Stefania Patrizi
We prove the existence of a principal eigenvalue associated to the $\infty$-Laplacian plus lower order terms and the Neumann boundary condition in a bounded smooth domain. As an application we get uniqueness and existence results for the Neumann problem and a decay estimate for viscosity solutions of the Neumann evolution problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We consider different fractional Neumann Laplacians of order s, 0<s<1, namely, the Restricted Neumann Laplacian, the Semirestricted Neumann Laplacian and the Spectral Neumann Laplacian. In particular, we are interested in attainability of Sobolev constants for these operators in half-spaces.
Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.
In this paper we study a variational Neumann problem for the higher order $s$-fractional Laplacian, with $s>1$. In the process, we introduce some non-local Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.
We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the $p^{th}$-Sub-Laplacian associated to a system of Hörmander vector fields
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 prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
We study the eigenvalue problem for the Neumann-Laplace operator in conformal regular planar domains $Ω\subset\mathbb{C}$. Conformal regular domains support the Poincaré inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.
We develop some properties of the $p-$Neumann derivative for the fractional $p-$Laplacian in bounded domains with general $p>1$. In particular, we prove the existence of a diverging sequence of eigenvalues and we introduce the evolution problem associated to such operators, studying the basic properties of solutions. Finally, we study a nonlinear problem with source in absence of the Ambrosetti-Rabinowitz condition.
The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitable normalized, converges as $p$ goes to $\infty$ to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian. We show among other things that the limit of the eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.
We investigate the limiting behavior of solutions to the inhomogeneous $p$-Laplacian equation $-Δ_p u = μ_p$ subject to Neumann boundary conditions. For right hand sides which are arbitrary signed measures we show that solutions converge to a Kantorovich potential associated with the geodesic Wasserstein-$1$ distance. In the regular case with continuous right hand sides we characterize the limit as viscosity solution to an infinity Laplacian / eikonal type equation.
On Neumann $p$-Laplacian Lane-Emden equations and their asymptotic relationship with relative isoperimetric problems
本文通过变分方法统一处理Neumann p-Laplacian问题族，建立解的存在性并刻画参数变化时的渐近行为，揭示其与Neumann-Cheeger问题的自然渐近联系，并描述不同几何区域中极小元的形状及正则性、唯一性、多重性、对称性与对称破缺现象。
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.
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szegö-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.
We consider nonlinear problems governed by the fractional $p-$Laplacian in presence of nonlocal Neumann boundary conditions. We face two problems. First: the $p-$superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.
Neumann $p$-Laplacian problems with a reaction term on metric spaces
本文利用变分方法，在具有倍增测度和Poincaré不等式的度量空间上研究带反应项的Neumann p-Laplacian问题，借助有界变差函数迹定理定义变分泛函，并证明极小元满足De Giorgi型条件，从而得到解的存在性与正则性。
Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
本文针对半空间中带非局部Neumann边界条件的分数阶拉普拉斯方程（含方程组），通过建立衰减原理和窄区域原理，结合移动平面法，首次证明了衰减解的对称性与单调性。
A supercritical nonlocal Neumann problem involving non-homogeneous fractional Laplacian
研究外部Neumann条件下分数阶g-Laplace算子支配的非局部非标准增长问题，利用分数阶Orlicz-Sobolev空间性质与非光滑泛函变分原理，证明正非减径向解的存在性。
Eigenvalues of the Neumann Laplacian with density and sharp Sobolev-Orlicz embeddings
本文针对特殊平面区域和权重估计加权Poincaré常数，进而证明带密度Neumann拉普拉斯算子第一非零特征值的下界，并指出该特征值可通过改变质量密度任意增大。
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Δ)^s u=f$ in $Ω$, $\mathcal N_s u=0$ in $Ω^c$, then $u$ is $C^α$ up tp the boundary for some $α>0$. Moreover, in case $s>\frac12$, we then show that $u\in C^{2s-1+α}(\overlineΩ)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary reg…
The anisotropic $\infty$-Laplacian eigenvalue problem with Neumann boundary conditions
本研究借助粘性解分析带诺伊曼边界的各向异性$p$-拉普拉斯当$p\rightarrow\infty$时的极限问题，证明特征值与特征函数的几何性质，验证Szegö-Weinberger型不等式的成立。
The Neumann condition for the superposition of fractional Laplacians
本文提出了一种新的函数设定，用于研究分数拉普拉斯算子叠加的Neumann条件，并探讨了相关数学性质及热方程的分析。
In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded $C^2$ domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.
Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions
本文研究由分数阶p-拉普拉斯算子叠加驱动的非线性非局部问题在Neumann边界条件下的存在性，通过主算子的谱分析结合山路引理或锥上链接技巧得到非平凡解，结果适用于广泛的局部-非局部模型。
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.
In this paper we study integral estimates of derivatives of conformal mappings $φ:\mathbb D\toΩ$ of the unit disc $\mathbb D\subset\mathbb C$ onto bounded domains $Ω$ that satisfy the Ahlfors condition. These integral estimates lead to estimates of constants in Sobolev-Poincaré inequalities, and by the Rayleigh quotient we obtain spectral estimates of the Neumann-Laplace operator in non-Lipschitz domains (quasidiscs) in terms of the (quasi)conformal geometry of the domains. Specifically, the lower estimates of the first non-trivial eigenvalues of the Neumann-Laplace operator in some fractal ty…
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 existence and multiplicity of positive solutions for a quasilinear problem with Neumann boundary conditions, set in a ball. The problem admits at least one constant non-zero solution and it involves a nonlinearity that can be supercritical in the sense of Sobolev embeddings. The main tools used are variational techniques and the shooting method for ODE's. These results are contained in [6,3].
For $p>2$, we consider the quasilinear equation $-Δ_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u|^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to\infty$.
In this paper we study spectral stability of the $\bar\partial$-Neumann Laplacian under the Kohn-Nirenberg elliptic regularization. We obtain quantitative estimates for stability of the spectrum of the $\bar\partial$-Neumann Laplacian when either the operator or the underlying domain is perturbed.
In this work we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of Rn which are invariant under the natural action of a compact subgroup G of O(n). We give a partial positive answer (in the Neumann case) to a conjecture of V. Arnold [1] on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show, for some classes of subgroups of O(n) that, generically in the set of G-invariant, C2-regions, the action is irreducible in each …
In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with non-homogeneous mixed boundary conditions. For a Lipschitz and positive function $g$ there exists a viscosity solution of the mixed boundary value problem, $$ \{\begin{array}{ll} \displaystyle -Δ_{\infty}u(x)=0\quad & \text{in} Ω, \displaystyle \frac{\partial u}{\partial n}(x)= g (x)\quad & \text{on} Γ_N, \displaystyle u(x)= 0 \quad & \text{on} Γ_D. \end{array}. $$
In this paper we study a non-homogeneous Neumann problem, where the $p(x)$-Laplacian is involved and $p=\infty$ in a subdomain. By considering a suitable sequence $p_k$ of bounded variable exponents such that $p_k \to p$ and replacing $p$ with $p_k$ in the original problem, we prove the existence of a solution $u_k$ for each of those intermediate ones. We show that the limit of the $u_k$ exists and after giving a variational characterization of it, in the part of the domain where $p$ is bounded, we show that it is a viscosity solution in the part where $p=\infty$. Finally, we formulate the pro…
This article gives a domain with a small compact set of removed and the magnetic Neumann Laplacian on such set. The main theorem of this article shows the description of the holes which do not change the spectrum drastically. In this article we prove that the spectrum of the magnetic Neumann Laplacian converges in the Hausdorff distance sense to the spectrum of the original operator defined on the unperturbed domain.
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.
We give a unified approach to strong maximum principles for a large class of nonlocal operators of the order $s\in(0,1)$, that includes the Dirichlet, the Neumann Restricted (or Regional) and the Neumann Semirestricted Laplacians.
Three non-zero solutions of a Neumann eigenvalue problems involving the fractional p-Laplacian
本文研究含分数阶p-Laplacian的非局部Neumann特征值问题，证明存在正特征值开区间，使问题在该区间内至少有三个非零解。
We discuss the Krein--von Neumann extensions of three Laplacian-type operators -- on discrete graphs, quantum graphs, and domains. In passing we present a class of one-dimensional elliptic operators such that for any $n\in \mathbb N$ infinitely many elements of the class have $n$-dimensional null space.
A smooth bounded pseudoconvex domain in two complex variables is of finite type if and only if the number of eigenvalues of the d-bar-Neumann Laplacian that are less than or equal to $λ$ has at most polynomial growth as $λ$ goes to infinity.
We study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.
We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-Δ_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The assumptions on the nonlinearity $f$ are very mild and allow it to be possibly supercritical in the sense of Sobolev embeddings. The main tools used are the truncation method and a mountain pass-type argument. In the pure power case, i.e., $f(u)=u^{q-1}$, we detect the limit profile of the solutions of the problems as $q\to\infty$.
We study the relationship between the symbol of the Dirichlet-to-Neumann operator associated with a connection Laplacian, and the geometry on and near the boundary. As a consequence, we show that the geometric data on the boundary, and when the dimension of the base is greater than two all corresponding normal derivatives, are determined by the symbol.
We deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region.
A note on the distribution of Neumann eigenvalues of the Laplacian on a Euclidean convex domain
本文研究了欧几里得凸域上拉普拉斯算子Neumann特征值的分布，建立了两个通用不等式。
Principal frequency of p-sub-Laplacians for general vector fields
本文证明了一般向量场的 Dirichlet p-次拉普拉斯算子主频率的唯一性与单重性，建立了 Caccioppoli 不等式，并讨论了 Grushin 平面和 Heisenberg 群上的特例。
We extend the results of [5], where we proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
Estimates on the Neumann and Steklov principal eigenvalues of collapsing domains
本文研究了 $\mathbb{R}^2$ 中塌缩凸域的 Neumann 和 Steklov 主特征值关系，部分证明了两者比值的估计猜想，指出变薄三角形最大化该比值，变薄矩形在具有对称性的凸变薄集中最小化该比值。
Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and boundary Harnack inequality are proved to support our analysis. This is a continuation of our first work [3] and a contribution in the development of the theory of \textit{generalized principal eigenvalue} beside the works [8, 13, 12, 9, 29]. We use these notions to characterize the validity of maximum principle and study the existence, nonexistence and uniquene…
Nonlinear Neumann boundary problems for $n$-Laplacian Liouville equation on a half space
本文研究了n-拉普拉斯李乌利维尔方程在半空间上正非线性Neumann边界条件下的解的分类，将二次阶结果推广到一般n≥2，并将临界p-拉普拉斯方程结果推广到p=n。
For $1\leq q \leq n-2$, we provide explicit examples to demonstrate non-compactness of the Neumann operator for the Kohn Laplacian acting on $L^2$ $(0,q)$-forms on the unit ball in $(2n+1)$-dimensional Heisenberg space.
We study the positive principal eigenvalue of a weighted problem associated with the Neumann spectral fractional Laplacian. This analysis is related to the investigation of the survival threshold in population dynamics. Our main result concerns the optimization of such threshold with respect to the fractional order $s\in(0,1]$, the case $s=1$ corresponding to the standard Neumann Laplacian: when the habitat is not too fragmented, the principal positive eigenvalue can not have local minima for $0<s<1$. As a consequence, the best strategy for survival is either following the diffusion with $s=1$…
Eigenvalues of the Magnetic Neumann Laplacian on Domains with Peaks
研究ℝ²带外尖峰有界区域上强磁场相关的磁Neumann拉普拉斯算子，分析λ→+∞时特征值行为，明确其渐近展开受尖峰锐度和几何影响，主项阶为λ^(2/(q+1))，拓展了前期相关研究。
Two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain
本文提出两个新通用不等式，用于平面凸域中拉普拉斯算子的Neumann特征值，扩展了相关理论的应用范围。
A note on domain monotonicity for the Neumann eigenvalues of the Laplacian
本文证明了凸区域及其子区域上拉普拉斯算子诺伊曼特征值的区域单调性变体，并基于该方法给出了凸区域诺伊曼特征值的上界。
Let $1<p<+\infty$ and let $Ω\subset\mathbb R^N$ be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type \[ -Δ_p u = f(u), \quad u>0 \mbox{ in } Ω, \quad \partial_νu = 0 \mbox{ on } \partialΩ. \] We suppose that $f(0)=f(1)=0$ and that $f$ is negative between the two zeros and positive after. In case $Ω$ is a ball, we also require that $f$ grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focusing in particular…
Principal eigenvalue for some elliptic operators with large drift: Neumann boundary conditions
本文研究二维带漂移线性椭圆算子的主特征值，精细刻画了漂移率趋于无穷时其渐近行为由对应常微分方程组ω极限集的连通分支决定的规律，并讨论了退化情形。
In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \math…
Spectral inequalities for magnetic Neumann Laplacian on rectangles
该研究探讨矩形上的磁Neumann拉普拉斯算子，证明弱磁场下固定面积矩形中正方形是最低本征值的局部极小值，还猜想其为全局极小值并部分证明了该结论。
The difference of the resolvents of two Laplacians on a half-space subject to Robin boundary conditions is studied. In general this difference is not compact, but it will be shown that it is compact and even belongs to some Schatten-von-Neumann class, if the coefficients in the boundary condition are sufficiently close to each other in a proper sense. In certain cases the resolvent difference is shown to belong even to the same Schatten-von Neumann class as it is known for the resolvent difference of two Robin Laplacians on a domain with a compact boundary.
The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size ε are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log(ε)), the other like O(ε^2).
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to t in C_+ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。