Related papers: Uniform Sobolev inequalities for second order non-…

In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case $p=2$, as well as an extension to the coefficient $p=1$ that was previously unknown.…

We prove certain $L^p$ Sobolev-type inequalities for twisted differential forms on real (and complex) manifolds for the Laplace operator $\Delta$, the differential operators $d$ and $d^*$, and the operator $\bar\partial$. A key tool to get…

We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(\Delta-\alpha)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $\Delta$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of…

The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)}…

For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x…

In this paper, we consider the Euclidean logarithmic Sobolev inequality \begin{eqnarray*} \int_{\mathbb{R}^d}|u|^2\log|u|dx\leq\frac{d}{4}\log\bigg(\frac{2}{\pi d e}\|\nabla u\|_{L^2(\mathbb{R}^d)}^2\bigg), \end{eqnarray*} where $u\in…

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the…

We give $L^p$ estimates for the second derivatives of weak solutions to the Dirichlet problem for equation $\Div(\mathbf{A}\nabla u) = f$ in $\Omega\subset \mathbb{R}^d$ with Sobolev coefficients. In particular, for $f\in L^2(\Omega)…

We show that on $\mathbb S^1(1/\sqrt{d-2})\times\mathbb S^{d-1}(1)$ the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This…

In this work, we consider a class of second order uniformly elliptic operators with smooth and bounded coefficients. We provide some estimates on the norm of the semigroup generated by these operators acting on weighted Sobolev spaces,…

We study a general class of parabolic equations $$ u_t-|Du|^\gamma\big(\Delta u+(p-2) \Delta_\infty^N u\big)=0, $$ which can be highly degenerate or singular. This class contains as special cases the standard parabolic $p$-Laplace equation…

We derive Lyapunov-type inequalities for general third order nonlinear equations involving multiple $\psi$-Laplacian operators of the form \begin{equation*} (\psi_{2}((\psi_{1}(u'))'))' + q(x)f(u) = 0, \end{equation*} where $\psi_{2}$ and…

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm…

We consider a second-order parabolic equation in $\bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally H\"older continuous in the space variables.…

Sobolev-type regularity results are proved for solutions to a class of second order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak…

We show that the inequality $$ \|D^{k-1}(u-\pi u)\|_{\mathrm{L}^{n/(n-1)}(\mathbb{R}^n)}\leq c\|\mathbb{B}(D) u\|_{\mathrm{L}^1(\mathbb{R}^n)} $$ holds for vector fields $u\in\mathrm{C}^\infty_c$ if and only if $\mathbb{B}$ is canceling.…

We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy…

We prove higher-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations in the presence of unbounded source terms. More precisely, we show the existence of a universal number $0< \varepsilon <1$, depending only…

These notes present Sobolev-Gagliardo-Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calder\'on-Zygmund estimates show that the ellipticity is necessary and…

This paper discusses the unique continuation principal of the solutions of the following perturbed fourth order elliptic differential operator $\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\ \sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n…