Related papers: $L^p$-regularity for fourth order elliptic systems…

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-\epsilon<p<2$, we obtain the solvability…

Maximal $L^p$-$L^q$ regularity is proved for the strong, weak and very weak solutions of the inhomogeneous Stokes problem with Navier-type boundary conditions in a bounded domain $\Omega$, not necessarily simply connected. This extends…

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

We consider the steady-state Boussinesq system in the whole three-dimensional space, with the action of external forces and the gravitational acceleration. First, for $3<p\leq +\infty$ we prove the existence of weak $L^p$-solutions.…

This paper establishes a complete framework for infinitely nested logarithmic improvements to regularity criteria for the three-dimensional incompressible Navier-Stokes equations. Building upon our previous works on logarithmically improved…

We consider the maximal regularity problem for non-autonomous evolution equations \begin{equation} \left\{ \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, \tau] u(0)&=&u_0. \end{array} \right. \end{equation} Each operator $A(t)$…

In this paper, we study the regularity of solutions to a linear elliptic equation involving a mixed local-nonlocal operator of the form $$Lu - \operatorname{div}\big(a(x)\nabla u(x)\big)= f, \quad \text{in } \Omega \subset \mathbb{R}^n,$$…

We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness,…

We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $\Delta_{p}u =…

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where…

In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it is a companion to arXiv:1604.02717 in…

We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in…

This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} {\Delta}^2u-\lambda u-\mu|u|^{p-2}u-(I_\alpha*|u|^{4^*_\alpha})|u|^{4^*_\alpha-2}u=0\ \ \mbox{in}\…

The first goal of our paper is to give a new type of regularity criterion for solutions $u$ to Navier-Stokes equation in terms of some supercritical function space condition $u \in L^{\infty}(L^{\alpha ,*})$ (with…

We study fourth-order quasilinear elliptic problems that involve the p-biharmonic operator and Navier boundary conditions. The nonlinear term grows at the critical Sobolev rate. Starting from a Hamiltonian system of two second-order…

We show that the new result on H\"older continuity of solutions to a class of nondiagonal elliptic systems with $p$-growth in [2] can be used to improve the $L^q$ theory for such systems.

In this paper, we consider the system $-\Delta u =\lambda (v+1)^p,\;\;-\Delta v = \gamma (u+1)^\theta$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$ with the Dirichlet boundary condition $u=v=0$ on $\partial \Omega.$ Here $…