Sub-criticality of Schroedinger Systems with Antisymmetric Potentials [PDF]
arXiv, 2009 Let $m$ be an integer larger or equal to 3. We prove that Schroedinger systems on $B^m$ with $L^{m/2}-$antisymmetric potential $\Omega$ of the form $$ -\Delta v=\Omega v $$ can be written in divergence form and we deduce that solutions $v$ in $L^{m/(m-2)}$ are in fact $W^{2,q}_{loc}$ for any $q
arxiv
$L^p$ solvability of the Stationary Stokes problem on domains with conical singularity in any dimension [PDF]
arXiv, 2010 The Dirichlet boundary value problem for the Stokes operator with $L^p$ data in any dimension on domains with conical singularity (not necessary a Lipschitz graph) is considered. We establish the solvability of the problem for all $p\in (2-\varepsilon,\infty]$ and also its solvability in $C(\overline{D})$ for the data in $C(\partial D)$
arxiv
Synchronized vector solutions for the nonlinear Hartree system with nonlocal interaction
Advanced Nonlinear StudiesWe are concerned with the following nonlinear Hartree system−Δu+P1(|x|)u=α1|x|−1∗u2u+β|x|−1∗v2u inR3,−Δv+P2(|x|)v=α2|x|−1∗v2v+β|x|−1∗u2v inR3, $$\begin{cases}-{\Delta}u+{P}_{1}\left(\vert x\vert \right)u={\alpha }_{1}\left(\vert x{\vert }^{-1}\ast {u}^{2}
Gao Fashun, Yang Minbo, Zhao Shunneng
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Extensions of Calabi's correspondence between minimal surfaces and maximal surfaces [PDF]
arXiv, 2011 The main goal of this survey is to illustrate geometric applications of the Poincar\'{e} Lemma to constant mean curvature equations. In 1970, Calabi introduced the duality between minimal graphs in three dimensional Euclidean space and maximal graphs in three dimensional Lorentz space. We construct two extensions of Calabi's correspondence.
arxiv
Energy quantization for biharmonic maps [PDF]
arXiv, 2011 In the present work we establish an energy quantization (or energy identity) result for solutions to scaling invariant variational problems in dimension 4 which includes biharmonic maps (extrinsic and intrinsic). To that aim we first establish an angular energy quantization for solutions to critical linear 4th order elliptic systems with antisymmetric ...
arxiv
The strong elliptic maximum principle for vector bundles and applications to minimal maps [PDF]
arXiv, 2012 Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein type theorems in higher ...
arxiv
Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition [PDF]
arXiv, 2012 In this article we present some results on the existence of positive and ground state solutions for the nonlinear Klein-Gordon-Maxwell equations. We introduce a general nonlinearity with subcritical and supercritical growth which does not require the usual Ambrosetti-Rabinowitz condition.
arxiv
On Lp Estimates in Homogenization of Elliptic Equations of Maxwell's Type [PDF]
arXiv, 2012 For a family of second-order elliptic systems of Maxwell's type with rapidly oscillating periodic coefficients in a $C^{1, \alpha}$ domain $\Omega$, we establish uniform estimates of solutions $u_\varep$ and $\nabla \times u_\varep$ in $L^p(\Omega)$ for $1
arxiv
On the structure of phase transition maps for three or more coexisting phases [PDF]
arXiv, 2013 This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.
arxiv
Elliptic and parabolic reguarity for second order divergence operators with mixed boundary conditions [PDF]
arXiv, 2013 We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. From this, elliptic and parabolic regularity results are deduced by means of Sneiberg's isomorphism theorem.
arxiv