Abstract

We establish compactness for nonnegative solutions of the fourth order constant Qcurvature equations on smooth compact Riemannian manifolds of dimension ≥ 5. If the Q-curvature equals −1, we prove that all solutions are universally bounded. If the Qcurvature is 1, assuming that Paneitz operator’s kernel is trivial and its Green function is positive, we establish universal energy bounds on manif...

In this work, we study the weighted Kirchhoff problem \[ \begin{cases} g\big( \int_{B} (w(x) |\Delta u|^{2}) \, dx \big) [\Delta \Delta u)] = f(x,u) &\textrm{in $B$}, \\ u > 0 \frac{\partial u}{\partial n} &\textrm{on $\partial B$}, \end{cases} \] where $B$ is unit ball of $\mathbb{R}^{4}$, $w(x) \big( \log \frac{e}{|x|} \big)^{\beta}$, singular logarithm weight in Adam's embedding, $g$ a conti...

In this note we investigate a fourth-order geometric flow on closed, compact Riemannian manifolds of dimension greater than four. In the presence of a positive conformal invariant it can be shown that this flow must be positivity preserving.

We consider a six-dimensional braneworld model and we study the cosmological evolution of a (4+1) braneuniverse. Introducing matter on the brane we show that the scale factor of the physical three-dimensional brane-universe is related to the scale factor of the fourth dimension on the brane, and the suppression of the extra dimension compared to the three dimensions requires the presence of dar...

We consider a six-dimensional braneworld model and we study the cosmological evolution of a (4+1) brane-universe. Introducing matter on the brane we show that the scale factor of the physical three-dimensional brane-universe is related to the scale factor of the fourth dimension on the brane, and the suppression of the extra dimension compared to the three dimensions requires the presence of da...

This note is inspired by the paper of Bialecki, Fairweather, and Bennett [1] which studies a method for the numerical solution of u00 = f on an interval, and u = f on a rectangle, with zero boundary data; their scheme calculates that C piecewise cubic U for which U 00 (or U) agrees with f at the two (or four) Gauss quadrature points of each subdivision of the original domain. They observe that ...