Cyril Imbert, Luis Silvestre
We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the H{ö}lder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
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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'.
A new proof of Hölder estimates for the gradient of quasilinear elliptic equations
本文受Evans-Krylov定理证明启发，采用覆盖方法给出拟线性椭圆方程梯度Hölder估计的新证明，并通过相同方法获得全非线性椭圆方程梯度的Hölder估计。
By borrowing ideas from the parabolic theory, we use a combination of De Giorgi's and Moser's methods to give some remarks on the proof of Hölder continuity of weak solutions of elliptic equations.
Local Schauder estimates hold in the nonuniformly elliptic setting. Specifically, first derivatives of solutions to nonuniformly elliptic variational problems and elliptic equations are locally Hölder continuous, provided coefficients are locally Hölder continuous.
In this note we study the global regularity in the Morrey spaces for the second derivatives for the strong solutions of non variational elliptic equations.
Quantitative unique continuation for elliptic equations with Hölder continuous potentials
本文研究了带有Holder正则低阶项的二阶椭圆方程的定量化唯一持续性，通过加权频率函数方法建立了三球不等式及消失阶界，填补了bounded势与C^1势之间定量化唯一持续性的空白。
We prove a sharp Hölder estimate for solutions of linear two-dimensional, divergence form elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has {\em unit determinant}. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.
In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.
In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point theorem.
We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-Hölder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of th…
For a family of second-order elliptic systems in divergence form with rapidly oscillating almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform Hölder estimates for the Dirichlet problem in a bounded $C^{1, α}$ domain.
The dependence of the smoothness of variational solutions to the first boundary value problems for second order elliptic operators are studied. The results use Sobolev-Slobodetskii and Nikolskii-Besov spaces and their properties. Methods are based on real interpolation technique and generalization of Savaré-Nirenberg difference quotient technique.
Estimates on elliptic equations that hold only where the Hessian is large
研究一类退化完全非线性椭圆方程的粘性解的赫尔德正则性，该方程算子仅在黑塞矩阵大处椭圆，结合修正尖点函数与接触集分解证明，得出自然假设下的内部赫尔德连续性结论。
We give existence and regularity results for solutions of some nonlinear elliptic problems. The equations we deal with are modeled on a problem which involves in its principal part an anisotropic operator, a Hardy-type potential, and a lower-order term depending on the gradient.
We extend several well-known tools from the theory of second-order divergence-form elliptic equations to the case of higher-order equations. These tools are the Caccioppoli inequality, Meyers's reverse Holder inequality for gradients, and the fundamental solution. Our construction of the fundamental solution may also be of interest in the theory of second-order operators, as we impose no regularity assumptions on our elliptic operator beyond ellipticity and boundedness of coefficients.
We study existence and Lorentz regularity of distributional solutions to elliptic equations with either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.
We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions
This paper is concerned with Hölder regularity of viscosity solutions of second-order, fully non-linear elliptic integro-differential equations. Our results rely on two key ingredients: first we assume that, at each point of the domain, either the equation is strictly elliptic in the classical fully non-linear sense, or (and this is the most original part of our work) the equation is strictly elliptic in a non-local non-linear sense we make precise. Next we impose some regularity and growth conditions on the equation. These results are concerned with a large class of integro-differential opera…
We study the local Hölder regularity of strong solutions $u$ of second-order uniformly elliptic equations having a gradient term with superquadratic growth $γ> 2$, and right-hand side in a Lebesgue space $L^q$. When $q > N\frac{γ-1}γ$ ($N$ is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent $α_q > \frac{γ-2}{γ-1}$. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of $u$ in $L^q$. Our methods are based on blow-up techniques and a Liouville theorem.
We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.
We consider the inverse problem of determining some class of nonlinear terms appearing in an elliptic equation from boundary measurements. More precisely, we study the stability issue for this class of inverse problems. Under suitable assumptions, we prove a Lipschitz and a Hölder stability estimate associated with the determination of quasilinear and semilinear terms appearing in this class of elliptic equations from measurements restricted to an arbitrary parts of the boundary of the domain. Besides their mathematical interest, our stability estimates can be useful for improving numerical re…
We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case of solutions that are not Lipschitz continuous at x=0. We show that our estimate is sharp.
We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order $2m$ with certain lower order terms.
We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.
Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of Hölder Continuous Solutions and Homogenization Results
本文综述CDC区域上一致椭圆Poisson型方程的全局Hölder连续解理论，并将其应用于ε周期系数均匀化，得到L^∞范数收敛速率估计。
In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.
Hölder stability for a semilinear elliptic inverse problem
本研究针对半线性椭圆方程非线性项的反演问题，改进了已有对数型稳定性结果，在更少边界测量和更低正则性条件下得到Hölder稳定性估计，采用构造子边界零值特殊解的方法完成证明。
In this paper we report the asymptotic behaviors of viscosity solutions of the following degenerate elliptic equations \begin{equation*}\label{main-Eq} Lu=x_n^{2α}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x) +2x_n^α\sum_{i=1}^{n-1}a_{in}(x)D_{in}u(x)+D_{nn}u(x)=0 \end{equation*} outside a bounded domain of the upper half space for $α>0$ with proper coefficients. Meanwhile, the Hölder estimates up to the boundary will be obtained using rescaling methods.
We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity.
We show that the new result on Hölder continuity of solutions to a class of nondiagonal elliptic systems with $p$-growth in [2] can be used to improve the $L^q$ theory for such systems.
We prove global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order differential operators in $\mathbb{R}^d$ with unbounded diffusion, drift and potential terms.
We study boundary gradient estimates for second-order divergence type parabolic and elliptic systems in $C^{1,α}$ domains. The coefficients and data are assumed to be Hölder in the time variable and all but one spatial variables. This type of systems arises from the problems of linearly elastic laminates and composite materials.
Gradient estimates for quasilinear elliptic Neumann problems with unbounded first-order terms
本文在源项满足适当可积性条件下，针对一阶项关于梯度多项式增长的p-Laplacian型散度方程，建立了凸域上Neumann问题的全局先验梯度估计。
We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is a measure. A global Calderón-Zygmund estimate for the gradient of an approximable solution is obtained in terms of the associated double obstacles and a given measure, identifying minimal requirements for the regularity estimate.
We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of non-negative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.
We discuss counterexamples to the validity of the weak Maximum Principle for linear elliptic systems with zero and first order couplings and prove, through a suitable reduction to a nonlinear scalar equation, a quite general result showing that some algebraic condition on the structure of gradient couplings and a cooperativity condition on the matrix of zero order couplings guarantee the existence of invariant cones in the sense of Weinberger [21].
We consider a nonlinear Neumann elliptic inclusion with a source (reaction term) consisting of a convex subdifferential plus a multivalued term depending on the gradient. The convex subdifferential incorporates in our framework problems with unilateral constraints (variational inequalities). Using topological methods and the Moreau-Yosida approximations of the subdifferential term, we establish the existence of a smooth solution.
In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=μ,$$ in a bounded domain $Ω\subset\mathbb{R}^{N}$, where $A(x,\nabla u)\nabla u \asymp |\nabla u|^p$, $p\in (1,2-\frac{1}{n}]$ and $μ$ is a Radon measure in $Ω$
Let $2\le n\le 5$. We establish an apriori interior Hölder regularity of $C^2$-stable solutions to the semilinear equation $-Δu=f(u)$ in any domain of $R^n$ for any nonlinearity $f\in C^{0,1}(R) $.If $f $ is nondecreasing and convex in addition,we obtain an interior Hölder regularity, and hence the local boundedness, of $W^{1,2}(Ω)$-stable solutions by locally approximating them via $C^2(Ω)$-stable solutions. In particular, we do not require any lower bound on $f$.
We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.
Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients
本文证明了一类具有低阶项、系数受边界距离负幂控制的椭圆方程全局Hölder连续解的存在性，改进了经典定理，并采用Ancona策略结合新Morrey型空间进行证明。
Higher Hölder regularity for degenerate elliptic PDEs with data in Morrey spaces
本文利用 Fefferman-Phong 不等式与非线性 PDE 正则性理论，证明了具有 Morrey 空间数据的 p-Laplacian 型退化椭圆方程弱解的精确局部 $C^{1,α}$-正则性。
We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for the solutions is proved through an argument which uses a boundary growth lemma, and consequences such as boundary "half"-Harnack inequalities, which are of independent interest. Our results are new even for linear equations.
A duality approach to gradient Hölder estimates for linear divergence form elliptic equations
本文提出了一种基于对偶性的方法，用于推导线性散度形式椭圆方程的梯度Hölder估计，通过Schauder理论与局部Hardy空间和Hölder空间的联系，展示了稀疏界和反Hölder不等式的推导方法。
We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.
This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data: \begin{cases} -\operatorname{div}(A(x,D u))=g-\operatorname{div} f \quad & \mathrm{in} \quad Ω\\ u= 0 \quad & \text{on} \ \partial Ω, \end{cases} where $Ω\subset \mathbb{R}^n$ is sufficiently flat in the sense of Reifenberg.
This paper investigates the relation between the boundary geometric properties and the boundary regularity of the solutions of elliptic equations. We prove by a new unified method the pointwise boundary Hölder regularity under proper geometric conditions. "Unified" means that our method is applicable for the Laplace equation, linear elliptic equations in divergence and non-divergence form, fully nonlinear elliptic equations, the $p-$Laplace equations and the fractional Laplace equations etc. In addition, these geometric conditions are quite general. In particular, for local equations, the meas…
On some singularly perturbed elliptic systems modeling partial segregation: uniform Hölder estimates and basic properties of the limits
本文针对具有三元相互作用的奇异摄动竞争-扩散椭圆系统，证明了一致Hölder估计，并确认极限轮廓部分隔离，带来几何图案形成与最优正则性等新现象。
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.
In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. As a result, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on $C^{1,α}$-regularity of solutions.
In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<α<1$, there exists $δ>0$ such that the solution is $C^α$ at $x_0\in \partial Ω$ provided that $Ω$ is $δ$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < α< 1$, if $\partial Ω$ is $C^1$ and $u=g$ on $\partial Ω$ with $g\in C^α(x_0)$, then $u\in C^α(x_0)$. A similar result for t…
We extend the Caffarelli-Świech-Winter $C^{1,α}$ regularity estimates to $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form with superlinear growth in the gradient and unbounded coefficients. As an application, in addition to the usual $W^{2,p}$ results, we prove the existence of positive eigenvalues for proper operators with nonnegative unbounded weight, in particular for Pucci's operators with unbounded coefficients.
This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where $1<p<N,0\leqμ<\left((N-p)/p\right)^{p}$ and $Q\in L^{\infty}(\R^{N})$. Optimal asymptotic estimates on the gradient of solutions are obtained both at the origin and at the infinity.
This paper is devoted to establishing the pointwise upper and lower bounds estimates of the gradient of the solutions to a class of general elliptic systems with Hölder continuous coefficients in a narrow region where the upper and lower boundaries is $C^{1,γ},$ $0<γ<1$, weaker than the previous $C^{2,γ}$ assumption. These estimates play a key role in the damage analysis of composite materials. From our results, the damage may initiate from the narrowest place.
We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given.
Higher Hölder regularity for degenerate fully nonlinear elliptic equations with Hamiltonian terms
针对带变指数双相退化律和哈密顿项的完全非线性椭圆方程，在哈密顿系数和源项消失的点处获得梯度Hölder正则性改进，并在局部极值处建立关于消失率的Sharp Schauder型估计。
In this paper, we prove global gradient estimates for solutions to linear elliptic and parabolic equations. For a sufficiently smooth bounded convex domain $Ω\subset \mathbb{R}^N$, we show that a solution $ϕ\in W_0^{1,\infty}(Ω)$ to an appropriate elliptic equation $\mathcal{L} ϕ= F$, with $F \in L^{\infty}(Ω;\mathbb{R})$, satisfies $|\nabla ϕ|_{\infty} \leq C |F|_{\infty}$, with a positive constant $C = \exp(C(\mathcal{L})\text{diam}(Ω))$. We also obtain similiar estimates in the parabolic setting. The proof of these exponential bounds relies on global gradient estimates inspired by a series …
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