C. Azizieh
We extend to the case of a system involving p-Laplacians, the monotonicity and symmetry results of Damascelli and Pacella obtained in the case of a scalar p-Laplace equation with $1<p<2$. For this purpose, we use the moving hyperplanes method and we suppose that the right hand sides are increasing and locally Lipschitz continuous.
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We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $σ_1 \geq σ_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.
We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.
Monotonicity of the Laplace Transform for Tomography in Dissipative Systems
本文研究了耗散材料内部的磁感应成像问题，提出基于拉普拉斯变换的单调性原理，用于解决MIT中非线性反问题的挑战，并给出实时成像方法。
For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the Laplace-Beltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be employed to prove the Lipschitz continuity for the solutions of a general class of two-phase free boundary problems on Riemannian manifolds.
Symmetry and Monotonicity Property of a Solution of (p,q) Laplace Equation with Singular Term
本文利用移动平面法研究带奇异项和零 Dirichlet 边界条件的 (p,q) Laplace 方程正解，证明其对称性与单调性，并在径向情形下建立强比较原理。
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。
Monotonicity Principle and "p-Laplace Signature" for Tomography in Nonlinear Elliptic Inverse Problems
本文结合单调性原理与p-拉普拉斯特征，提出新成像方法，扩大了非线性椭圆逆障碍问题可处理的非线性性类别，无噪声时p=2可外支撑重构、p≠2可凸包重构。
We consider the inverse problems of for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension $n\geq 2$ and only requires the background solution of the fractional Schrödinger equation.
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
In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
In this paper we study weighted singular $p$-Laplace equations involving a bounded weight function which can be discontinuous. Due to its discontinuity classical regularity results cannot be applied. Based on Nehari manifolds we prove the existence of at least two positive bounded solutions of such problems.
Monotonicity of non-negative solutions of quasilinear elliptic equations in a cylindrical domain
研究柱形域中p-Laplace方程在混合边界条件下的正解，证明对任意p>1解在x_N方向单调递增，并应用于Allen-Cahn和Lane-Emden型方程。
We consider weak distributional solutions to the equation $-Δ_pu=f(u)$ in half-spaces under zero Dirichlet boundary condition. We assume that the nonlinearity is positive and superlinear at zero. For $p>2$ (the case $1<p\leq2$ is already known) we prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary of the half-space. As a consequence we deduce some Liouville type theorems for the Lane-Emden type equation. Furthermore any nonnegative solution turns out to be $C^{2,α}$ smooth.
Monotonicity of positive solutions for an indefinite logarithmic Laplacian equation
研究含不定非线性项的对数拉普拉斯方程，通过反对称函数边界估计和移动平面法，证明有界正解的单调性与不存在性。
A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.
We provide the classification of the positive solutions to $-Δ_p u =u^{p^*-1}$ in $\mathcal {D}^{1,p}(\R^N)$ in the case $2<p<N$. Since the case $1<p\leq2$ is already known this provides the complete classification for $1<p<N$.
Classification results for bounded positive solutions to the critical $p$-Laplace equation
本文通过提供最优或接近最优的积分估计，证明了临界p-拉普拉斯方程在R^n中（n≥3）的每一个正、有界或适度增长的局部弱解，若其在球上的下确界行为适当，则必须是气泡解。
Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles
本文通过移动平面方法和最大原理，研究了三角形中混合边界条件下半线性椭圆方程正解的单调性，揭示了特征函数极值的位置特性。
In this paper, we first establish a narrow region principle and a decay at infinity theorem to extend the direct method of moving planes for general fractional $p$-Laplacian systems. By virtue of this method, we can investigate the qualitative properties of the following Schrödinger system with fractional $p$-Laplacian \begin{equation*} \left\{\begin{array}{r@{\ \ }c@{\ \ }ll} \left(-Δ\right)_{p}^{s}u+au^{p-1}& =&f(u,v), \\[0.05cm] \left(-Δ\right)_{p}^{t}v+bv^{p-1}& =&g(u,v), \end{array}\right. \end{equation*} where $0<s,\,t<1$ and $2<p<\infty$. We obtain the radial symmetry in the unit ball o…
Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition
针对带不定超线性边界条件的 Laplace 方程可能存在多正解的问题，本文通过不稳定正解处线性化特征值问题的谱分析，给出隐函数定理适用的充分条件以确立唯一性，并应用于群体遗传学的 logistic 边界条件。
We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved moving plane procedure.
Monotonicity of the period and positive periodic solutions of a quasilinear equation
本文研究了准线性微分方程中周期解的单调性，扩展了经典结果到 $p\ge2$，并推广了相关单调性结果。
We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.
We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2} u + β|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $Ω$. The main result of our research is the construction of a continuous curve in $(α,β)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $Ω$ for which the corresponding first Dirichlet eigenvalue of $-Δ_p$ is not monotone w.r.t. $p > 1$.
Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli
该研究证明分数阶$p$-Laplace算子在环域上的首个$q$-特征值随内球外移严格递减，通过极化下的严格Faber-Krahn型不等式推导，推广了已有结果并给出一般对称差集区域的相关结论。
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-Δ_p u-\fracμ{|x|^p}u^{p-1}=\frac{C}{|x|^σ}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $μ\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,σ)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator wi…
Sharp monotonicity for closed three-manifolds with positive scalar curvature
该研究针对具有正标量曲率的闭三维流形的自然格林函数，证明了两个严格单调性公式，并将其解释为格林函数下水平集的包围质量的单调性，推广了非负标量曲率情形的相关结果。
In this paper, we introduce concepts of separable functions in balls and in the whole space, and develop a new method to investigate the qualitative properties of separable functions. We first study the axial symmetry and monotonicity of separable functions in unit circles by geometry analysis, and we prove the uniqueness of the symmetry axis for nontrivial separable functions. Then by using reduction dimension and convex analysis, we get the axial symmetry and monotonicity of separable functions in high dimensional spheres. Based on the above results on unit circles and spheres, we deduce the…
We analyze a non-linear elliptic boundary value problem, that involves $(p, q)$ Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a continuous function in $(0,\infty)$ which is singular, monotonically increasing and eventually positive. We prove the existence of a positive solution of this problem using a fixed point theorem due to Amann\cite{amann1976fixed}. In addition, for a specific nonlinearity we derive that the obtained solution is maximal in nature. The main results obtained here are first of its kind…
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
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.
We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.
In this paper we consider classical solutions $u$ of the semilinear fractional problem $(-Δ)^s u = f(u)$ in $\mathbb{R}^N_+$ with $u=0$ in $\mathbb{R}^N \setminus \mathbb{R}^N_+$, where $(-Δ)^s$, $0<s<1$, stands for the fractional laplacian, $N\ge 2$, $\mathbb{R}^N_+=\{x=(x',x_N)\in \mathbb{R}^N:\ x_N>0\}$ is the half-space and $f\in C^1$ is a given function. With no additional restriction on the function $f$, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $\mathbb{R}^N_+$ and verify $$ \frac{\partial u}{\partial x_N}>0 \quad \hbox{in } \mathbb{R}^N_+.…
For any $Ω\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $Ω$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(Ω)\hookrightarrow L^{2p}(Ω)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonici…
We study a class of fractional elliptic problems of the form $\Ds u= f(u)$ in the half space $\R^N_+:=\{x \in \R^N\::\: x_1>0\}$ with the complementary Dirichlet condition $u \equiv 0$ in $\R^N \setminus \R^N_+$. Under mild assumptions on the nonlinearity $f$, we show that bounded positive solutions are increasing in $x_1$. For the special case $f(u)=u^q$, we deduce nonexistence of positive bounded solutions in the case where $q \ge 1$ and $q<\frac{N-1+2s}{N-1-2s}$ if $N \ge 1+2s$. We do not require integrability assumptions on the solutions we study.
A note on the classification of positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$
本文在 $n\ge4$ 且 $p>p_n$（其中 $p_n$ 略大于 $n/3$）的条件下，改进了 Ou 对 $\mathbb{R}^n$ 中临界 p-Laplace 方程正解的分类结果。
We study the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = α|u|^{p-2}u+β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(α,β)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(α,β) = \left(\|\nabla φ_p\|_p^p/\|φ_p\|_p^p, \|\nabla φ_p\|_q^q/\|φ_p\|_q^q\right)$, where $φ_p$ is the first eigenfunction of the $p$-Laplacian, we show the existence of two and, which is rather unexpected, three distinct posi…
We first give some apriori estimates of positive radial solutions of $p$-Laplace Hénon equation. Then we study the local and global properties of those solutions. Finally, we generalize some radial results to the nonradial case.
Positive Solutions for Fractional p- Laplace Semipositone Problem with Superlinear Growth
本文研究含分数阶 $p$-Laplacian 的超线性半正问题，通过分析障碍函数并结合度理论证明小参数下正解存在，且用变分方法处理临界增长多参数情形的基态正解。
We prove a weak comparison principle in narrow unbounded domains for solutions to $-Δ_p u=f(u)$ in the case $2<p< 3$ and $f(\cdot)$ is a power-type nonlinearity, or in the case $p>2$ and $f(\cdot)$ is super-linear. We exploit it to prove the monotonicity of positive solutions to $-Δ_p u=f(u)$ in half spaces (with zero Dirichlet assumption) and therefore to prove some Liouville-type theorems.
Sharp Hessian inequalities and monotonicity for Green's functions of positively curved Einstein manifolds
该研究针对正曲率爱因斯坦流形的格林函数，证明了其尖锐黑塞不等式及相关单调性公式，扩展了Park的结果，并给出几何应用。
We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: \[ \begin{cases} -Δu = -u v^2 & \text{in $\R^N$}\\ -Δv= -u^2 v & \text{in $\R^N$}, \end{cases} \] for every dimension $N \ge 2$. In particular, we prove a Gibbons-type conjecture proposed by H. Berestycki, T. C. Lin, J. Wei and C. Zhao.
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by $p$-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below.
We consider a quasilinear elliptic equation involving a first order term, under zero Dirichlet boundary condition in half spaces. We prove that any positive solution is monotone increasing w.r.t. the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.
Classification of positive solutions to a class of Laplace equations with a gradient term
本文首次在第二临界情况下证明了带有梯度项的拉普拉斯方程正解的分类结果，展示了环境流形的刚性及解的结构。
Monotonicity results in half spaces for quasilinear elliptic equations involving a singular term
本文研究半空间中带奇异非线性项的 p-Laplace 椭圆方程正解，结合先验估计与移动平面法证明解在边界法向上单调递增。
Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p Δ_{p}u + u^{p-1} = f(u) & \hbox{in} \ Ω \newline u>0 & \hbox{in} \ Ω \newline u=0 & \hbox{on} \ \partial Ω \end{array} \right. \] where $Ω$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ is a parameter and $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least …
Sharp gradient estimates and monotonicity in positive Ricci curvature
本文在正 Ricci 曲率闭流形上证明了尖锐梯度估计，并将其与单调性公式联系起来，同时给出了 Bishop 定理的新证明。
Existence and convergence of solutions to $p$-Laplace equations on locally finite graphs
本文研究局部有限图上非线性p-Laplace方程，利用山路定理和Nehari流形证明正解与正基态解的存在性并分析其渐近行为，在p的取值范围、图与位势条件及非线性项增长条件三方面作出改进。
Under the assumption of finite energy, positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ for $1< p<n$ have been classified completely by moving plane method. In this paper, the author provide a new approach to obtain the same classification results for $\frac{n+1}{3}\leq p<n$, without any further assumptions.
Liouville theorems for anisotropic $p$-Laplace equations with a semilinear term
针对带半线性项的各向异性 $p$-Laplace 方程，结合 Serrin 技巧与不变张量方法，建立正解、超/子解及稳定变号解的 Liouville 不存在性与分类结果。
In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -Δu=|x|^β v^{\vartheta},\\ -Δv=|x|^α |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ Ω, \end{equation*}where $Ω$ is a subset of $\mathbb{R}^N$, $N \ge 5$, $α>-4$, $0 \le β\le \dfrac{N-4}{2}$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and find their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems of stable solutions …
In this paper, we consider nonnegative solutions of spatially heterogeneous Fisher-KPP type reaction-diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the solution is increasing in time at large times. Furthermore, in the one-dimensional case, we prove that, if the equation is homogeneous outside a bounded interval and the reaction is linear around the zero state, then the solution is time-increasing in the whole line at large tim…
We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our r…
Parabolic frequency monotonicity for two nonlinear equations under Ricci flow
本文研究闭流形Ricci流下两类非线性抛物方程正解的抛物频率单调性，并据此推导出积分型Harnack不等式。
On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity
该研究针对含对数非线性与Trudinger-Moser临界指数非线性的奇异摄动$(p,N)$-拉普拉斯薛定谔方程，结合变分法、惩罚技术与畴数理论，建立了正解数量与势函数最小值集拓扑性质的关联，明确了基态的存在性、多重性等特性。
Let $n \geq 2$ and $Ω\subset \mathbb{R}^n$ be a bounded domain. Then by Trudinger-Moser embedding, $W_0^{1,n}(Ω)$ is embedded in an Orlicz space consisting of exponential functions. Consider the corresponding semi linear $n$-Laplace equation with critical or sub-critical exponential nonlinearity in a ball $B(R)$ with dirichlet boundary condition. In this paper, we prove that under suitable growth conditions on the nonlinearity, there exists an $γ_0 > 0$, and a corresponding $R_0(γ_0 ) > 0$ such that for all $0 < R < R_0$ , the problem admits a unique non degenerate positive radial solution $u$…
In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.
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