problem is degenerate. The adopted techniques are variational and the
main theorems extend in several directions previous results recently appeared
in the literature
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains
We investigate quantitative properties of the nonnegative solutions
to the nonlinear fractional diffusion equation, , posed in a bounded domain, with for . As we use one of the most common
definitions of the fractional Laplacian , , in a bounded
domain with zero Dirichlet boundary conditions. We consider a general class of
very weak solutions of the equation, and obtain a priori estimates in the form
of smoothing effects, absolute upper bounds, lower bounds, and Harnack
inequalities. We also investigate the boundary behaviour and we obtain sharp
estimates from above and below. The standard Laplacian case or the linear
case are recovered as limits. The method is quite general, suitable to be
applied to a number of similar problems
Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains
We consider nonlinear diffusive evolution equations posed on bounded space
domains, governed by fractional Laplace-type operators, and involving porous
medium type nonlinearities. We establish existence and uniqueness results in a
suitable class of solutions using the theory of maximal monotone operators on
dual spaces. Then we describe the long-time asymptotics in terms of
separate-variables solutions of the friendly giant type. As a by-product, we
obtain an existence and uniqueness result for semilinear elliptic non local
equations with sub-linear nonlinearities. The Appendix contains a review of the
theory of fractional Sobolev spaces and of the interpolation theory that are
used in the rest of the paper.Comment: Keywords: Fractional Laplace operators, Porous Medium diffusion,
Existence and uniqueness theory, Asymptotic behaviour, Fractional Sobolev
Space
Stationarity-conservation laws for certain linear fractional differential equations
The Leibniz rule for fractional Riemann-Liouville derivative is studied in
algebra of functions defined by Laplace convolution. This algebra and the
derived Leibniz rule are used in construction of explicit form of
stationary-conserved currents for linear fractional differential equations. The
examples of the fractional diffusion in 1+1 and the fractional diffusion in d+1
dimensions are discussed in detail. The results are generalized to the mixed
fractional-differential and mixed sequential fractional-differential systems
for which the stationarity-conservation laws are obtained. The derived currents
are used in construction of stationary nonlocal charges.Comment: 28 page
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