Nikolaos S. Papageorgiou, Patrick Winkert
We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.
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We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations.
Global asymptotic expansion for anisotropic singularly perturbed elliptic problems
针对一类线性椭圆问题的各向异性奇异扰动，本文给出并证明了解在特定函数空间中的全局渐近展开。
We consider a class of singular weighted anisotropic $p$-Laplace equations. We provide sufficient condition on the weight function that may vanish or blow up near the origin to ensure the existence of at least one weak solution in the purely singular case and at least two different weak solutions in the purturbed singular case.
The singular anisotropic Adams' type inequality in $\mathbb{R}^n$
本文研究了奇异各向异性Adams型不等式在$\mathbb{R}^n$中的最佳常数及在有界域上的应用。
In this paper, we deal with anisotropic singular perturbations of some class of elliptic problem. We study the asymptotic behavior of the solution in certain second order pseudo Sobolev space.
In this article we study the asymptotic behavior, of the solution of a nonlinear elliptic, anisotropic singular perturbations problem in cylindrical domain, the limit problem is given and strong convergences are proved, we also give an application to intergo-differential problems.
Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity
本文针对扩散系数张量具有时间依赖、各向异性且可能奇异的热方程，在弱解或分布解可能不存在的情况下，利用很弱解框架证明了解的存在性与唯一性。
Very weak solutions of the heat equation with anisotropically singular time-dependent diffusivity
研究热方程在各向异性且时间依赖的奇异扩散张量下的非常弱解存在性和唯一性。
We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity, and some integral and pointwise Harnack inequalities.
We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.
Existence and uniqueness of weak solutions to singular anisotropic elliptic problems
本文研究了一类带p次次线扰动的非线性各向异性椭圆问题，证明了弱解的唯一性，并给出了存在性的必要充分条件。
Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations
本文研究了具有各向异性和奇异核的非局部Cahn-Hilliard方程的非局部到局部收敛性，并证明了在特定条件下非局部方程弱解的存在性。
We establish existence results for a class of mixed anisotropic and nonlocal $p$-Laplace equation with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, Harnack inequality for solutions and weak Harnack inequality for supersolutions.
In this article we give an extention of the L^2-theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear problems involving L^p data (1<p<2). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains
Existence and uniqueness of solutions to some anisotropic elliptic equations with a singular convection term
研究具有奇异对流项的各类各向异性椭圆方程，证明了弱解的存在唯一性，并推导了先验估计与正则性结果。
This article consists of study of anisotropic double phase problems with singular term and sign changing subcritical as well as critical nonlinearity. Seeking the help of well known Nehari manifold technique, we establish existence of at least two opposite sign energy solutions in the subcritical case and one negative energy solution in the critical case. The results in the critical case is even new in the classical $p$-Laplacian case.
In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.
In this article, we deal with some problems involving a class of singularly perturbed elliptic operator. We prove the asymptotic preserving of a general Galerkin method associated to a semilinear problem. We use a particular Galerkin approximation to estimate the convergence rate on the whole domain, for the linear problem. Finally, we study the asymptotic behavior of the semigroup generated.
The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, is…
We consider the Wulff-type energy functional $$ \mathcal{W}_Ω(u) := \int_ΩB(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.
Singular elliptic problems with convection term in anisotropic media
我们关心的奇异椭圆问题的形式$-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$在$Ω,$其中$Ω$是一个光滑的有界区域在$\RR^N$，$d(x)={\rm dist}(x,\partialΩ),$ $\la>0,$ $μ\in\RR$，$0<a\leq 2$，和$f,k$是非负和非减函数。我们假设$p(d(x))$是一个正的权重，在$Ω$的边界上可能有奇异行为，并且非线性$g$在原点周围是无界的。考虑到各向异性势$p(d(x))$、对流项$|\nabla u|^a$和奇异非线性项$g$之间的竞争，我们建立了各种存在性和不存在性结果。
Propagation of anisotropic Gabor singularities for Schrödinger type equations
本文研究了一类Schrödinger型演化方程解的各向异性Gabor波前集传播，证明了传播子在Shubin--Sobolev空间上的连续性，并揭示了波前集传播遵循主符号的Hamilton流。
We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. We prove a weak Harnack inequality and Hölder regularity results for solutions to corresponding integro-differential equations.
We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $λ$ varies. We also show the existence of minimal positive solutions $u_λ^*$ and determine the monotonicity and continuity propert…
In this paper we consider a very singular elliptic equation that involves an anisotropic diffusion operator, including one-Laplacian, and is perturbed by a $p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems analytically difficult to handle near a facet, the place where the gradient vanishes. Our main purpose is to prove that weak solutions are continuously differentiable even across the facet. Here it is of interest to know whether a gradient is continuous when it is truncated near a facet. To answer this affirmatively, we consider an approximation problem, and use st…
Local regularity for solutions to quasi-linear singular parabolic equations with anisotropic weights
本文提出了一种基于改进能量不等式和内在尺度因子的 De Giorgi 截断法，用于研究 $p$-Laplacian 型各向异性加权拟线性奇异抛物型方程解的局部正则性，并给出了无权情形下局部 Hölder 连续性的新证明。
For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0…
Regularity for Minimizers of non Autonomous Singular Functionals with Anisotropic Growth
研究非自治凸积分泛函在次二次各向异性增长条件下局部最小值的更高可微性，通过指数间隙和能量密度振荡的Sobolev假设得到结果。
Let $p_i\geq 2$ and consider the following anisotropic $p$-Laplace equation $$ -\sum_{i=1}^{N}\frac{\partial}{\partial x_i}\Big(\Big|\frac{\partial u}{\partial x_i}\Big|^{p_i-2}\frac{\partial u}{\partial x_i}\Big)=g(x)f(u),\,\,u>0\text{ in }Ω. $$ Under suitable hypothesis on the weight function $g$ we present an existence result for $f(u)=e^\frac{1}{u}$ in a bounded smooth domain $Ω$ and nonexistence results for $f(u)=-e^\frac{1}{u}$ or $-(u^{-δ}+u^{-γ}),$ $δ,γ>0$ with $Ω=\mathbb{R}^N$ respectively.
This paper concerns with the heat equation in the half-space $\mathbb{R}_{+}^{n}$ with nonlinearity and singular potential on the boundary $\partial\mathbb{R}_{+}^{n}$. We develop a well-posedness theory (without using Kato and Hardy inequalities) that allows us to consider critical potentials with infinite many singularities and anisotropy. Motivated by potential profiles of interest, the analysis is performed in weak $L^{p}$-spaces in which we prove key linear estimates for some boundary operators arising from the Duhamel integral formulation in $\mathbb{R}_{+}^{n}$. Moreover, we investigate…
We consider the time-harmonic acoustic wave scattering by a bounded {\it anisotropic inhomogeneity} embedded in an unbounded {\it anisotropic} homogeneous medium. The material parameters may have discontinuities across the interface between the inhomogeneous interior and homogeneous exterior regions. The corresponding mathematical problem is formulated as a transmission problems for a second order elliptic partial differential equation of Helmholtz type with discontinuous variable coefficients. Using a localised quasi-parametrix based on the harmonic fundamental solution, the transmission prob…
In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P)\left\{\begin{split} -Δ_{H,p}u&=\frac{\la f(x)}{u^{q(x)}}+g(u)\text{ in }Ω,\\ u&>0\text{ in }Ω,\,u=0\text{ on }\partialΩ, \end{split}\right. \end{equation*} under the assumption $Ω$ is a bounded smooth domain in $\mathbb{R}^N$ with $p,N\geq 2$, $\la>0$ and $0<q \in C(\bar \Om)$. For the purely singular case that is $g\equiv 0$, we proved existence and uniqueness of solution. We also demonstrate the existence of multiple solution to $(P)$ provided $f\equiv 1$ and $g(u)=u…
On the existence of ground state solutions to nonlinear Schoedinger equations with multisingular inverse-square anisotropic potentials
研究了一类具有临界幂非线性项和多个各向异性反平方奇性势的非线性Schroedinger方程。强度，位置和方向的奇异性的条件给出了最小的相关瑞利商要实现的，无论是在整个$\R^N$和有界域。
Compactness of Extremals for Singular Anisotropic Trudinger-Moser functionals on bounded domain
本文研究了临界奇异各向异性Trudinger-Moser泛函在有界域上的极值函数的紧致性，通过吹胀分析证明极值函数收敛到达到上确界的函数。
Singular limits of anisotropic weak solutions to compressible magnetohydrodynamics
本文严格证明了强各向异性可压缩磁流体动力学弱解收敛至简化磁流体动力学模型弱解的奇异极限，揭示了系统在垂直磁场方向不可压缩而在平行方向可压缩的特征。
The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences
本文构造了各向异性快速扩散方程的特异解，揭示其质量沿不同方向按一维VSS最小值分配的公式，并证明基本解尾部渐近收敛于该特异解尾部，进而得到全局Harnack原理和相对误差渐近收敛。
We study a large family of axisymmetric Riesz-type singular interaction potentials with anisotropy in three dimensions. We generalize some of the results of our recent work in two dimensions to the present setting. For potentials with linear interpolation convexity, their associated global energy minimizers are given by explicit formulas whose supports are ellipsoids. We show that for less singular anisotropic Riesz potentials, the global minimizer may collapse into one or two dimensional concentrated measures which minimize restricted isotropic Riesz interaction energies. Some partial aspects…
Uniform estimates for conforming Galerkin method for anisotropic singularly perturbed elliptic problems
该研究针对一类线性椭圆的各向异性奇异摄动问题，推导协调$Q_1$有限元方法的一致估计，同时证明连续问题的收敛性与正则性结果。
In this paper we study a gradient flow generated by the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the space of $Q$-tensors. This free energy density functional is composed of three quadratic terms as the elastic energy density part, and a singular potential in the bulk part that is considered as a natural enforcement of a physical constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approaches the physical boundary. We give a r…
In this paper we consider the 2D Ericksen-Leslie equations which describes the hydrodynamics of nematic Liquid crystal with external body forces and anisotropic energy modeling the energy of applied external control such as magnetic or electric field. Under general assumptions on the initial data, the external data and the anisotropic energy, we prove the existence and uniqueness of global weak solutions with finitely many singular times. If the initial data and the external forces are sufficiently small, then we establish that the global weak solution does not have any singular times and is r…
We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov-Poisson equation with measure density. The proof exploits an anisotropic variant of the argument of Crippa and De Lellis and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantita…
Anisotropic elliptic equations involving unbounded coefficients and singular nonlinearities
本文研究了涉及无界系数和奇非线性项的各向异性椭圆方程解的存在性和正则性。
The purpose of this article is to study very weak solutions of elliptic equation $$ -Δu+g(u)=2k\frac{\partial δ_0}{\partial x_N }+jδ_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0), $$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $δ_0$ is the Dirac mass concentrated at the origin and $\frac{\partialδ_0}{\partial x_N}$ is defined in the distribution sense that $$ \langle\frac{\partial δ_0}{\partial x_N},ζ\rangle=\frac{\partialζ(…
A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms
本文研究了一类具有无界系数和奇异二次低阶项的非线性各向异性椭圆问题的存在性和正则性，通过分析模型问题得出了相关结论。
Anisotropic Function Spaces on Singular Manifolds
本文对（可能）带边界的非紧黎曼流形上圆柱上的各向异性Bessel势、Besov和Hölder空间进行了较全面的研究。几何的基础流形附近的“两端”是由奇异函数自然导致加权函数空间的研究。除了Sobolev型嵌入结果的推导外，还特别强调了由边界条件区分的空间的尖迹定理、逐点乘子性质和插值特征。这一工作为奇异流形上含时偏微分方程的分析奠定了基础。
Anisotropic regularity for elliptic problems with Dirac measures as data
本文研究三维区域中一维曲线上的狄拉克测度作为源项的泊松问题，利用正则化方法在各向同性与各向异性Kondratiev型加权空间中证明了解的正则性，并推广到二维及闭曲线、多边形线支集情形。
We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $Ω$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{ \begin{array}{ll} \mathcal A u+ Φ(u,\nabla u)=Ψ(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } Ω,\\ u=0 & \mbox{on } \partialΩ. \end{array} \right. \end{equation} Here $ \mathcal A u=-\sum_{j=1}^N |\partial_j u|^{p_j-2}\partial_j u$ is the anisotropic $\overrightarrow{p}$-Laplace operator, while $\mathfrak B$ is an operator from $W_0^{1,\overrightarrow{p}}(Ω)$ into $W^{-1,\o…
Volume Integral Equations for Scattering from Anisotropic Diffraction Gratings
我们分析了电磁散射的TM偏振波从衍射光栅组成的周期性，各向异性，并可能负折射率的电介质材料。这样的散射问题对于例如，纳米光学元件和超材料中的光传播。周期性散射问题可以转化为一个强奇异体积积分方程，这是一种在工程界引起持续关注的技术，但很少得到严格的理论处理。本文证明了加权和非加权索伯列夫空间中强奇异积分方程的新的（广义）Garding不等式.这些不等式也适用于实部在衍射光栅内取负值的材料，与虚部的值无关。
Gradient rearrangement estimates for singular and degenerate anisotropic $p$-Laplace equations
本文针对各向异性 $p$-拉普拉斯方程的SOLA解，证明了其梯度的全局点态重排估计，将相关重排理论扩展到奇异范围，完善了退化范围的结果并得到端点弱型梯度估计与梯度差稳定性估计。
Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation
研究了一类空间非齐次各向异性反应扩散方程的奇异极限。我们使用Finsler度量相关的各向异性扩散项和工作在相对几何。我们证明了一个弱比较原理，并进行分析的产生和运动的接口。极限问题涉及各向异性中曲率运动。我们还证明了过渡层厚度的最佳估计。
We study the perfect conductivity problem when two perfectly conducting inclusions are closely located to each other in an anisotropic background medium. We establish optimal upper and lower gradient bounds for the solution in any dimension which characterize the singular behavior of the electric field as the distance between the inclusions goes to zero.
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels
本文研究了具有奇异各向异性核的非局部卷积算子向局部微分算子的非局部到局部收敛性，并在一般 $L^p$ 空间中证明了强收敛性及收敛速率。
We study Sobolev spaces with weights in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the singular elliptic operators \begin{equation*} \mathcal L =y^{α_1}Δ_{x} +y^{α_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right). \end{equation*}
We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the euclidian space, and in Carnot groups is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the euclidian setting. These results simplify the handling o…
Anisotropic finite elements for elliptic problems with singular data
本文研究了具有奇异测度的椭圆问题，通过适当梯度网格实现有限元方法的最优收敛率，探讨了各向同性和各向异性梯度网格的应用及数值实验结果。
Singular perturbations models in phase transitions for anisotropic higher-order materials
本文通过Γ-收敛分析高阶各向异性材料相变中的奇异摄动泛函，克服系数负性与张量范数各向异性困难，证明Γ-极限为各向异性周长。
We establish small energy Hölder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_ΩW(\nabla u)+ \frac{1}{\varepsilon^2} \int_Ωf(u),\] where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are kno…
Dimensional estimates for singular sets in geometric variational problems with free boundaries
我们证明了余维1各向异性几何变分问题中自由边界的奇异集是$\mathcal H ^{n-3}$ -可忽略的，其中$n$是环境空间维数.特别是我们的结果适用于毛细作用型问题，并建立无处不在的三维情况下的规律性。
We prove Strichartz estimates for Maxwell equations in media in the fully anisotropic case with Hölder-continuous coefficients. To this end, we use the FBI transform to conjugate the problem to phase space. After reducing to a scalar estimate by means of a matrix symmetrizer, we show oscillatory integral estimates for a variable-coefficient Fourier extension operator. The characteristic surface has conical singularities for any non-vanishing time frequency. Combined with energy estimates, we improve the local well-posedness for certain fully anisotropic quasilinear Maxwell equations.
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