Ying Wang, Hongxing Chen, Hichem Hajaiej
We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
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Eigenvalue estimates for the fractional Laplacian on lattice subgraphs
本文研究图子图上带 Dirichlet 边界条件的分数阶 Laplace 算子，并针对格子图建立前 k 个特征值之和的上下界估计，推广了 Li-Yau 与 Kröger 的经典结论。
We provide bounds for the sequence of eigenvalues $\{λ_i(Ω)\}_i$ of the Dirichlet problem $$ L_Δu=λu\ \ {\rm in}\ \, Ω,\quad\quad u=0\ \ {\rm in}\ \ \mathbb{R}^N\setminus Ω,$$ where $L_Δ$ is the logarithmic Laplacian operator with Fourier transform symbol $2\ln |ζ|$. The logarithmic Laplacian operator is not positively definitive if the volume of the domain is large enough. In this article, we obtain the upper and lower bounds for the sum of the first $k$ eigenvalues by extending the Li-Yau method and Kröger's method respectively. Moreover, we show the limit of the sum of the first $k$ eigenva…
The existence of at least two solutions to superlinear integral equation in cone is proved using the Krasnosielskii Fixed Point Theorem. The result is applied to the Dirichlet BVPs with the fractional Laplacian.
In the present paper we obtain growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue in bounded non-Lipschitz domains. The proposed method is based on composition operators generated by quasiconformal mappings and their applications to weighted Sobolev inequalities.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.
We prove the multiplicity of spike-layer type solutions to the Dirichlet problem for the equation $-Δ_p u = u^{q-1}$ in expanding annuli.
We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.
Counting function and lower bound for Dirichlet eigenvalues of the m-order logarithmic Laplacian
本文研究有界Lipschitz域上m阶对数拉普拉斯算子Dirichlet特征值的计数函数极限及下界。
Inverse problems for the spectral fractional Laplacian with inhomogeneous Dirichlet boundary data
本文研究了具有非均匀狄利克雷边界数据的谱分数阶拉普拉斯算子，提出狄利克雷到诺伊曼映射并分析反问题，同时建立谱分数阶拉普拉斯算子的密度结果。
We study spectral stability estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains $Ω\subset\mathbb R^2$. With the help of these estimates we obtain asymptotically sharp inequalities of ratios of eigenvalues in the frameworks of the Payne-Pólya-Weinberger inequalities. These estimates are equivalent to spectral gap estimates of the Dirichlet eigenvalues of the Laplacian in non-convex domains in terms of conformal (hyperbolic) geometry.
We construct Green functions of Dirichlet boundary value problems for sub-Laplacians on certain unbounded domains of a prototype Heisenberg-type group (prototype H-type group, in short). We also present solutions in an explicit form of the Dirichlet problem for the sub-Laplacian with non-zero boundary datum on half-space, quadrant-space, etc, as well as in a strip.
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space which strengthens the well-known Ashbaugh-Beguria inequality conjectured by Payne-Pólya-Weinberger on the ratio of the first two Dirichlet eigenvalues and makes an important step toward the proof of a conjecture by Ashbaugh-Benguria.
In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-Δ_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $Γ$-convergence.
In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.
On the solutions of a double-phase Dirichlet problem involving the 1-Laplacian
针对含1-Laplacian的非齐次Dirichlet双相问题，论文证明了弱解的存在唯一性，并通过相应极小化问题给出其变分刻画。
A monotonicity result for the first Steklov-Dirichlet Laplacian eigenvalue
本文研究带球形洞的环形区域中拉普拉斯算子的第一Steklov-Dirichlet特征值，证明了外部区域中心对称时特征值关于洞的单调性结果。
Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains
本文改进了Dirichlet特征值的下界，通过推导两种二元多项式展开式，优化了现有下界，并证明所获下界在正项捕获上更全面。
In the whole space $\mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $\mathbb R^d$.
In this paper, we prove the existence of a weak solution for the Dirichlet boundary value problem related to the $p(x)-$Laplacian $$ -\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)+u\in -[\underline{g}(x,u),\overline{g}(x,u)], $$ by using the degree theory after turning the problem into a Hammerstein equation. The right hand side $g$ is a possibly discontinuous function in the second variable satisfying some non-standard growth conditions..
Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic Laplacian
本文研究涉及对数拉普拉斯算子的狄利克雷问题，利用Kelvin变换构造障碍函数证明最优边界正则性，并由此得到非负弱解的Hopf型引理及非线性问题解的唯一性。
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
We obtain a criterion for the existence of solutions of the problem $$ Δ_p u = 0 \quad \mbox{in } M \setminus \partial M, \quad \left. u \right|_{ \partial M } = h, $$ with the bounded Dirichlet integral, where $M$ is an oriented complete Riemannian manifold with boundary and $h \in W_{p, loc}^1 (M)$, $p > 1$.
In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-nor…
In this paper we study the nonlinear elliptic problem involving p(x)-Laplacian with nonsmooth potential, where the weighted function may change sign. By using critical point theory for locally Lipschitz functionals due to Chang, we obtain conditions which ensure the existence of a solution for our problem.
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $Γ$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $Γ$-converges to a minimizer of Hölder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-fun…
Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs
本文引入了具有 Dirichlet 边界条件的子图上的离散多重 Laplace 算子，推导了格图 $\mathbb{Z}^{d}$ 有限子图上其前 $k$ 个特征值之和的上下界，并证明了高阶特征值与低阶特征值平方之间的不等式关系。
Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian
本文研究了非局部狄利克雷问题中对数$p$-拉普拉斯算子的极小特征值存在性，并利用$p$-对数赛博利夫不等式证明了新存在性结果。
Let $Ω$ be a bounded annular $C^{1,1}$ domain in $\mathbb{R}^2$ which is left invariant under the action of the dihedral group $D_n$ of isometries of $\mathbb{R}^2$ .We show that the nodal line of a second Dirichlet eigenfunction must intersect the boundary of $Ω$, under suitable conditions on $\frac{\partial}{\partial θ}$ .
The Dirichlet Problem For the Logarithmic p-Laplacian
本文引入对数p-拉普拉斯算子，建立变分框架研究其狄利克雷问题，并推导出第一特征值的Faber-Krahn不等式及边界Hardy型不等式等结果。
We consider an optimal control problem of diffusion equation with missing data governed by the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary interaction domain disjoint from the domain of the state equation. We assume that the unknown initial condition belongs to an appropriate space of infinite dimension, the so-called space of uncertainties. The key tools we used in order to characterize the optimal control is the no-regret and low-regret control developed by J.L Lions.
We present explicit formulas for solutions to nonhomogeneous boundary value problems involving any positive power of the Laplacian in the half-space. For non-integer powers the operator becomes nonlocal and this requires a suitable extension of Dirichlet-type boundary conditions. A key ingredient in our proofs is a point inversion transformation which preserves harmonicity and allows us to use known results for the ball. We include uniqueness statements, regularity estimates, and describe the growth or decay of solutions at infinity and at the boundary.
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 consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.
The purpose of this article is to establish new lower bounds for the sums of powers of eigenvalues of the Dirichlet fractional Laplacian operator $(-Δ)^{α/2}|_Ω$ restricted to a bounded domain $Ω\subset{\mathbb R}^d$ with $d=2,$ $1\leq α\leq 2$ and $d\geq 3,$ $0< α\le 2$. Our main result yields a sharper lower bound, in the sense of Weyl asymptotics, for the Berezin-Li-Yau type inequality improving the previous result in [36]. Furthermore, we give a result improving the bounds for analogous elliptic operators in [19].
We study the existence of positive radially symmetric solution for the singular $p$-Laplacian Dirichlet problem, $-\bigtriangleup_p u =λ|u|^{p-2} u-γu^{-α}$ where $λ>0,γ>0$ and, $0<α<1$, are parameters and $Ω$, the domain of the equation, is a ball in $\mathbb{R}^N$. By using some variational methods we show that, if $λ$ is contained in some interval, then the problem has a radially symmetric positive solution on the ball. Moreover, we obtain a nonexistence result, whenever $λ\leq 0, γ<0$ and $Ω$ is a bounded domain, with smooth boundary.
On the Dirichlet Fractional Laplacian and Applications to the SQG Equation on Bounded Domains
本文研究有界区域上的分数阶 Dirichlet Laplace 算子，建立乘积估计与非线性 Poincaré 不等式，并用于证明受稳态外力驱动的 SQG 方程存在有限维全局吸引子。
On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
本文研究分数-对数拉普拉斯算子的Faber-Krahn问题，证明小体积时球体最小化第一Dirichlet特征值且具有平移刚性。
This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations wi…
A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $λ>0$ varies, is proven. Since for every admissible $λ$ the problem has a smallest positive solution $\bar u_λ$, both monotonicity and continuity of the map $ λ\mapsto \bar u_λ$ are studied.
A stability result in terms of the perimeter is obtained for the first Dirichlet eigenvalue of the Laplacian operator. In particular, we prove that, once we fix the dimension $n\geq2$, there exists a constant $c>0$, depending only on $n$, such that, for every $Ω\subset\mathbb{R}^n$ open, bounded and convex set with volume equal to the volume of a ball $B$ with radius $1$, it holds \begin{equation*} λ_1(Ω)-λ_1(B)\geq c\left(P(Ω)-P(B) \right)^{2}, \end{equation*} where by $λ_1(\cdot)$ we denote the first Dirichlet eigenvalue of a set and by $P(\cdot)$ its perimeter. The hearth of the present pap…
In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit Neumann problem. The first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. Then we focus on the case of Diri…
In this paper we look for the domains minimizing the h-th eigenvalue of the Dirichlet-Laplacian $λ$ h with a constraint on the diameter. Existence of an optimal domain is easily obtained, and is attained at a constant width body. In the case of a simple eigenvalue, we provide non standard (i.e., non local) optimality conditions. Then we address the question whether or not the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane…
In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the e…
This paper deals with the eigenvalue problem for the operator $L=-Δ-x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $λ_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N} $ the following minimization problem $$ \min\left\{λ_k(Ω): \> Ω\>\mbox{quasi-open} \>\mbox{set}, \> \int_Ωe^{|x|^2/2}dx\le c\right\} $$ has a solution.
Optimal solvability for the fractional p-Laplacian with Dirichlet conditions
本文研究含 (p-1)-次线性反应项的分数阶 p-Laplacian 非局部 Dirichlet 问题，利用弱比较原理与渐近加权特征值问题刻画解的唯一性及存在充要条件，推广了经典结果。
Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian
该研究针对分数阶拉普拉斯方程基本Dirichlet型问题的非唯一性现象，建立了锐利的唯一性条件，推导了算子的$L^p$估计，引入加权分数阶Sobolev空间并证明其嵌入关系，为相关问题研究提供基础性质支撑。
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-Δ)^s$ on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(Ω)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(Ω)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working dir…
In a recent paper D. D. Hai showed that the equation $ -Δ_{p} u = λf(u) \mbox{in} Ω$, under Dirichlet boundary condition, where $Ω\subset {\bf R^N}$ is a bounded domain with smooth boundary $\partialΩ$, $Δ_{p}$ is the p-Laplacian, $f : (0,\infty) \rightarrow {\bf R} $ is a continuous function which may blow up to $\pm \infty$ at the origin, admits a solution if $λ> λ_0$ and has no solution if $0 < λ< λ_0$. In this paper we show that the solution set $\mathcal{S}$ of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.
In this article, we study the following fractional $p$-Laplacian equation with critical growth singular nonlinearity \begin{equation*} \quad (-\De_{p})^s u = \la u^{-q} + u^α, u>0 \; \text{in}\; \Om,\quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om. \end{equation*} where $\Om$ is a bounded domain in $\mb{R}^n$ with smooth boundary $\partial \Om$, $n > sp, s \in (0,1), \la >0, 0 < q \leq 1 $ and $α\le p^*_s-1$. We use variational methods to show the existence and multiplicity of positive solutions of above problem with respect to parameter $\la$.
We consider a model of fractional diffusion involving the natural nonlocal version of the $p$-Laplacian operator. We study the Dirichlet problem posed in a bounded domain $Ω$ of ${\mathbb{R}}^N$ with zero data outside of $Ω$, for which the existence and uniqueness of strong nonnegative solutions is proved, and a number of quantitative properties are established. A main objective is proving the existence of a special separate variable solution $U(x,t)=t^{-1/(p-2)}F(x)$, called the friendly giant, which produces a universal upper bound and explains the large-time behaviour of all nontrivial nonn…
Let $(M,g)$ be a complete manifold of nonpositive scalar curvature, let $Ω\subset M$ be a suitable domain, and let $λ(Ω)$ be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on $Ω$. We prove several bounds for the rate of decrease of $λ(Ω)$ and $Ω$ increases, and a result comparing the rate of decrease of $λ$ before and after a conformal diffeomorphism. Along the way, we prove a reverse-Holder inequality for the first eigenfunction, which generalizes results of Chiti to the monifold setting and may be of independent interest
Our purpose in this paper is to provide a self contained account of the inhomogeneous Dirichlet problem $Δ_\infty u=f(x,u)$ where $u$ takes a prescribed continuous data on the boundary of bounded domains. We employ a combination of Perron's method and a priori estimates to give general sufficient conditions on the right hand side $f$ that would ensure existence of viscosity solutions to the Dirichlet problem. Examples show that these sufficient conditions may not be relaxed. We also identify a class of inhomogeneous terms for which the corresponding Dirichlet problem has no solution in any dom…
In this paper, we study the logarithmic Laplacian operator $L_Δ$, which is a singular integral operator with symbol $2\log |ζ|$. We show that this operator has the integral representation $$L_Δu(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + ρ_N u(x) $$ with $c_N = π^{- \frac{N}{2}} Γ(\frac{N}{2})$ and $ρ_N=2 \log 2 + ψ(\frac{N}{2}) -γ$, where $Γ$ is the Gamma function, $ψ= \frac{Γ'}Γ$ is the Digamma function and $γ= -Γ'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-Δ)^s$ of fractional Laplacians at $s…
We derive spectral estimates of the Lieb-Thirring type for eigenvalues of Dirichlet Laplacians on strictly shrinking spiral-shaped domains.
We study the Dirichlet problem for an elliptic equation involving the $1$-Laplace operator and a reaction term, namely: $$ \left\{\begin{array}{ll} \displaystyle -Δ_1 u =h(u)f(x)&\hbox{in }Ω\,,\\ u=0&\hbox{on }\partialΩ\,, \end{array}\right. $$ where $ Ω\subset \mathbb{R}^N$ is an open bounded set having Lipschitz boundary, $f\in L^1(Ω)$ is nonnegative, and $h$ is a continuous real function that may possibly blow up at zero. We investigate optimal ranges for the data in order to obtain existence, nonexistence and (whenever expected) uniqueness of nonnegative solutions.
We consider the Dirichlet-energy maximization problem of the solution $u_f$ of (\ref{main}), among all functions $f\in L^2(D)$, such that $\|f\|_2= 1$. We show that the two maximizers are the first eigenfunctions of the Laplacian with Dirichlet boundary condition $f=\pm u_1$.
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