Related papers: L^p norms of eigenfunctions in the completely inte…

In this paper, using the method of blow-up analysis, we obtained a Trudinger-Moser inequality involving L p -norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger-Moser functional.…

We look at the $L^p$ bounds on eigenfunctions for polygonal domains (or more generally Euclidean surfaces with conic singularities) by analysis of the wave operator on the flat Euclidean cone $C(\mathbb{S}^1_\rho) := \mathbb{R}_+ \times…

In this short survey, we derive some weyl-type universal inequalities of eigenvalues of the Laplacian on a closed Riemannian manifold of nonnegative Ricci curvature. We also give upper bounds for the $L_{\infty}$ norm of eigenfunctions of…

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(\Omega)$ for every $p \in (1,\infty)$, where $\Omega := \mathbb{R}^{n+1}…

We prove a blow up criterion in terms of the Hessian of the pressure of smooth solutions $u\in C([0, T); W^{2,q} (\mathbb R^3))$, $q>3$ of the incompressible Euler equations. We show that a blow up at $t=T$ happens only if $$\int_0 ^T…

If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2<p\ne p_c=\tfrac{2(n+1)}{n-1}$, the critical exponent. Since improved…

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We…

Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $\Gamma$ of $M$. The sets $\Gamma$ that we consider are Borel…

Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-\lambda^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$…

We establish sharp algebraic criteria for the $L^{p}$-integrability, for $p = 1, 2, \infty$, of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic $\mathbb{Q}$-groups,…

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown…

We establish a priori estimates showing the propagation and generation of $L^p$-norms for solutions to the non-cutoff spatially homogeneous Boltzmann equation with soft potentials. The singularity of the collision kernel is key to generate…

On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…

Motivated by the question of existence of global solutions, we obtain pointwise upper bounds for radially symmetric and monotone solutions to the homogeneous Landau equation with Coulomb potential. The estimates say that blow up in the…

We construct a real sequence $\{\lambda_n\}_{n=1}^{\infty}$ satisfying $\lambda_n = n + o(1)$, and a Schwartz function $f$ on $\mathbb{R}$, such that for any $N$ the system of translates $\{f(x - \lambda_n)\}$, $n > N$, is complete in the…

In this note we provide a new proof of the $W^{2,p}$ Calder\'on-Zygmund regularity estimates for the Laplacian, i.e., $\Delta u=f$ and its parabolic counterpart $\partial_t u-\Delta u=f$. Our proof is an adaptation of a contradiction and…

Motivated by Thurston and Daskalopoulos--Uhlenbeck's approach to Teichm\"uller theory, we study the behavior of $q$-harmonic functions and their $p$-harmonic conjugates in the limit as $q \to 1$, where $1/p + 1/q = 1$. The $1$-Laplacian is…

In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla…

We prove Liouville type theorems for $p$-harmonic functions on exterior domains of the $d$-dimensional Euclidean space, where $1<p<\infty$ and $d\geq 2$. We show that every positive $p$-harmonic function satisfying zero Dirichlet, Neumann…

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window…