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

Multiple solutions of the quasirelativistic Choquard equation

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

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

Existence of solutions to a higher dimensional mean-field equation on manifolds

For $m\geq 1$ we prove an existence result for the equation $$(-\Delta_g)^m u+\lambda=\lambda\frac{e^{2mu}}{\int_M e^{2mu}d\mu_g}$$ on a closed Riemannian manifold $(M,g)$ of dimension $2m$ for certain values of $\lambda$.Comment: 15 Page

On a functional satisfying a weak Palais-Smale condition

In this paper we study a quasilinear elliptic problem whose functional satisfies a weak version of the well known Palais-Smale condition. An existence result is proved under general assumptions on the nonlinearities.Comment: 18 page

On Singularity formation for the L^2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a "finite speed of propagation" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.Comment: 24 pages. Accepted in Nonlinearit

Separation of isomeric glycans by ion mobility spectrometry–the impact of fluorescent labelling

The analysis of complex oligosaccharides is traditionally based on multidimensional workflows where liquid chromatography is coupled to tandem mass spectrometry (LC-MS/MS). Due to the presence of multiple isomers, which cannot be distinguished easily using tandem MS, a detailed structural elucidation is still challenging in many cases. Recently, ion mobility spectrometry (IMS) showed great potential as an additional structural parameter in glycan analysis. While the time-scale of the IMS separation is fully compatible to that of LC-MS-based workflows, there are very few reports in which both techniques have been directly coupled for glycan analysis. As a result, there is little knowledge on how the derivatization with fluorescent labels as common in glycan LC-MS affects the mobility and, as a result, the selectivity of IMS separations. Here, we address this problem by systematically analyzing six isomeric glycans derivatized with the most common fluorescent tags using ion mobility spectrometry. We report >150 collision cross-sections (CCS) acquired in positive and negative ion mode and compare the quality of the separation for each derivatization strategy. Our results show that isomer separation strongly depends on the chosen label, as well as on the type of adduct ion. In some cases, fluorescent labels significantly enhance peak-to-peak resolution which can help to distinguish isomeric specie

N-glycan microheterogeneity regulates interactions of plasma proteins

Altered glycosylation patterns of plasma proteins are associated with autoimmune disorders and pathogenesis of various cancers. Elucidating glycoprotein microheterogeneity and relating subtle changes in the glycan structural repertoire to changes in protein–protein, or protein–small molecule interactions, remains a significant challenge in glycobiology. Here, we apply mass spectrometry-based approaches to elucidate the global and site-specific microheterogeneity of two plasma proteins: α1-acid glycoprotein (AGP) and haptoglobin (Hp). We then determine the dissociation constants of the anticoagulant warfarin to different AGP glycoforms and reveal how subtle N-glycan differences, namely, increased antennae branching and terminal fucosylation, reduce drug-binding affinity. Conversely, similar analysis of the haptoglobin–hemoglobin (Hp–Hb) complex reveals the contrary effects of fucosylation and N-glycan branching on Hp–Hb interactions. Taken together, our results not only elucidate how glycoprotein microheterogeneity regulates protein–drug/protein interactions but also inform the pharmacokinetics of plasma proteins, many of which are drug targets, and whose glycosylation status changes in various disease states

Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part I

In this paper (Part I) and its sequels (Part II and Part III), we analyze the structure of the space of solutions to the epsilon-Dirichlet problem for the Yang-Mills equations on the 4-dimensional disk, for small values of the coupling constant epsilon. These are in one-to-one correspondence with solutions to the Dirichlet problem for the Yang Mills equations, for small boundary data. We prove the existence of multiple solutions, and, in particular, non minimal ones, and establish a Morse Theory for this non-compact variational problem. In part I, we describe the problem, state the main theorems and do the first part of the proof. This consists in transforming the problem into a finite dimensional problem, by seeking solutions that are approximated by the connected sum of a minimal solution with an instanton, plus a correction term due to the boundary. An auxiliary equation is introduced that allows us to solve the problem orthogonally to the tangent space to the space of approximate solutions. In Part II, the finite dimensional problem is solved via the Ljusternik-Schirelman theory, and the existence proofs are completed. In Part III, we prove that the space of gauge equivalence classes of Sobolev connections with prescribed boundary value is a smooth manifold, as well as some technical lemmas used in Part I. The methods employed still work when the 4-dimensional disk is replaced by a more general compact manifold with boundary, and SU(2) is replaced by any compact Lie group

Singular kernels, multiscale decomposition of microstructure, and dislocation models

We consider a model for dislocations in crystals introduced by Koslowski, Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel behaving as the $H^{1/2}$ norm of the slip. We obtain a sharp-interface limit of the model within the framework of $\Gamma$-convergence. From an analytical point of view, our functional is a vector-valued generalization of the one studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement argument no longer applies. Instead we show that the microstructure must be approximately one-dimensional on most length scales and exploit this property to derive a sharp lower bound