我们建立了与分数拉普拉斯算子相关的狄利克雷问题解的形状导数的点wise公式。具体来说，我们考虑方程$(-Δ)^s u = h$在$Ω$内，$u=0$在$Ω^c$的情况，其中右边$h$要么是狄拉克δ分布，要么是利普希茨函数。在两种情况下，我们证明了相应的解在每个方向上都是形状可微的，并推导了其形状导数点wise值的公式。这些公式涉及领域边界的积分和分数诺伊曼迹。这将经典的哈达玛变分公式扩展到了分数拉普拉斯算子的情况。我们的论证精神与\cite{Ushikoshi, Kozono-Ushikoshi}相似，并基于偏微分方程技术。
英文摘要：
We establish pointwise formulas for the shape derivative of solutions to the Dirichlet problem associated with the fractional Laplacian. Specifically, we consider the equation $(-Δ)^s u = h$ in $Ω$ and $u=0$ in $Ω^c$, where the right-hand side $h$ is either a Dirac delta distribution or a Lipschitz function. In both cases, we prove that the corresponding solution is shape differentiable in every direction and we derive a formula for the pointwise value of its shape derivative. These formulas involve integral on the domain's boundary and fractional Neumann's traces. This extends to the case of the fractional Laplacian the well-known Hadamard variational formula for the standard Laplacian. Our argument is in the spirit of \cite{Ushikoshi, Kozono-Ushikoshi} and is based on PDEs techniques.
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.
This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for $C^1$-domains with $C^1$-perturbations. We also derive an optimal estimate for the remainder term in the $C^{1,α}$-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.
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…
We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frechét differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.
In this paper, we present an approach to the fractional Dunkl Laplacian in a framework emerging from certain reflection symmetries in Euclidean spaces. Our main result is pointwise formulas, Bochner subordination, and an extension problem for the fractional Dunkl Laplacian as well
We extend and survey results in the theory of analysis on fractal sets from the standard Laplacian on the Sierpiński gasket to the energy Laplacian, which is defined weakly by using the Kusuoka energy measure. We also extend results from the Sierpiński gasket to level-$k$ Sierpiński gaskets, for all $k\geq 2$. We observe that the pointwise formula for the energy Laplacian is valid for all level-$k$ Sierpiński gaskets, $SG_k$, and we provide a proof of a known formula for the renormalization constants of the Dirichlet form for post-critically finite self-similar sets along with a probabilistic …
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…
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.
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 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.
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class $C^1$. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case however, in the $C^1$-case we derive an asymptotic formula for the eigenvalues when the domains are of class $C^1$. Moreover, as an application we consider the case of a $C^1$-perturbation when the reference domain is of class $C^{1,α}$.
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 prove a fractional Pohozaev type identity in a generalized framework and discuss its applications. Specifically, we shall consider applications to nonexistence of solutions in the case of supercritical semilinear Dirichlet problems and regarding a Hadamard formula for the derivative of Dirichlet eigenvalues of the fractional Laplacian with respect to domain deformations. We also derive the simplicity of radial eigenvalues in the case of radial bounded domains and apply the Hadamard formula to this case.
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…
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 introduce n/p-harmonic maps as critical points of E(v) the Lp-Norm of the alpha-laplacian of v, where pointwise v maps Rn into a sphere, and alpha = n/p. This energy combines the non-local behaviour of the fractional harmonic maps introduced by Riviere and the first author with the degenerate arguments of the n-laplacian. In this setting, we will prove Holder continuity.
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.
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…
In a Hadamard manifold $M$, it is proved that if $u$ is a $λ$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $λ$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,λ$ and $H.$
We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-Δ)^s u =f$ in $Ω$, $u \equiv 0$ on $\mathbb{R}^N\setminus Ω$ in an open bounded set $Ω\subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $Ω$ is of class $C^2$ and $f \in C^α(\barΩ)$ for some $α>0$, we show that the map $(0,1) \to L^\infty(Ω)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions…
We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ λ_{s,p}(Ω):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(Ω),\,\, \|u\|_{L^p(Ω)}=1 \right\}, $$ where $s\in (0,1)$, $Ω$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $Ω\mapsto λ_{s,p}(Ω)$ under domain perturbations. In the case where $ λ_{s,p}(Ω)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal v…
This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n.
In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus $\mathbb{T}^n$ from the fractional Laplacian on $\mathbb{R}^n$. Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that $L^2$ functions on the torus can not be identified with $L^2$ functions on $\mathbb{R}^n$. The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise…
We study the fractional Laplacian $(-Δ)^{σ/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^γ$, $γ>0$, where $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. This generalizes to all $γ>0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for $(-Δ)^{σ/2}$, when $0<σ<2$. We dedu…
The \textit{method of semigroups} is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators $L$ and their regularity properties in related functional spaces. The approach was introduced by the author and José L.~Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian $L^s=(-Δ)^s$, $0<s<1$. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise for…
We set up a general framework for Calderón projectors (and their generalization to non-compact manifolds), associated with complex Laplacians e.g. obtained by Wick rotation of a Lorentzian metric. In the analytic case, we use this to show that the Laplacian's Green's functions have analytic continuations whose boundary values are two-point functions of analytic Hadamard states. The result does not require the metric to be stationary. As an aside, we describe how thermal states are obtained as a special case of this construction if the coefficients are time-independent.
We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $Ω$ of $\R^n$. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a $L^1$ constraint on densities, the so-called {\it Rellich functions} maximize this functional.Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that …
We study quasilinear elliptic equations of the type $-Δ_{p} u = σu^{q} + μ\; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $μ$ and $σ$ are nonnegative measurable functions, or locally finite measures, and $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian. Similar equations with more general local and nonlocal operators in place of $Δ_{p}$ are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions $u$: $$ u(x) \approx (\bf{W}_p σ(x))^{\frac{p-q}{p-q-1}} + \bf{K}_{p,q} σ(x) + \bf{W}_p μ(x), \quad x \in \bf{R…
The Green function has a complex dependence upon its underlying domain and differential operator. We briefly review Hadamard's formula for the first variation of the Green function due to a perturbation of the domain. We then take a different avenue and approximate the change in the Green function when the Laplacian is perturbed into a number of different operators: Helmholtz, Schrödinger and Laplace--Beltrami.
Let $Ω\subseteq \mathbb{R}^n$ be an open set, where $n \geq 2$. Suppose $ω$ is a locally finite Borel measure on $Ω$. For $α\in (0,2)$, define the fractional Laplacian $(-\triangle )^{α/2}$ via the Fourier transform on $\mathbb{R}^n$, and let $G $ be the corresponding Green's operator of order $α$ on $Ω$. Define $T(u) = G(u ω).$ If $\Vert T \Vert_{L^2(ω) \rightarrow L^2 (ω)} <1$, we obtain a representation for the unique weak solution $u$ in the homogeneous Sobolev space $L^{α/2, 2}_0 (Ω)$ of \[ (-\triangle)^{α/2} u = u ω+ ν\,\,\, \mbox{on} \,\,\, Ω, \,\,\, u=0 \,\,\, \mbox{on} \,\,\, Ω^c, \] …
