For all functions on an arbitrary open set $Ω\subset\R^3$ with zero boundary values, we prove the optimal bound \[ \sup_Ω|u| \leq (2π)^{-1/2} \left(\int_Ω|\nabla u|^2 \,dx\, \int_Ω|Δu|^2 \,dx\right)^{1/4}. \] The method of proof is elementary and admits generalizations. The inequality is applied to establish an existence theorem for the Burgers equation.
The fractional laplacian is an operator appearing in several evolution models where diffusion coming from a Lévy process is present but also in the analysis of fluid interphases. We provide an extension of a pointwise inequality that plays a rôle in their study. We begin recalling two scenarios where it has been used. After stating the results, for fractional Laplace-Beltrami and Dirichlet-Neumann operators, we provide an sketch of their proofs, unravelling the underlying principle to such inequalities.
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Δ_{p}u=0,\ x\inΩ$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Δ_{p}u=|u|^{λ-1}u, \ x\inΩ$, which are then generalized for the equation with gen…
We derive a priori bounds for positive supersolutions of $ - Δ_{p} u = ρ(x) f(u) $, where $p>1$ and $Δ_{p}$ is the $p$-Laplace operator, in a smooth bounded domain of $R^{N}$ with zero Dirichlet boundary conditions. We apply the results to nonlinear elliptic eigenvalue problem $ - Δ_{p} u = λf(u) $, with Dirichlet boundary condition, where $ f $ is a nondecreasing continuous differentiable function on $[0,\infty]$ such that $ f(0) > 0 $, $ f(t)^{\frac{1}{p-1}} $ is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter $ λ_{p}^{*} $. In particular, we conside…
In this paper, we study an inhomogeneous variant of the normalized $p$-Laplacian evolution which has been recently treated in \cite{BG1}, \cite{Do}, \cite{MPR} and \cite{Ju}. We show that if the initial datum satisfies the pointwise gradient estimate \eqref{e:main1} a.e., then the unique solution to the Cauchy problem \eqref{main5} satisfies the same gradient estimate a.e. for all later times, see \eqref{e:main} below. A general pointwise gradient bound for the entire bounded solutions of the elliptic counterpart of equation \eqref{main5} was first obtained in \cite{CGS}. Such estimate general…
Any positive power of the Laplacian is related via its Fourier symbol to a hypersingular integral with finite differences. We show how this yields a pointwise evaluation which is more flexible than other notions used so far in the literature for powers larger than 1; in particular, this evaluation can be applied to more general boundary value problems and we exhibit explicit examples. We also provide a natural variational framework and, using an asymptotic analysis, we prove how these hypersingular integrals reduce to polyharmonic operators in some cases. Our presentation aims to be as self-co…
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
We study quasilinear elliptic equations of the type $$-Δ_pu=σ\, u^q \quad \text{in} \, \, \, \mathbb{R}^n,$$ where $Δ_p u=\nabla \cdot(\nabla u |\nabla u|^{p-2})$ is the $p$-Laplacian (or a more general $\mathcal{A}$-Laplace operator $\text{div} \, \mathcal{A}(x, \nabla u)$), $0<q < p-1$, and $σ\ge 0$ is an arbitrary locally integrable function or measure on $\mathbb{R}^n$. We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis-Kamin type given in terms of Wolff potentials. Similar probl…
In this note we obtain some Strichartz estimates for the Schrödinger equation associated to the twisted Laplacian on $\mathbb{C}^{n}\cong \mathbb{R}^{2n}$. The initial data will be considered in suitable Sobolev spaces associated to the twisted Laplacian.
We deal with solutions of the Cauchy problem to linear both homogeneous and nonhomogeneous parabolic second order equations with real constant coefficients in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$, where $n\geq 1$ and $T<\infty$. The homogeneous equation is considered with initial data in $L^p({\mathbb R}^n)$, $1\leq p \leq \infty $. For the nonhomogeneous equation we suppose that initial function is equal to zero and the function in the right-hand side belongs to $f\in L^p({\mathbb R}^{n+1}_T)\cap C^α\big (\bar{{\mathbb R}^{n+1}_T} \big ) $ , $p>n+2$ and $α\in (0, 1)$. Ex…
We consider Laplacian eigenfunctions on a $d-$dimensional bounded domain $M$ (or a $d-$dimensional compact manifold $M$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions $(e_\ell)_{\ell \in \mathbb{N}}$. We study the subspace of all pointwise products $$ A_n = \mbox{span} \left\{ e_i(x) e_j(x): 1 \leq i,j \leq n\right\} \subseteq L^2(M).$$ Clearly, that vector space has dimension $\mbox{dim}(A_n) = n(n+1)/2$. We prove that products $e_i e_j$ of eigenfunctions are simple in a certain sense: for any $\varepsilon > 0$, there exists a low-dimensional vector spac…
We deal with $m$-component vector-valued solutions to the Cauchy problem for linear both homogeneous and nonhomogeneous weakly coupled second order parabolic system in the layer ${\mathbb R}^{n+1}_T={\mathbb R}^n\times (0, T)$. We assume that coefficients of the system are real and depending only on $t$, $n\geq 1$ and $T<\infty$. The homogeneous system is considered with initial data in $[L^p({\mathbb R}^n)]^m$, $1\leq p \leq \infty $. For the nonhomogeneous system we suppose that the initial function is equal to zero and the right-hand side belongs to $[L^p({\mathbb R}^{n+1}_T)]^m\cap [C^α\bi…
We consider local weak solutions of the Poisson equation for the p--Laplace operator. We prove a higher differentiability result, under an essentially sharp condition on the right-hand side. The result comes with a local scaling invariant a priori estimate.
We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.
Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt's inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide short proofs of the Frank-Seiringer "Hardy inequalities" where fractional smoothness is measured by a Besov norm.
We prove a new asymptotic mean value formula for the $p$-Laplace operator, $$ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), $$ valid in the viscosity sense. In the plane, and for a certain range of $p$, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.
We provide a Hopf boundary lemma for the regional fractional Laplacian $(-Δ)^s_Ω$, with $Ω\subset\mathbb{R}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation $(-Δ)^s_Ω u = c(x)u$ in $Ω$, we show that the ratio $u(x)/(\mathrm{dist}(x,\partialΩ))^{2s-1}$ is strictly positive as $x$ approaches the boundary $\partialΩ$ of $Ω$. We also prove a strong maximum principle for distributional super-solutions.
We derive sharp Adams inequalities for the Riesz and more general Riesz-like potentials on the whole of R^n. As a consequence, we obtain sharp Moser-Trudinger inequalities for the critical Sobolev spaces W^{a,n/a}(R^n), 0<a<n. These inequalities involve fractional Laplacians, higher order gradients, general homogeneous elliptic operators with constant coefficients, and general trace type Borel measures.
We are interested in the three-dimensional quasilinear wave equations with null condition. Global existence and pointwise decay for this model have been proved in the celebrated works of Klainerman \cite{Klainerman86} and Christodoulou \cite{Christodoulou86} for small smooth initial data. In this work, we illustrate the precise pointwise asymptotic behavior of the solutions for initial data posed on a hyperboloid and show that the decay rate $v^{-1}u^{-1}$ is optimal for a generic set of initial data.
We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $Ω$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.
Let $u(t,x)$ be a solution of the heat equation in $\mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-Δϕ_k = λ_k ϕ_k$ with Dirichlet conditions on sm…
By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well as Hölder continuity of the gradient of the heat kernel.
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
The solutions of boundary value problems for the Laplacian and the bilaplacian exhibit very different qualitative behaviors. Particularly, the failure of general maximum principles for the bilaplacian implies that solutions of higher-order problems are less rigid and more complex. One way to better understand this transition is to study the intermediate Dirichlet problem in terms of fractional Laplacians. This survey aims to be an introduction to these type of problems; in particular, the different pointwise notions for these operators is introduced considering a suitable natural extension of …
We study the asymptotic properties of the small data solutions of the Vlasov-Maxwell system in dimension three. No neutral hypothesis nor compact support assumptions are made on the data. In particular, the initial decay in the velocity variable is optimal. We use vector field methods to obtain sharp pointwise decay estimates in null directions on the electromagnetic field and its derivatives. For the Vlasov field and its derivatives, we obtain optimal pointwise decay estimates by a vector field method where the commutators are modification of those of the free relativistic transport equation.…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $\mathbb{R}^n$ are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the $p$-Laplace system, our conclusions broaden the range of the admissible values of the exponent $p$ previously known.
Combined with our previous work \cite{LW19eigenvalue}, we prove sharp lower bound estimates for the first nonzero eigenvalue of the weighted $p$-Laplacian with $1< p< \infty$ on a compact Bakry-Émery manifold $(M^n,g,f)$, without boundary or with a convex boundary and Neumann boundary condition, satisfying $\text{Ric}+\nabla^2 f \geq κ\, g$ for some $κ\in \mathbb{R}$.
We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.
In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by Minakshisundaram's ideas, we find a precise pointwise description of $(-Δ_{\mathbb{S}^{n-1}})^s u(x)$ in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, addition…
We use the mean value property in an asymptotic way to provide a notion of a pointwise Laplacian, called AMV Laplacian, that we study in several contexts including the Heisenberg group and weighted Lebesgue measures. We focus especially on a class of metric measure spaces including intersecting submanifolds of $\mathbb{R}^n$, a context in which our notion brings new insights; the Kirchhoff law appears as a special case. In the general case, we also prove a maximum and comparison principle, as well as a Green-type identity for a related operator.
Let $(Ω,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial Ω.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp $h$-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosph…
In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.
The core result of this paper is an upper bound for the ground state energyof the magnetic Laplacian with constant magnetic field on cones that are contained in ahalf-space. This bound involves a weighted norm of the magnetic field related to momentson a plane section of the cone. When the cone is sharp, i.e. when its section is small, thisupper bound tends to 0. A lower bound on the essential spectrum is proved for familiesof sharp cones, implying that if the section is small enough the ground state energy is aneigenvalue. This circumstance produces corner concentration in the semi-classical …
A representation of the sharp coefficient in a pointwise estimate for the gradient of the generalized Poisson integral of a function $f$ on ${\mathbb R}^n$ is obtained under the assumption that $f$ belongs to $L^p$. The explicit value of the coefficient is found for the cases $p=1$ and $p=2$.
