A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

Global pointwise decay estimates for defocusing radial nonlinear wave equations

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.Comment: 9 pages, 1 figur

The Construction of a Partially Regular Solution to the Landau-Lifshitz-Gilbert Equation in R2\mathbb{R}^2

We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in $\mathbb{R}^2$. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution from the continuous setting into the discrete setting. This method is quite general and accommodates more general geometries involving targets that are compact smooth hypersurfaces.Comment: 43 pages, 2 figure

A convergent and constraint-preserving finite element method for the p-harmonic flow into spheres

Published versio

Multiple solutions of the quasirelativistic Choquard equation

We prove existence of multiple solutions to the quasirelativistic Choquard equation with a scalar potential

Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system

We describe the asymptotic behavior as time goes to infinity of solutions of the 2 dimensional corotational wave map system and of solutions to the 4 dimensional, radially symmetric Yang-Mills equation, in the critical energy space, with data of energy smaller than or equal to a harmonic map of minimal energy. An alternative holds: either the data is the harmonic map and the soltuion is constant in time, or the solution scatters in infinite time

Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<\lambda_k\to 0$. Assuming that $$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we prove that $\Lambda$ is an integer multiple of \Lambda_1:=(2m-1)!\vol(S^{2m}), the total $Q$-curvature of the standard $2m$-dimensional sphere.Comment: 33 page

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow up solutions for the wave map problem from 2+1 dimensions into the 2-sphere. These solutions are the sum of a dynamically rescaled ground-state harmonic map plus a radiation term. The local energy of the latter tends to zero as time approaches blow up time. This is accomplished by first "renormalizing" the rescaled ground state harmonic map profile by solving an elliptic equation, followed by a perturbative analysis