### A ${C}^{0}$-theory for the blow-up of second order elliptic equations of critical Sobolev growth.

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For $n=2m\ge 4$, let $\Omega \in {\mathbb{R}}^{n}$ be a bounded smooth domain and $\mathcal{N}\subset {\mathbb{R}}^{L}$ a compact smooth Riemannian manifold without boundary. Suppose that $\left\{{u}_{k}\right\}\in {W}^{m,2}(\Omega ,\mathcal{N})$ is a sequence of weak solutions in the critical dimension to the perturbed $m$-polyharmonic maps $$\frac{\mathrm{d}}{\mathrm{d}t}{|}_{t=0}{E}_{m}\left(\Pi (u+t\xi )\right)=0$$ with ${\Phi}_{k}\to 0$ in ${\left({W}^{m,2}(\Omega ,\mathcal{N})\right)}^{*}$ and ${u}_{k}\rightharpoonup u$ weakly in ${W}^{m,2}(\Omega ,\mathcal{N})$. Then $u$ is an $m$-polyharmonic map. In particular, the space of $m$-polyharmonic maps is sequentially compact for the weak-${W}^{m,2}$ topology.

In this paper we consider Riemannian manifolds $({M}^{n},g)$ of dimension $n\ge 5$, with semi-positive $Q$-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$-curvature. Modifying the test function construction of Esposito-Robert, we show...

In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.

Spaces with corner singularities, locally modelled by cones with base spaces having conical singularities, belong to the hierarchy of (pseudo-) manifolds with piecewise smooth geometry. We consider a typical case of a manifold with corners, the so-called "edged spindle", and a natural algebra of pseudodifferential operators on it with special degeneracy in the symbols, the "corner algebra". There are three levels of principal symbols in the corner algebra, namely the interior,...

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis and Kral-Wedland . We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the...