Linearity of homogeneous order one solutions to elliptic equations in dimension three

Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-Δu=|x|^{-s} u^{2^\star(s)-1} -μu^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $μ>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three diff…

Regularity for solutions of fully nonlinear elliptic equations with non-homogeneous degeneracy

We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally $C^{1,γ}$-regular.

Singular quasilinear elliptic systems with (super-) homogeneous conditions

In this paper we establish existence, nonexitence and regularity of positive solutions for a class of singular quasilinear elliptic systems subject to (super-) homogeneous condition. The approach is based on sub-supersolution methods for systems of quasilinear singular equations combined with perturbation arguments involving singular terms.

Strongly singular convective elliptic equations in RN driven by a non-homogeneous operator

Existence of a generalized solution to a strongly singular convective elliptic equation in the whole space is established. The differential operator, patterned after the (p,q)-Laplacian, can be non-homogeneous. The result is obtained by solving some regularized problems through fixed point theory, variational methods and compactness results, besides exploiting nonlinear regularity theory and comparison principles.

有界函数类中椭圆与消没齐次常系数微分算子可去集合的豪斯多夫维数

Hausdorff dimension of removable sets for elliptic and canceling homogeneous differential operators in the class of bounded functions

本文通过推广Frostman引理并引入额外幂衰减，证明了有界函数类中椭圆与消没齐次常系数微分算子可去集合的豪斯多夫维数上界。

Fundamental matrices and Green matrices for non-homogeneous elliptic systems

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi-Nash-Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions.

On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions

We extend the result of D. Phillips (On one-homogeneous solutions to elliptic systems in two dimensions. C. R. Math. Acad. Sci. Paris 335 (2002), no. 1, 39-42) by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to one-homogeneous solutions belonging to the Sobolev space H^{1}, from which his treatment of Lipschitz solutions follows as…

A note on Lebesgue solvability of elliptic homogeneous linear equations with measure data

In this work, we present new results on solvability of the equation $A^{*}(D)f=μ$ for $f \in L^{p}$ and positive measure data $μ$ associated to an elliptic homogeneous linear differential operator $A(D)$ of order m. Our method is based on $(m,p)-$energy control of $μ$ giving a natural characterization for solutions when $1\leq p < \infty$. We also obtain sufficient conditions in the limiting case $p=\infty$ using {new $L^{1}$ estimates on measures for elliptic and canceling operators.

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $Ω$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum princi…

齐次分数阶 Sobolev 空间中的非局部临界椭圆方程

Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

本文基于 ${\mathbb Z}_2$-上同调指标的抽象临界点定理、分数阶 $p$-Laplacian 方程的正则性结果与紧嵌入，证明了齐次分数阶 Sobolev 空间中非局部临界增长椭圆方程的多重性结果。

Fundamental solutions for a class of non-elliptic homogeneous differential operators

We compute temperate fundamental solutions of homogeneous differential operators with real-principal type symbols. Via analytic continuation of meromorphic distributions, fundamental solutions for these non-elliptic operators can be constructed in terms of radial averages and invariant distributions on the unit sphere.

Ground states of elliptic problems involving non homogeneous operators

We investigate the existence of ground states for functionals with nonhomogenous principal part. Roughly speaking, we show that the Nehari manifold method requires no homogeinity on the principal part of a functional. This result is motivated by some elliptic problems involving nonhomogeneous operators. As an application, we prove the existence of a ground state and infinitely many solutions for three classes of boundary value problems.

Fundamental solutions of homogeneous elliptic differential operators

We compute fundamental solutions of homogeneous elliptic differential operators, with constant coefficients, on $\mathbb{R}^n$ by mean of analytic continuation of distributions. The result obtained is valid in any dimension, for any degree and can be extended to pseudodifferential operators of the same type.

拟线性椭圆方程的非齐次共法向导数问题（具有Morrey数据）

Non-homogeneous conormal derivative problem for quasilinear elliptic equations with Morrey data

本文研究了拟线性椭圆方程在Morrey空间框架下的非齐次共法向导数问题，证明了弱解的全局本质有界性，推广了经典L^p结果。

Nonlinear elliptic equations with a singular perturbation on compact Lie groups and Homogeneous spaces

This paper is devoted to the study of a class of singular perturbation elliptic type problems on compact Lie groups or homogeneous spaces $\mathcal{M}$. By constructing a suitable Nash-Moser-type iteration scheme on compact Lie groups and homogeneous spaces, we overcome the clusters of "small divisor" problem, then the existence of solutions for nonlinear elliptic equations with a singular perturbation is established. Especially, if $\mathcal{M}$ is the standard torus $\textbf{T}^n$ or the spheres $\textbf{S}^n$, our result shows that there is a local uniqueness of spatially periodic solutions…

A Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space with Homogeneous Neumann Boundary Data

In this note we derive large-scale regularity properties of solutions to second-order linear elliptic equations with random coefficients on the half- space with homogeneous Neumann boundary data; it is a companion to arXiv:1604.02717 in which the situation for homogeneous Dirichlet boundary data was addressed. Similarly to arXiv:1604.02717, the results in this contribution are expressed in terms of a first-order Liouville principle. It follows from an excess-decay that is shown through means of a stochastic homogenization-inspired Campanato iteration. The core of this contribution is the const…

具有齐次临界非线性的各向异性拟线性椭圆系统

Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities

研究各向异性 p-拉普拉斯拟线性椭圆系统，给出保证非平凡解存在与不存在的条件，涉及临界 Sobolev 增长。

Lawruk elliptic boundary-value problems for homogeneous differential equations

We investigate Lawruk elliptic boundary-value problems for homogeneous differential equations in a two-sided refined Sobolev scale. These problems contain additional unknown functions in the boundary conditions of arbitrary orders. The scale consists of inner-product Hörmander spaces whose orders of regularity are given by any real number and a function which varies slowly at infinity in the sense of Karamata. We establish theorems on the Fredholm property of the problems in the refined Sobolev scale and on local regularity and local a priori estimate (up to the boundary of the domain) of thei…

Existence and multiplicity results for a class of non-linear Schrödinger equations with magnetic potential involving sign-changing non linearity

In this work we consider the following class of elliptic problems $$- Δ_A u + u = a(x) |u|^{q-2}u+b(x) |u|^{p-2}u , \mbox{ in } \mathbb{R}^N, $$ $u\in H^1_A (\mathbb{R}^N)$, with $2<q<p<2^*= \frac{2N}{N-2}$, $a(x)$ and $b(x)$ are functions that can change signal and satisfy some additional conditions; $u \in H^1_A(\mathbb{R}^N)$ and $A:\mathbb{R}^N \rightarrow \mathbb{R}^N$ is a magnetic potential. Also using the Nehari method in combination with other complementary arguments, we discuss the existence of infinite solutions to the problem in question, varying the assumptions about the weight fu…

The Brunn-Minkowski inequality for the principal eigenvalue of fully nonlinear homogeneous elliptic operators

We prove that the principal eigenvalue of any fully nonlinear homogeneous elliptic operator which fulfills a very simple convexity assumption satisfies a Brunn-Minkowski type inequality on the class of open bounded sets in $\mathbb{R}^n$ satisfying a uniform exterior sphere condition. In particular the result applies to the (possibly normalized) $p$-Laplacian, and to the minimal Pucci operator. The proof is inspired by the approach introduced by Colesanti for the principal frequency of the Laplacian within the class of convex domains, and relies on a generalization of the convex envelope metho…

Finsler p-Laplacian 方程具有多项式非线性时正解的唯一性

Uniqueness of positive solutions for finsler p-Laplacian equations with polynomial non-linearity

本文研究Finsler p-Laplacian方程在多项式非线性下正解的唯一性，利用线性化方法推广了Brasco和Lindgren的结论。

具有非局部齐次临界非线性项的半线性椭圆方程组

Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities

本文研究推广Brezis-Nirenberg问题的半线性椭圆方程组，确定了其非平凡解的存在条件与仅存在平凡解的条件，丰富了非局部临界非线性椭圆问题的理论研究。

Multiplicity of solutions for homogeneous elliptic systems with critical growth

In this paper we are concerned with the number of nonnegative solutions of the elliptic system $$ {array}{ll} -Δu = Q_u(u,v) + 1/2{2^*} H_u(u,v),& {in} Ω,\vdois\ -Δv = Q_v(u,v) + 1/{2^*} H_v(u,v),& {in} Ω,\vdois\ u=v=0,& {on} \partialΩ. \leqno{(P)} $$ where $Ω\subset \mathbb{R}^N$ is a bounded smooth domain, $N \geq 3$, $2^*:=2N/(N-2)$ and $Q_u, H_u$ and $Q_v$, $H_v$ are the partial derivatives of the homogeneous functions $Q,\,H \in C^1(\mathbb{R}^2_+,\mathbb{R})$, where $ \mathbb{R}^2_+ := [0,\infty) \times [0,\infty)$. In the proofs we apply variational methods and Ljusternik-Schnirelmann t…

Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $Σ$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $Φ(H,K_e,K)=0$. In this paper we prove that $Σ$ is a sphere of revolution, provided that the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times \mathbb{R}$ i…

Singular Solutions of Hessian Elliptic Equations in Five Dimensions

We show that for any $δ\in [0,1)$ there exists a homogeneous order $2-δ$ analytic outside zero solution to a uniformly elliptic Hessian equation in R^5.

具有变指数、齐次Neumann边界条件和广义logistic源的二重非线性椭圆问题

A doubly nonlinear elliptic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source

本文研究一类源于Rothe时间离散化的变指数二重非线性椭圆问题，通过弱化源项假设并利用Nemytskii算子连续性，建立了在广义logistic源下的解的存在性分析。

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: Off-diagonal estimates on spaces of homogeneous type

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant $L^p-L^q$ estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators.

Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems

We give examples of systems of Partial Differential Equations that admit non-trivial, Lipschitz and one-homogeneous solutions in the form $u(R,θ) = Rg(θ)$, where $(R,θ)$ are plane polar coordinates and $g: \mathbb{R}^{2} \to \mathbb{R}^{m}$, $m \geq 2$. The systems are singular in the sense that they arise as the Euler-Lagrange equations of the functionals $I(u) = \int_{B}W(x,\nabla u(x))\,dx$, where $D_{F}W(x,F)$ behaves like $\frac{1}{|x|}$ as $|x| \to 0$ and $W$ satisfies an ellipticity condition. Such solutions cannot exist when $|x|D_{F}W(x,F) \to 0$ as $|x| \to 0$, so the condition is op…

L^p regularity of homogeneous elliptic differential operators with constant coefficients on R^N

Let $A$ be a homogeneous elliptic differential operator of order $m$ on $% \Bbb{R}^{N}$ with constant complex coefficients. A partial version of the main result is as follows: Suppose that $u\in L_{loc}^{1}$ and that $Au\in L^{p}$ for some $1<p<\infty .$ Then, all the partial derivatives of order $m$ of $u$ are in $L^{p}$ if and only if $|u|$ grows slower than $|x|^{m}$ at infinity, provided that growth is measured in an $L^{1}$-averaged sense over balls with increasing radii. The necessity provides an alternative answer to the pointwise growth question investigated with mixed success in the l…

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

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…

利用 $p(x)$-Díaz-Saa 不等式的推广研究具变指数和齐次 Neumann 边界条件的一般拟线性椭圆问题强正解的唯一性

Uniqueness of the strong positive solution for a general quasilinear elliptic problem with variable exponents and homogeneous Neumann boundary conditions using a generalization of the $p(x)$-Díaz-Saa inequality

本文推广了 $p(x)$-Díaz-Saa 不等式，使其适用于具变指数和齐次 Neumann 边界条件的一般拟线性椭圆问题，并由此证明了强正解的唯一性，同时给出多相问题和图像处理中的具体应用。

$L^\infty$ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity

We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We …

椭圆齐次线性方程在加权Lebesgue空间中带有测度数据的可解性

Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces

本文研究了在加权Lebesgue空间中，椭圆齐次线性微分方程在给定测度数据下的可解性条件，通过控制测度的Riesz势提出充分条件，并在p=∞时引入抵消条件。

与Monge-Ampère方程可比的凹齐次复退化椭圆方程的一致估计

Uniform estimates for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation

该研究依托Kuo和Trudinger的思路、复Monge-Ampère算子多重位势理论及定制的$L^p$-粘性下解理论，证明了一大类完全非线性退化椭圆复方程强上解的精确一致估计。

带齐次边界条件的椭圆型偏微分方程的Barron空间表示

Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

该研究针对带齐次边界条件的高维二阶椭圆型偏微分方程，证明在系数和强迫项满足Barron假设时，其解可由参数规模为O(d^{C|logε|})的两层神经网络逼近，且不受维数灾难影响。

与Monge-Ampère方程可比的凹齐次复退化椭圆方程的局部正则性

Local regularity for concave homogeneous complex degenerate elliptic equations comparable to the Monge-Ampère equation

该研究针对与Monge-Ampère方程可比的复退化非线性椭圆方程，建立了其$W^{2,p}_{\mathrm{loc}}$解的局部正则性结果，并将成果应用于$k$-Monge-Ampère方程，推进了复退化椭圆方程的正则性理论研究。

Existence and homogenization of nonlinear elliptic systems in nonreflexive spaces

We consider a strongly nonlinear elliptic problem with the homogeneous Dirichlet boundary condition. The growth and the coercivity of the elliptic operator is assumed to be indicated by an inhomogeneous anisotropic $\mathcal{N}$-function. First, an existence result is shown under the assumption that the $\mathcal{N}$-function or its convex conjugate satisfies $Δ_2$-condition. The second result concerns the homogenization process for families of strongly nonlinear elliptic problems with the homogeneous Dirichlet boundary condition under above stated conditions on the elliptic operator, which is…

Revised regularity results for quasilinear elliptic problems driven by the $Φ$-Laplacian operator

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known $Φ$-Laplacian operator given by \begin{equation*} \left\{\ \begin{array}{cl} \displaystyle-Δ_Φu= g(x,u), & \mbox{in}~Ω, u=0, & \mbox{on}~\partial Ω, \end{array} \right. \end{equation*} where $Δ_Φu :=\mbox{div}(ϕ(|\nabla u|)\nabla u)$ and $Ω\subset\mathbb{R}^{N}, N \geq 2,$ is a bounded domain with smooth boundary $\partialΩ$. Our work concerns on nonlinearities $g$ which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a re…

A lower bound for the principal eigenvalue of fully nonlinear elliptic operators

In this article we present a new technique to obtain a lower bound for the principal Dirichlet eigenvalue of a fully nonlinear elliptic operator. We ilustrate the construction of an appropriate radial function required to obtain the bound in several examples. In particular we use our results to prove that $\lim_{p\to \infty}λ_{1,p}=λ_{1,\infty}=\left(\fracπ{2R}\right)^2$ where $λ_{1,p}$ and $λ_{1,\infty}$ are the principal eigenvalue for the homogeneous $p$-laplacian and the homogeneous infinity laplacian respectively.

Fundamental solutions of homogeneous fully nonlinear elliptic equations

We prove the existence of two fundamental solutions $Φ$ and $\tilde Φ$ of the PDE \[ F(D^2Φ) = 0 \quad {in} \mathbb{R}^n \setminus \{0 \} \] for any positively homogeneous, uniformly elliptic operator $F$. Corresponding to $F$ are two unique scaling exponents $α^*, \tildeα^* > -1$ which describe the homogeneity of $Φ$ and $\tilde Φ$. We give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation $F(D^2u) = 0$, which is bounded on one side. A Liouville-type result demonstrates that the two fundamental solutions are the unique nontrivial…

Existence of Positive Solution of a Class of Semi-linear Sub-elliptic Equation in the Entire Space $H^n$

In this paper, we study the following problem $$ \{{ll} Δ_{H^n} u-u+u^p=0 & in H^n u>0& in H^n u(x)\to 0 &ρ(x)\to\infty}. $$ where $1<p < \frac{Q+2}{Q-2}$, Q is the homogeneous dimension of Heisenberg group $H^n$. Our main result is that this problem has at least one positive solution.

Blow-up time of strong solutions to a biological network formation model in high space dimensions

We investigate the possible blow-up of strong solutions to a biological network formation model originally introduced by D. Cai and D. Hu \cite{HC}. The model is represented by an initial boundary value problem for an elliptic-parabolic system with cubic non linearity. We obtain an algebraic equation for the possible blow-up time of strong solutions. The equation yields information on how various given data may contribute to the blow-up of solutions. As a by-product of our development, we establish a $W^{1,q}$ estimate for solutions to an elliptic equation which shows the explicit dependence o…

Sound travellingwaves in wind instruments as solutions to non linear homogeneous gas dynamics equations

The sound propagation is usually described by a linear homogeneous wave equation, though the air flow in a duct is described by the gas dynamics equations, using a variable cross section, which corresponds to a non linear non homogneous system. The aim of this paper is to exhibit a common periodic solution to both models, with several free parameters such as frequency or amplitude, able to represent any sound. By taking in account a friction term linked to the material (wood or brass for instance) of the duct, it is possible to build an analytic such solution when the cross section fullfills s…

On the principal frequency of non-homogeneous membranes

We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $Ω\subset \mathbb C$ satisfying quasihyperbolic boundary conditions. The suggested method is based on the quasiconformal composition operators on Sobolev spaces and their applications to constant estimates in the corresponding Sobolev-Poincaré inequalities. We also prove a variant of the Rayleigh-Faber-Khran inequality for a special case of these elliptic operators.

A priori bounds for positive solutions of subcritical elliptic equations

We provide a-priori $L^\infty$ bounds for positive solutions to a class of subcritical elliptic problems in bounded $C^2$ domains. Our arguments rely on the moving planes method applied on the Kelvin transform of solutions. We prove that locally the image through the inversion map of a neighborhood of the boundary contains a convex neighborhood; applying the moving planes method, we prove that the transformed functions have no extremal point in a neighborhood of the boundary of the inverted domain. Retrieving the original solution $u$, the maximum of any positive solution in the domain $\Om,$ …

Nonlinear bound states on weakly homogeneous spaces

We prove the existence of ground state solutions for a class of nonlinear elliptic equations, arising in the production of standing wave solutions to an associated family of nonlinear Schrödinger equations. We examine two constrained minimization problems, which give rise to such solutions. One yields what we call $F_λ$-minimizers, the other energy minimizers. We produce such ground state solutions on a class of Riemannian manifolds called weakly homogeneous spaces, and establish smoothness, positivity, and decay properties. We also identify classes of Riemannian manifolds with no such minimiz…

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.

Saddle point solutions for non-local elliptic operators

The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theorem.

Maximal Function and Riesz Transform Characterizations of Hardy Spaces Associated with Homogeneous Higher Order Elliptic Operators and Ball Quasi-Banach Function Spaces

Let $L$ be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on $\mathbb{R}^n$ and $X$ a ball quasi-Banach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Denote by $H_{X,\, L}(\mathbb{R}^n)$ the Hardy space, associated with both $L$ and $X$, which is defined via the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish both the maximal function and the Riesz transform characterizations of $H_{X,\, L}(\mathbb{R}^n)$. The results obtained in this article have a wide range o…

The Liouville theorems for elliptic equations with nonstandard growth

We study solutions and supersolutions of homogeneous and nonhomogeneous $\mathcal{A}$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is illustrated by a number of examples.

关于具有椭圆退化的分数阶拟线性方程

On fractional quasilinear equations with elliptic degeneracy

本文提出系统方法，研究具有局部梯度退化的非局部拟线性方程解的存在性、多重性与梯度正则性，通过交互式几何论证与齐次问题唯一性结合，获得梯度Hölder估计，并展示多重解与唯一性情形。

带有卷积项的非齐次四阶椭圆不等式

Non-homogeneous fourth order elliptic inequalities with the convolution term

本文研究一类带卷积项的非齐次四阶椭圆不等式，刻画了非负解存在与不存在的参数条件，揭示了维数、指数及位势参数对解的影响。

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted Sobolev classes. We establish mapping properties for the double and single layer potentials, as well as the Newton potential, on Besov spaces. We prove extrapolation-type solvability results: that is, we show that solvability of the Dirichlet or Neumann boundary value problem at any given L^p space automatically assures their solvability in an extended range of Besov spaces. We also esta…

Geometric Gradient Estimates For Fully Nonlinear Models With Non-Homogeneous Degeneracy and Applications

We establish sharp geometric Holder regularity estimates for Gradient for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy. Such regularity estimates simplify and generalize, to some extent, earlier ones via a totally different modus operandi. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. In the end, we present some connections of our findings with a variety of nonlinear geometric free boundary problems and relevant nonlinear problems in t…