LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.
Combining the approaches of functionals associated with h-concave functions
and fixed point techniques, we study the existence and uniqueness of a
solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) +
? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R)
denotes the space of all continuous functions on [0,T] equipped with
the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0,
T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued
functions in [0,T]xR.
In this paper, we study the singularly perturbed problem for the Schrödinger–Poisson equation with critical growth. When the perturbed coefficient is small, we establish the relationship between the number of solutions and the profiles of the coefficients. Furthermore, without any restriction on the perturbed coefficient, we obtain a different concentration phenomenon. Besides, we obtain an existence result.
Abstract
We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold
$\mathcal {M}$
. We obtain several results for this setting. If a cocycle is bounded in
$C^{1+\gamma }$
, we show that it has a continuous invariant family of
$\gamma $
-Hölder Riemannian metrics on
$\mathcal {M}$
. We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in
$C^0$
for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.
AbstractWe consider a Cauchy–Dirichlet problem for a quasilinear second order parabolic equation with lower order term driven by a singular coefficient. We establish an existence result to such a problem and we describe the time behavior of the solution in the case of the infinite–time horizon.
Duality by reproducing kernels
