Samuel Littig, Fridemann Schuricht
We consider perturbed eigenvalue problems of the 1-Laplace operator and verify the existence of a sequence of solutions. It is shown that the eigenvalues of the perturbed problem converge to the corresponding eigenvalue of the unperturbed problem as the perturbation becomes small. The results rely on nonsmooth critical point theory based on the weak slope.
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In this paper we prove a concentration result and an error estimate for a model of heat conduction in composite materials having a microscopic structure arranged in a perodic array and thermally active membranes separating the heat conductive phases.
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear eigenvalue problem. We also show the existence of multiple solutions of the nonlinear problem using variational methods.
We establish some existence results for Schrödinger $p(x)$-Laplace equations in $\mathbb{R}^N$ with various potentials and critical growth of nonlinearity that may occur on some nonempty set, although not necessarily the whole space $\mathbb{R}^N$. The proofs are mainly based on concentration-compactness principles in a suitable weighted variable exponent Sobolev space and its imbeddings.
In this work we use variational methods to prove results on existence and concentration of solutions to a problem in $\mathbb{R}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV(\mathbb{R}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.
This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace operator, i.e. $Δ_{p(\cdot)}u={\rm div}(|\nabla u|^{p(\cdot)-2}\nabla u)$, where $p(\cdot)$ is a continuous function. The proof of the main result is based on Schauder's fixed point theorem combined with adequate variational arguments. The function space setting used here makes appeal to the variable exponent Lebesgue and Sobolev spaces.
We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace-Beltrami pseudo-parabolic equation. This pseudo-parabolic character is present also in the homogenized equation, which is obtained by the unfolding technique.
We analyze the interior controllability problem for a nonlocal Schrödinger equation involving the fractional Laplace operator $(-Δ)^s$, $s\in(0,1)$, on a bounded $C^{1,1}$ domain $Ω\subset\mathbb{R}^n$. The controllability from a neighborhood of the boundary of the domain is obtained for exponents $s$ in the interval $[1/2,1)$, while for $s<1/2$ the equation is shown to be not controllable. The results follow applying the multiplier method, joint with a Pohozaev-type identity for the fractional Laplacian, and from an explicit computation of the spectrum of the operator in the one-dimensional c…
We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.
We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |Δu(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{loc}(Ω)$ is postive and obeys some additional assumptions, $Δu$ is the Laplace operator, ${\cal T}_{h,C}(\cdot )$ is certain transformation of the continuous function $h(\cdot)$. We also explain how to apply such inequality to deduce regularity for solutions of nonlinear eigenvalue problems of elliptic type for degenerated PDEs, with the illustration within the model of electros…
In this paper, we study the existence of multiple solutions to a generalized $p(\cdot)$-Laplace equation with two parameters involving critical growth. More precisely, we give sufficient "local" conditions, which mean that growths between the main operator and nonlinear term are locally assumed for the cases $p(\cdot)$-sublinear, $p(\cdot)$-superlinear, and sandwich-type. Compared to constant exponent problems (for examples, $p$-Laplacian and $(p,q)$-Laplacian), this characterizes the study of variable exponent problems. We show this by applying variants of the Mountain Pass Theorem for $p(\cd…
In this paper we prove a strong comparison principle for radially decreasing solutions $u,v\in C_{0}^{1,α}(\Bar{B_R})$ of the singular equations $-Δ_p u-\frac{1}{u^δ}=f(x)$ and $-Δ_p v-\frac{1}{v^δ}=g(x)$ in $B_R$. Here we assume that $ 1<p<2 , \; δ\in (0,1)$ and $f,g$ are continuous, radial functions such that $0 \leq f \leq g$ but $f\not \equiv g$ in $B_R.$ For the case $p>2$ a counterexample is provided where the strong comparison principle is violated. As an application of strong comparison principle, we prove a three solution theorem for p-Laplace equation and illustrate with an example.
In this work we study the existence of solutions $u \in W^{1,p}_0(Ω)$ to the implicit elliptic problem $ f(x, u, \nabla u, Δ_p u)= 0$ in $ Ω$, where $ Ω$ is a bounded domain in $ \mathbb R^N $, $ N \ge 2 $, with smooth boundary $ \partial Ω$, $ 1< p< +\infty $, and $ f\colon Ω\times \mathbb R \times \mathbb R^N \times \R \to \R $. We choose the particular case when the function $ f $ can be expressed in the form $ f(x, z, w, y)= φ(x, z, w)- ψ(y) $, where the function $ ψ$ depends only on the $p$-Laplacian $ Δ_p u $. We also present some applications of our results.
Least energy solutions for affine $p$-Laplace equations involving subcritical and critical nonlinearities
本文研究含仿射$p$-拉普拉斯非局部算子的两类椭圆问题，探讨其最小能量型正$C^1$解的存在性与不存在性，分析了由能量泛函非凸及不等式严格性带来的困难。
By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the $p$-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.
We study Hölder continuity for solutions of the $p$-Laplace equation. This is established through a method involving an ordinary differential inequality, in contrast to the classical proof of the De Giorgi-Nash-Moser Theorem which uses iteration of an inequality through concentric balls.
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.
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.
The stability for the viscosity solutions of a differential equation with a perturbation term added to the Infinity-Laplace Operator is studied. This is the so-called Infinity-Laplace Equation with variable exponent infinity. An approximation of the identity is crucial for the proofs.
The existence of a nontrivial solution is proved for a class of quasilinear elliptic equations involving, as principal part, either the p-Laplace operator or the operator related to the p-area functional, and a nonlinearity with p-linear growth at infinity. To this aim, Morse theory techniques are combined with critical groups estimates.
This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_β}}{|x|^β} \mathrm{d}x\right)^\frac{p}{p^*_β},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_β>0$, where $1<p<N$, $0\leq β<p$, $p^*_β=\frac{p(N-β)}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $|x|^{-β}$, some new estimates are established. Based on …
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.
We adapt in the present note the perturbation method introduced in [3] to get a Gaussian lower bound for the Neumann heat kernel of the Laplace-Beltrami operator on an open subset of a compact Riemannian manifold.
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if said perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any assumption on the topology of the nodal sets. As an indirect application, we are able to show that a certain "Payne property" concerning the second nodal line remains stable under controlled perturbations of the domain.
We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.
We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.
We study the following singular problem involving the p$(x)$-Laplace operator $Δ_{p(x)}u= div(|\nabla u|^{p(x)-2}\nabla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {(\rm P_λ)} \left\{\begin{aligned} - Δ_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{λb(x)}{u^{δ(x)}} \quad\mbox{in}\,Ω,\\ u &>0 \quad\mbox{in}\,Ω, \\ u & =0 \quad\mbox{on}\,\partialΩ.\end{aligned} \right. \end{equation} Here, $Ω$ is a bounded domain in $\mathbb{R}^{N\geq2}$ with $C^2$-boundary, $λ$ is a positive parameter, $a(x), b(x) \in C(\overlineΩ)$ are positive weight functions with compact su…
Fractional Sobolev Spaces for the Singular-perturbed Laplace Operator in the $L^p$ setting
本文研究了二维和三维欧几里得空间中Laplace算子奇异扰动Δ_α的分数Sobolev空间H^{s,p}_α(R^d)，扩展了L^2理论至L^p情况，并利用压缩方法处理非线性Schrödinger方程的局部初值问题。
Sobolev spaces for singular perturbation of Laplace operato
该研究针对二维拉普拉斯算子奇异摄动对应的摄动Sobolev空间，将其L²理论推广至L^r情形，给出r>2时空间函数的正则-奇异分解，并应用于相关NLS方程的局部适定性分析。
For a Jordan domain with sufficiently smooth boundaries, the solution to the Dirichlet problem for second order skew-symmetric strongly elliptic system with constant coefficients and regular enough boundary data is constructed in the form of a power series of a small parameter describing the perturbation of the given system from the Laplace one. The coefficients of this series are the functions that are determined sequentially as solutions to special Dirichlet problems for the usual Laplace and Poisson equations. The obtained series converges uniformly in the closure of the domain under consid…
We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in and around the Payne property, opening angle estimates of nodal domains, saturation of (fundamental) spectral gaps etc., and behaviour of all of the above under small scale perturbations. We aim to highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions, as opposed to asymptotic results.
The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary v…
In this paper we deal with an inhomogeneous parabolic Dirichlet problem involving the 1-Laplacian operator. We show the existence of a unique solution when data belong to $L^1(0,T;L^2(Ω))$ for every $T>0$. As a consequence, global existence and uniqueness for data in $L^1_{loc}(0,+\infty;L^2(Ω))$ is obtained. Our analysis retrieves the results of \cite{SW} in a correct and complete way.
We study the global solution curves, and prove the existence of infinitely many positive solutions for three classes of self-similar equations, with $p$-Laplace operator. In case $p=2$, these are well-known problems involving the Gelfand equation, the equation modeling electrostatic micro-electromechanical systems (MEMS), and a polynomial nonlinearity. We extend the classical results of D.D. Joseph and T.S. Lundgren \cite{JL} to the case $p \ne 2$, and we generalize the main result of Z. Guo and J. Wei \cite{GW} on the equation modeling MEMS.
The eigenvalue problem for the p-Laplace operator with Robin boundary condition is considered in this paper. A Faber-Krahn type inequality is proved. More precisely, it is shown that amongst all the domains of fixed volume, the ball has the smallest first eigenvalue.
Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace
本文研究含加权 p-Laplace 的 CKN 不等式，通过线性化问题分类证明极值元非退化，并结合谱估计与紧性论证建立梯度型余项稳定性。
A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.
In this paper we study the nonlinear Neumann boundary value problem of the following equations -\text{div}(|\nabla u|^{p_{1}(x)-2}\nabla u)-\text{div}(|\nabla u|^{p_{2}(x)-2}\nabla u)+|u|^{p_{1}(x)-2}u+|u|^{p_{2}(x)-2}u=λf(x,u) in a bounded smooth domain $Ω\subset\mathbb{R}^{N}$ with Neumann boundary condition given by |\nabla u|^{p_{1}(x)-2}\frac{\partial u}{\partialν}+|\nabla u|^{p_{2}(x)-2}\frac{\partial u}{\partialν}=μg(x,u) on $\partialΩ$. Under appropriate conditions on the source and boundary nonlinearities, we obtain a number of results on existence and multiplicity of solutions by var…
The Dirichlet problem on a bounded planar domain is more readily understood and solved for the Laplace operator than it is for a Schrodinger operator. When the potential function is small, we might hope to approximate the solution to the Schrodinger equation with the solution to the Laplace equation. In this vein we develop a series expansion for the solution and give explicit bounds on the error terms when truncating the series. We also examine a handful of examples and derive similar results for the Green function and Dirichlet-to-Neumann map.
Approximate radial symmetry for $p$-Laplace equations via the moving planes method
本文利用移动平面法研究 $p$-Laplacian 的 Gidas-Ni-Nirenberg 问题及 $p>2$ 临界 $p$-Laplace 方程在小扰动下的近似径向对称性，通过定量回顾弱 Harnack 比较不等式和局部有界性比较不等式，并证明小区域比较原理来实现。
On a shape derivative formula for the Robin $p$-Laplace eigenvalue
本文推导了Robin $p$-Laplace算子第一特征值的形状导数公式，并利用该公式证明在边界参数较大时，光滑区域上的第一特征值关于区域包含关系具有单调性。
This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-Δ)^s$ and involving a critical Sobolev term. In particular, we consider $$\begin{cases} (-Δ)^su=γ|u|^{2^*-2}u+f(x,u) & \mbox{in } Ωu=0 & \mbox{in } \mathbb R^n\setminus Ω, \end{cases}$$ where $Ω\subset\mathbb R^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in(0,1)$, $γ$ is a positive real parameter, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $f$ is a Carathéodory function satisfying different subcritical conditions.
Existence, non-existence and degeneracy of limit solutions to $p-$Laplace problems involving Hardy potentials as $p\to1^+$
本文研究$p\to1^+$时含Hardy势的$p$-Laplace问题解的极限行为，在尖锐小性条件下证明收敛到$1$-Laplace齐次Dirichlet问题的解，并通过例子验证假设最优性。
We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to o…
We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator $\mathcal L_{g,X,q}$, with $X$ being a $W^{1,\infty}$ vector field and $q$ being an $L^\infty$ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann determines $X$ and $q$ uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace-Beltrami operator due to Dos Santos Ferr…
Concentration Phenomena for $(p,N)$-Laplace Equation Under Discontinuous Nonlinearities and Penalization Method
本文研究了(p,N)-拉普拉斯方程在不连续非线性和惩罚方法下的解的存在性与集中现象，结合变分方法和Moser型迭代技术，揭示了临界增长下的解行为。
By using a perturbation technique in critical point theory, we prove the existence of solutions for two types of nonlinear equations involving fractional differential operators.
In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet-Neumann boundary data which extends the one proved by J. Dávila to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non-local counterpart to a Hopf's Lemma for fractional elliptic problems with mixed boundary data.
This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces …
We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equati…
We consider radial solutions of a general elliptic equation involving a weighted $p$-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or w…
Square integrable surface potentials on non-smooth domains and application to the Laplace equation in $L^2$
本文旨在将单层与双层位势理论从 $H^1_{loc}$ 正则性函数推广至局部平方可积函数，并使其适用于具有 Lipschitz 连续或多边形边界的非光滑区域，以满足流体动力学与数值模拟的需求。
Fractional Laplace operator on finite graphs
本文在任意有限图上离散化分数阶拉普拉斯算子，利用特征值与特征函数给出显式表示，并研究其收敛性质，进而通过变分原理和上下解方法得到分数阶 Kazdan-Warner 方程的存在性结果。
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。
We consider a Neumann problem for the Laplace equation in a periodic domain. We prove that the solution depends real analytically on the shape of the domain, on the periodicity parameters, on the Neumann datum, and on its boundary integral.
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 consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$, $-Δ_p u =u^{p^*-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$ then all the solutions in ${\mathcal D}^{1,p}(\R^N)$ are radial (and radially decreasing) about some point.
The C^r dependence problem of eigenvalues of the Laplace operator on domains in the plane
本文讨论椭圆算子多重狄利克雷特征值对区域的C^r依赖性问题，通过主定理对圆盘、正方形上拉普拉斯算子所有特征值及n≥3时球上第二特征值提供完整描述，核心工具为巴拿赫空间退化隐函数定理。
Tug-of-war games associated with boundary value problems involving derivatives
该研究聚焦一类可阐释1<p<2时规范化p-Laplace方程边值问题的带噪声拔河博弈，分析其值函数的边界正则性及向粘性解的收敛性，为相关偏微分方程问题提供博弈论视角的研究支撑。
Properties of fractional p-Laplace equations with sign-changing potential
本文研究受De Giorgi猜想启发的带变号势分数阶p-Laplacian方程，分别用移动平面法和滑动法证明有界域解的径向对称性与无界域解的方向性单调递增，揭示其性质差异源于算子的非局部性。
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