Homogeneous Solutions to Fully Nonlinear Elliptic Equations

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。

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.

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。

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…

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…

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…

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

We prove that any homogeneous order one solution to 3-d nondivergence elliptic equations must be linear.

Homogeneous Solutions of Fully Nonlinear Elliptic Equations in Four Dimensions

We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.

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。

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…

Borderline estimates for fully nonlinear elliptic equations

We prove new borderline regularity results for solutions to fully nonlinear elliptic equations together with pointwise gradient potential estimates.

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.

Smooth approximations for fully nonlinear nonlocal elliptic equations

We show that any viscosity solution to a general fully nonlinear nonlocal elliptic equation can be approximated by smooth ($C^\infty$) solutions.

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。

完全非线性椭圆方程正则性理论的视觉方法

A Visual Approach to the Regularity Theory for Fully Nonlinear Elliptic Equations

本文通过视觉方法重新审视一致椭圆方程的正则性理论，旨在提供直观理解。

ABP and global Holder estimates for fully nonlinear elliptic equations in unbounded domains

We prove global Holder estimates for solution of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions on the domain a' la Cabre'.

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。

Estimates for fully anisotropic elliptic equations with a zero order term

Integral estimates for weak solutions to a class of Dirichlet problems for nonlinear, fully anisotropic, elliptic equations with a zero order term are obtained using symmetrization techniques.

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.

完全非线性退化椭圆方程的Schauder型估计

Schauder-type estimates for fully nonlinear degenerate elliptic equations

该文针对完全非线性退化椭圆方程，研究其解在内部源项消失点处的正则性，得到了依赖类Hölder源项-椭圆性消失速率的Schauder型正则性估计。

On Axially Symmetric Solutions of Fully Nonlinear Elliptic Equations

We show that in dimension 3 axial-symmetric viscosity solutions of uniformly elliptic Hessian equations are in fact the classical ones.

C^{1,α}-regularity for a class of degenerate/singular fully nonlinear elliptic equations

We establish an optimal C^{1,α}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.

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…

二维情况下全非线性均匀椭圆方程的内部$C^{2,α}$正则性

Interior $C^{2,α}$ regularity for fully nonlinear uniformly elliptic equations in dimension two

本文研究了二维全非线性均匀椭圆方程粘性解的内部$C^{2,α}$正则性问题。

Pointwise Boundary Differentiability for Fully Nonlinear Elliptic Equations

In this paper, we prove the pointwise boundary differentiability for viscosity solutions of fully nonlinear elliptic equations. This generalizes the previous related results for linear equations. The geometrical conditions in this paper are pointwise and more general than before. Moreover, our proofs are relatively simple.

半空间上全非线性椭圆方程的Liouville定理

Liouville theorems for fully nonlinear elliptic equations on half spaces

本文通过边界正则性、Hopf和Carleson估计，简洁证明了半空间上全非线性椭圆方程的两个Liouville定理。

Fully nonlinear elliptic and parabolic equations in weighted and mixed-norm Sobolev spaces

We prove weighted and mixed-norm Sobolev estimates for fully nonlinear elliptic and parabolic equations in the whole space under a relaxed convexity condition with almost VMO dependence on space-time variables. The corresponding interior and boundary estimates are also obtained.

Necessary and sufficient conditions on existence of radial solutions for exterior Dirichlet problem of fully nonlinear elliptic equations

In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(λ(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior at infinity.

Dini estimates for nonlocal fully nonlinear elliptic equations

We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{σ+α}$ estimate in [8].

Overdetermined problems for fully nonlinear elliptic equations

We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $Ω$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $Ω$ to be a ball. This is a generalization of Serrin's classical result from 1971.

Free boundary regularity for a degenerate fully non-linear elliptic problem with right hand side

We consider an one-phase free boundary problem for a degenerate fully non-linear elliptic operators with non-zero right hand side. We use the approach present in \cite{DeSilva} to prove that flat free boundaries and Lipschitz free boundaries are $C^{1, γ}$.

Global regularity results for a class of singular/degenerate fully nonlinear elliptic equations

We provide the Alexandroff-Bakelman-Pucci estimate and global $C^{1, α}$-regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator.

On $W^{2,p}$-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $W^{2,p}$-regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.

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.

The Neumann problem for a class of fully nonlinear elliptic partial differential equations

In this paper, we establish a global $C^2$ estimates to the Neumann problem for a class of fullly nonlinear elliptic equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann problems.

Principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term

We study existence of principal eigenvalues of fully nonlinear integro-differential elliptic equations with a drift term via the Krein-Rutman theorem which based on regularity up to boundary of viscosity solutions. We also show the simplicity of the eigenfunctions in viscosity sense by a nonlocal version of ABP estimate.

Polynomial decay in $W^{2,\varepsilon}$ estimates for viscosity supersolutions of fully nonlinear elliptic equations

We prove $W^{2,\varepsilon}$ estimates for viscosity supersolutions of fully nonlinear, uniformly elliptic equations where $\varepsilon$ decays polynomially with respect to the ellipticity ratio of the equations. Our result is related to a conjecture of Armstrong-Silvestre-Smart [Comm. Pure Appl. Math. 65 (2012), no. 8, 1169--1184] which predicts a linear decay for $\varepsilon$ with respect to the ellipticity ratio of the equations.

Concentration and energy invariance for a class of fully nonlinear elliptic equations

We study the asymptotic behaviour of positive solutions of fully nonlinear elliptic equations in a ball, as the exponent of the power nonlinearity approaches a critical value. We show that solutions concentrate and blow up at the center of the ball, while a suitable associated energy remains invariant.

Entire subsolutions of fully nonlinear degenerate elliptic equations

We prove existence and non existence results for fully nonlinear degenerate elliptic inequalities, by showing that the classical Keller--Osserman condition on the zero order term is a necessary and sufficient condition for the existence of entire sub solutions.

Asymptotics and regularity of flat solutions to fully nonlinear elliptic problems

In this work we establish local $C^{2,α}$ regularity estimates for flat solutions to non-convex fully nonlinear elliptic equations provided the coefficients and the source function are of class $C^{0,α}$. For problems with merely continuous data, we prove that flat solutions are locally $C^{1,\text{Log-Lip}}$.

Boundary regularity for viscosity solutions of fully nonlinear elliptic equations

We provide regularity results at the boundary for continuous viscosity solutions to nonconvex fully nonlinear uniformly elliptic equations and inequalities in Euclidian domains. We show that (i) any solution of two sided inequalities with Pucci extremal operators is $C^{1,α}$ on the boundary; (ii) the solution of the Dirichlet problem for fully nonlinear uniformly elliptic equations is $C^{2,α}$ on the boundary; (iii) corresponding asymptotic expansions hold. This is an extension to viscosity solutions of the classical Krylov estimates for smooth solutions.

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…

Optimal C^{1,\apha} regularity for degenerate fully nonlinear elliptic equations with Neumann boundary condition

In the present paper, we study sharp C^{1;α} regularity results with boundary Neumann condition for viscosity solutions for a class of degenerate fully non-linear elliptic equations with Neumann boundary conditions.

Global branches of solutions for a class of non uniformly fully nonlinear elliptic equations

We consider singular perturbations of eigenvalue problems. We prove that to these problems correspond simple eigenvalues and we study their asymptotic behavior. As a result, we prove global bifurcation results for non uniformly and fully nonlinear elliptic equations. These equations are defined on smooth enough bounded domains and the solutions belonging in these branches are smooth and positive.

The Wiener criterion for fully nonlinear elliptic equations

We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle.

带Dirichlet边界条件的完全非线性退化椭圆障碍问题系统

Systems of Fully Nonlinear Degenerate Elliptic Obstacle problems with Dirichlet boundary conditions

本文研究带Dirichlet边界条件的完全非线性退化椭圆障碍问题系统，证明了含互联障碍极小化问题对应完全非线性椭圆系统粘性解的存在唯一性，该系统可应用于有界域最优切换问题等领域。

Symmetry of global solutions to a class of fully nonlinear elliptic equations in 2D

We prove that entire bounded monotone solutions to a certain class of fully nonlinear equations in 2D are one-dimensional. Our result also gives a new (non-variational) proof of the well known De Giorgi's conjecture.

Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.