Félix del Teso, Erik Lindgren
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.
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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
In this paper we study the obstacle problems for the Navier (spectral) fractional Laplacian $\left(-Δ_Ω\right)^{\!s}$ of order $s\in(0,1)$, in a bounded domain $Ω\subset\mathbb R^n$.
Variational eigenvalues of the subelliptic $p$-Laplacian
本文利用 Lusternik-Schnirelman 理论，证明了次椭圆 $p$-Laplacian 在 Dirichlet 边界条件下存在非递减的变分特征值序列。
By virtue of $Γ-$convergence arguments, we investigate the stability of variational eigenvalues associated with a given topological index for the fractional $p$-Laplacian operator, in the singular limit as the nonlocal operator converges to the $p$-Laplacian. We also obtain the convergence of the corresponding normalized eigenfunctions in a suitable fractional norm.
Three tupes of self-similar blow-up for the fourth-order p-Laplacian equaiton with source: variational and branching approaches
证明了带p-Laplacian算子的四阶拟线性反应扩散方程允许三种类型的爆破.自相似模式首先构造的区域爆破的情况下，重新标度的问题承认一个变分设置。接下来，通过分支方法将这些扩展到非变分问题。
Let $1<p \leq \infty$. We provide an asymptotic characterization of continuous viscosity solutions $u$ of the normalized $p$-Laplacian $Δ_{p\,\mathbb{G}}^N u=0$ in any Carnot group $\mathbb{G}$.
In the present work we study existence of sequences of variational eigenvalues to non-local non-standard growth problems ruled by the fractional $g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann and Robin). Due to the non-homogeneous nature of the operator several drawbacks must be overcome, leading to some results that contrast with the case of power functions.
We study a nonlinear, nonlocal Dirichlet problem driven by the degenerate fractional p-Laplacian via a combination of topological methods (degree theory for operators of monotone type) and variational methods (critical point theory). We assume local conditions ensuring the existence of sub- and supersolutions. So we prove existence of two positive solutions, in both the coercive and noncoercive cases.
In this article, we prove a minimax characterization of the second eigenvalue of the p-Laplacian operator on p-quasi-open sets, using a construction based on minimizing movements. This leads also to an existence theorem for spectral functionals depending on the first two eigenvalues of the p-Laplacian.
In this paper we study the obstacle problems for the fractional Lapalcian of order $s\in(0,1)$ in a bounded domain $Ω\subset\mathbb R^n$, under mild assumptions on the data.
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.
Pointwise Hadamard variational formula for the fractional Laplacian
本文推导了分数拉普拉斯算子狄利克雷问题解的点wise形状导数公式，扩展了经典的哈达玛变分公式。
We consider a nonlinear Robin problems driven by the $p$-Laplacian plus an indefinite potential. The reaction is resonant with respect to a variational eigenvalue. For the principal eigenvalue we assume strong resonance. Using variational tools and critical groups we prove existence and multiplicity theorems.
An $L^\infty$-variational problem involving the Fractional Laplacian
本文研究了在给定Dirichlet数据条件下，分数Laplacian相关 supremal 函数的绝对极小解的存在性和唯一性，并证明了极小解满足一个分数PDE。
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。
Existence of solutions for a class of $p(x)$-laplacian equations involving a concave-convex nonlinearity with critical growth in $\mathbb{R}^{N}$
证明了一类具有临界增长的变指数非线性拟线性问题解的存在性。所使用的主要工具是变分方法，更准确地说，是Ekeland的变分原理和山路定理。
Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights
本文研究了带有奇异或退化矩阵权的 $p(x)$-拉普拉斯变分不等式的全局正则性，通过构造性水平集方法，得出了加权 Calderón-Zygmund 和 Orlicz 型估计，并展示了结果的最优缩放参数特性。
The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
We propose a new monotone finite difference discretization for the variational $p$-Laplace operator, \[ Δ_p u=\text{div}(|\nabla u|^{p-2}\nabla u), \] and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational $p$-Laplacian and also the first time that nonhomogeneous pr…
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.
Ground state solutions of a class of (2,q)-Laplacian Schrödinger equations with inhomogeneous nonlinearity
本文通过全局与局部约束变分技术，系统地研究了一类具有非均匀非线性的(2,q)-Laplacian Schrödinger方程的基态解，建立了其在质量次临界、临界及超临界情形下的存在性、不存在性与渐近行为。
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.
Asymptotics of variational eigenvalues for a general nonlocal $p$-Laplacian with varying horizon
本文研究了具有可变视距的通用非局部 $p$-拉普拉斯算子的变分特征值渐进行为，利用 $\Gamma$-收敛论证，分析了视距变化极限情况下的解稳定性，恢复了局部 $p$-拉普拉斯和 $H^{s,p}$-拉普拉斯特征值问题的解。
The pseudorelativistic Hartree equation with a general nonlinearity: existence, non existence and variational identities
证明了一类具有一般非线性项的伪相对论Hartree方程的孤波的存在性和不存在性。我们使用变分方法和一些新的变分恒等式涉及半拉普拉斯。
Non-standard eigenvalue problems for perturbed $p$-Laplacians
本文研究了扰动$p$ -Laplacian算子的多参数本征值问题，模拟了一类非线性发展偏微分方程的行波解。本征参数之间的色散关系、本征向量和正本征向量的存在性、扰动$p$ -Laplacian本征值的变分原理和构造解析解是本文的主要内容。除了$p$ -Laplacian类本征值问题外，我们还处理了新的非标准本征值问题，这些问题不能用$p$ -Laplacian算子和类似算子的非线性本征值问题的方法来解决.我们两样都做：扩展和使用经典的变分和分析技术来解决标准的本征值问题，并提出新的变分和分析方法来解决非标准的本征值问题，我们遇到的搜索行波。
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.
We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.
In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.
Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms
研究一类含双非局部项的$p(x)$-基尔霍夫问题，利用变分方法证明两个非平凡解及无穷多解的存在性，推广了已有结果。
We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \beg…
In this note we present upper bounds for the variational eigenvalues of the Steklov $p$-Laplacian on domains of $\mathbb R^n$, $n\geq 2$. We show that for $1<p\leq n$ the variational eigenvalues $σ_{p,k}$ are bounded above in terms of $k,p,n$ and $|\partialΩ|$ only. In the case $p>n$ upper bounds depend on a geometric constant $D(Ω)$, the $(n-1)$-distortion of $Ω$ which quantifies the concentration of the boundary measure. We prove that the presence of this constant is necessary in the upper estimates for $p>n$ and that the corresponding inequality is sharp, providing examples of domains with …
Hadamard's variational formula for simple eigenvalues
本文研究动态与共形形变下简单特征值的 Hadamard 变分公式，并证明二维区域中混合边界条件下 Laplacian 第一特征值的调和凸性，得到若干新不等式。
We prove a nonlocal, nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions. For the fractional $p$-Laplace operator it implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weak fractional $p$-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded $\frac{n}{s}$-harmonic maps converge strongly outside at most finitely many points.
Nodal domain theorems for $p$-Laplacians on signed graphs
本文在带符号图上建立p-拉普拉斯算子的节点域定理，统一了现有图p-拉普拉斯算子与对称矩阵的节点域结果，并由此得到联系变分特征值与多路Cheeger常数的高阶Cheeger不等式，还导出了通常图上的新结论。
In this paper, we prove that the second eigenfunctions of the $p$-Laplacian, $p>1$, are not radial on the unit ball in $\mathbb{R}^N,$ for any $N\ge 2.$ Our proof relies on the variational characterization of the second eigenvalue and a variant of the deformation lemma. We also construct an infinite sequence of eigenpairs $\{τ_n,Ψ_n\}$ such that $Ψ_n$ is nonradial and has exactly $2n$ nodal domains. A few related open problems are also stated.
We study a nonlinear boundary value problem driven by the $p$-Laplacian plus an indefinite potential with Robin boundary condition. The reaction term is a Carathéodory function which is asymptotically resonant at $\pm\infty$ with respect to a nonprincipal Ljusternik-Schnirelmann eigenvalue. Using variational methods, together with Morse theory and truncation-perturbation techniques, we show that the problem has at least three nontrivial smooth solutions, two of which have a fixed sign.
Some Remarks on the Fucik Spectrum of the p-Laplacian and Critical Groups
我们计算临界群的变分泛函所产生的拟线性椭圆边值问题跳跃非线性，当方程的渐近极限位于不同区域的平面上的某些曲线的Fucik谱。作为应用，通过Morse理论和扰动参数建立了一些存在性和多重性结果。
We show that a Weyl law holds for the variational spectrum of the $p$-Laplacian. More precisely, let $(λ_i)_{i=1}^\infty$ be the variational spectrum of $Δ_p$ on a closed Riemannian manifold $(X,g)$ and let $N(λ) = \#\{i:\, λ_i < λ\}$ be the associated counting function. Then we have a Weyl law $N(λ) \sim c \operatorname{vol}(X) λ^{n/p}$. This confirms a conjecture of Friedlander. The proof is based on ideas of Gromov and Liokumovich, Marques, Neves.
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 …
Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian
研究了涉及向量p-拉普拉斯算子的椭圆系统存在性和正则性，考虑了特殊情形并证明了分类结果，显示最小能量解的形式。
This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, the indefinite nonlinearity, and depend on the real parameter $λ$. A special focus is made on the extreme value of Nehari manifold $λ^*$, which determines the threshold of applicability of Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where parameter $λ$ lies above the threshold $λ^*$ is obtained.
This paper is concerned with a nonlinear Steklov boundary-value problem involving weighted $p(.)$-Laplacian. Using the Ricceri's variational principle, we obtain the existence of at least three weak solutions in double weighted variable exponent Sobolev space.
Subelliptic $p$-Laplacian spectral problem for Hörmander vector fields
该研究利用变分方法探究光滑Hörmander向量场导出的次椭圆$p$-Laplacian谱问题，得到最小特征值的相关性质与特征函数的正则性，还顺带确定了对应$L^p$-Poincaré-Friedrichs不等式的最佳常数。
Variational Methods and Planar Elliptic Growth
如果每个区域边界的法向速度正比于区域内部具有固定奇点的格林函数的梯度，则一个嵌套的增长或收缩平面区域族称为拉普拉斯增长过程。在本文中，我们回顾了拉普拉斯增长模型及其关键的基本假设，使我们可以考虑推广到所谓的椭圆增长，其中的格林函数被替换为一个更一般的椭圆算子-这个模型，例如，在底层平面的不均匀性。在本文中，我们继续发展的基础数学椭圆增长，考虑扰动的格林函数，由于那些驱动运营商，推导出的特征和增长的例子，开发一个弱制定的增长通过平衡，并讨论了一对夫妇的反问题的卡尔德龙的精神。最后，我们推导出一个更微妙的，重新调整模型的Hele-Shaw流。
Existence, uniqueness, and decay results for singular $Φ$-Laplacian systems in $\mathbb{R}^N$
本文利用移位法、变分法及多种分析技术，研究了奇异 $\Phi$-Laplacian 系统解的存在性、先验估计、衰减性与唯一性，并给出了 Diaz-Saa 型结果。
Using a variational technique we guarantee the existence of a solution to the \emph{resonant Lane-Emden} problem $-Δ_p u=λ|u|^{q-2}u$, $u|_{\partialΩ}=0$ if and only if a solution to $-Δ_p u=λ|u|^{q-2}u+f$, $u|_{\partialΩ}=0$, $f\in L^{p'}(Ω)$ ($p'$ being the conjugate of $p$), exists for $q\in (1,p)\bigcup (p,p^{*})$ under a certain condition for both the cases, i.e., $1<q<p<p^{*}$ and $1< p < q < p^{*}$ - the sub-linear and the super-linear cases.
We study a general class of discrete $p$-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional $p$-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.
In this note we introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called "fundamental solutions". This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to "understand" what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes…
We consider the nonlinear half laplacian heat equation $$ u_t+(-Δ)^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad \mathbb{R}^n\times (0, T). $$ We prove that all blows-up are type I, provided that $n \leq 4$ and $ 1<p<p_{*} (n)$ where $ p_{*} (n)$ is an explicit exponent which is below $\frac{n+1}{n-1}$, the critical Sobolev exponent. Central to our proof is a Giga-Kohn type monotonicity formula for half laplacian and a Liouville type theorem for self-similar nonlinear heat equation. This is the first instance of a monotonicity formula at the level of the nonlocal equation, without invoking the extension …
Multiple solutions for elliptic equations driven by higher order fractional Laplacian
本文研究由高阶分数阶拉普拉斯算子驱动、带齐次狄利克雷边界条件的椭圆方程，在超线性、凹-凸及对称等条件下，利用山路定理、Ekeland 变分原理和 Fountain 定理证明其存在两个非平凡弱解以及无穷多个正能量和负能量弱解。
We study the relationship of higher order variational eigenvalues of p-Laplacian and the higher order Cheeger constants. The asymptotic behavior of the k-th Cheeger constant is investigated. Using methods of decompostion of the domain with respect to the eigenfunctions, we obtain the high-order Cheeger's inequality of p-Laplacian on domain.
Continuity of weak solutions to an elliptic problem on $p$-fractional Laplacian
该文研究$\u211d^N$空间中$p$-分数阶拉普拉斯算子椭圆变分问题弱解的连续性，给出弱解连续的充分条件，同时修正了两篇相关文献中特定参数范围下引理的证明。
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。
Superlinear problems involving nonlinear superposition operators of mixed fractional order
本文研究了一类涉及混合分数阶非线性叠加算子的超线性椭圆问题，通过变分方法和Fountain定理证明了无限多个弱解的存在性。
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 consider the fractional Laplacian operator $(-Δ)^s$ (let $ s \in (0,1) $) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $ L^2(\mathbb{R}^d) $ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space $ \dot{H}^s(\mathbb{R}^d) $. More precisely, we focus on functions belonging to some weighted $ L^2 $ space whose fractional Laplacian belongs to another weighted $ L^2 $ space: we prove and disprove the validity of the integration-by-parts formula depending on the behavio…
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 …
A nonsmooth variational approach to differential problems. A case study of nonresonance under the first eigenvalue for a strongly nonlinear elliptic problem
我们适应的非光滑临界点理论的技术开发的Degiovanni-Zani的半线性问题，涉及拉普拉斯算子的情况下的$p$ -拉普拉斯算子。我们只假设势函数满足非线性条件，而不对非线性项施加增长条件。在两个不同的结果中，使用类似的非共振条件[Mawhin-Ward-Willem]和[Landesman-Lazer]，并在第三个结果中使用一些比较函数和比较空间，得到了非共振性。这三个定理都不蕴含另外两个定理。
Weighted isoperimetric ratios and extension problems for fractional conformal Laplacians
本研究建立了加权等周问题与分数阶共形拉普拉斯算子延拓问题的关联，推导了半空间与单位球上加权泊松积分的精确不等式并分类极值，构造CCE流形上的加权等周变分问题并证明其与Chang-González延拓的联系，给出极值存在准则并提出猜想。
A variational approach to the hot spots conjecture
本文回顾了一种基于非标准变分原理研究拉普拉斯特征函数临界点的新方法，并用其为热点猜想的若干已知结果提供简洁证明。
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