Related papers: Improved decoupling for the parabola

We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension $3$ and higher and for coefficients having a finite range of dependence, we prove a pointwise version of…

We prove doubling inequalities for solutions of elliptic systems with an iterated Laplacian as diagonal principal part and for solutions of the Lame' system of isotropic linearized elasticity. These inequalities depend on global properties…

A decoupling type inequality for a sum of functions of Guassian vectors is established.

We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data.

Given a two-loop beta function for multiple marginal coupling constants, we derive an asymptotic formula for the running coupling constants driven to an infrared fixed point. It can play an important role in universal loglog corrections to…

We prove sharp $\ell^2$-decoupling inequalities for non-degenerate complex curves via the bilinear argument due to Guo--Li--Yung--Zorin-Kranich, which in turn is inspired by the efficient congruencing argument of Wooley. Secondly,…

We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the…

In this article, we establish an $\ell^2$ decoupling inequality for the surface $$F_4^2:=\Big\{(\xi_1,\xi_2,\xi_1^4+\xi_2^4): (\xi_1,\xi_2) \in [0,1]^2\Big\}$$ associated with the decomposition adapted to finite type geometry from our…

We obtain a series improvement to higher-order $L^p$-Rellich inequalities on a Riemannian manifold $M$. The improvement is shown to be sharp as each new term of the series is added.

For each $d\geq 0$, we prove decoupling inequalities in $\mathbb R^3$ for the graphs of all bivariate polynomials of degree at most $d$ with bounded coefficients, with the decoupling constant depending uniformly in $d$ but not the…

We consider a class of of massless gradient Gibbs measures, in dimension greater or equal to three, and prove a decoupling inequality for these fields. As a result, we obtain detailed information about their geometry, and the percolative…

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…

For the paraboloid decomposition $F=\sum_{\Theta} F_{\Theta}$ with $\Theta\subset{|\xi|\sim\lambda}$ and radius $r=\lambda^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}}…

We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $\Omega_T := \Omega\times (0,T)$,…

The moving coframe method is applied to solve the local equivalence problem for the class of linear parabolic equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and…

The finite operators are derived for the nonrecoil (leading order in the $(m/M)$ expansion) relativistic corrections in hydrogen-like atoms and ions at orders $m\alpha^6$ in the two- and three-body formalism beyond the adiabatic…

We analyze $p$-Laplace operators with degenerate elliptic coefficients. This investigation includes Gru\v{s}in type $p$-Laplace operators. We describe a \emph{separation phenomenon} in elliptic and parabolic $p$-Laplace type equations,…

We introduce an analogue of Calder\'on's first commutator along a parabola, and establish its $L^2$ boundedness under essentially sharp hypotheses.

We prove a sharp $l^{10}(L^{10})$ decoupling for the moment curve in $\mathbb{R}^3$. The proof involves a two-step decoupling combined with new incidence estimates for planks, tubes and plates.

We improve the Gagliardo-Nirenberg inequality \[ \|\varphi\|_{L^q(\mathbb{R}^n)} \le C \|\nabla\varphi\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nabla\varphi\|_{L^r(\mathbb{R}^n)}), \] $r=2$,…