Existence of Strong and Nontrivial Solutions to Strongly Coupled Elliptic Systems [PDF]
arXiv, 2016 We establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations. The existence of of nontrivial and non constant solutions (or pattern formations) will also be studied.
arxiv
The $\infty$-Besov Capacity Problem [PDF]
, 2015 A theory of $\infty$-Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the $\infty$-Besov semi-norms, we obtain new restriction-extension inequalities and we characterize the point-wise multipliers acting on the $\infty$-Besov spaces.
arxiv +1 more source
Approximate tangents, harmonic measure, and domains with rectifiable boundaries [PDF]
arXiv, 2016 Let $\Omega \subset \mathbb{R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal{T}_n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial\Omega$ at $\xi$ and $\partial\Omega$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $\xi$.
arxiv
Green function estimates on complements of low-dimensional uniformly rectifiable sets. [PDF]
Math Ann, 2023 David G, Feneuil J, Mayboroda S.
europepmc +1 more source
Global well-posedness and scattering for Derivative Schrödinger equation [PDF]
arXiv, 2009 In this paper we study the Cauchy problem for the elliptic and non-elliptic derivative nonlinear Schr\"odinger equations in higher spatial dimensions ($n\geq 2$) and some global well-posedness results with small initial data in critical Besov spaces $B^s_{2,1}$ are obtained.
arxiv
A fractional version of Rivière's GL(n)-gauge. [PDF]
Ann Mat Pura Appl, 2022 Da Lio F, Mazowiecka K, Schikorra A.
europepmc +1 more source
The fundamental solution of nonlinear equations with natural growth terms [PDF]
arXiv, 2010 We find bilateral global bounds for the fundamental solutions associated with some quasilinear and fully nonlinear operators perturbed by a nonnegative zero order term with natural growth. In addition, we consider the Sobolev regularity of the fundamental solution away from its pole.
arxiv
A weighted $L_p$-theory for second-order elliptic and parabolic partial differential systems on a half space [PDF]
arXiv, 2012 In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
arxiv
Strichartz estimates for Schrödinger equations on irrational tori [PDF]
arXiv, 2013 In this paper, we prove new Strichartz estimates for linear Schrodinger equations posed on d-dimensional irrational tori. Then, we use these estimates to prove subcritical and critical local well-posedness results for nonlinear Schrodinger equations (NLS) on irrational tori.
arxiv
On the Strichartz estimates for the kinetic transport equation [PDF]
arXiv, 2013 We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present a new approach to proving the non-endpoint cases using multilinear analysis.
arxiv