Fausto Ferrari, Giulio Galise
We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations leaded by operator that are neither convex/concave nor uniformly elliptic.
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Optimal Boundary Regularity for Uniformly Degenerate Elliptic Equations
本文综述了有界区域中一致退化椭圆方程解的最优正则性，建立了解及其导数直至边界的Hölder连续性。
Global Schauder Regularity and Convergence for Uniformly Degenerate Parabolic Equations
该文研究一致退化抛物方程解的全局Hölder正则性，并在极限一致退化椭圆方程特征指数的附加假设下，分析时间趋于无穷时解的收敛性。
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。
We report on new techniques and results in the regularity theory of general non-uniformly elliptic variational integrals. By means of a new potential theoretic approach we reproduce, in the non-uniformly elliptic setting, the optimal criteria for Lipschitz continuity known in the uniformly elliptic one and provide a unified approach between non-uniformly and uniformly elliptic problems.
This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a nonlinear, quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of the Fabes-Stroock estimate. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of f…
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。
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
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.
Boundary regularity of uniformly rotating vortex patches and an unstable elliptic free boundary problem
本文研究了与二维不可压缩欧拉方程均匀旋转涡旋片解相关的变号自由边界问题，证明了涡旋片边界在奇异边界点附近局部形成90度角，若涡旋片为Lipschitz连续。
We prove $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients
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.
We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is $C^{2,α}$ on the compliment of a closed set of Hausdorff dimension at most $ε$ less than the dimension. The equation is assumed to be $C^1$, and the constant $ε> 0$ depends only on the dimension and the ellipticity constants. The argument combines the $W^{2,ε}$ estimates of Lin with a result of Savin on the $C^{2,α}$ regularity of viscosity solutions which are close to quadratic polynomials.
Interior $BMO$ regularity for elliptic equations in divergence form
本文研究了椭圆方程在散度形式下弱解的内部BMO正则性，并证明了系数假设的最优性。
Uniformly Degenerate Elliptic Equations with Varying Characteristic Exponents
本文研究有界区域中特征多项式具变特征指数的一致退化椭圆方程，核心关注解的正则性问题。
We give full boundary extensions to two fundamental estimates in the theory of elliptic PDE, the weak Harnack inequality and the quantitative strong maximum principle, for uniformly elliptic equations in non-divergence form.
We show, in a borderline case which was not covered before, the validity of nonlinear Calderón-Zygmund estimates for a class of non-uniformly elliptic problems driven by double phase energies.
In this paper, we obtain a uniform $W^{2,\varepsilon}$-estimate of solutions to the fully nonlinear uniformly elliptic equations on Riemannian manifolds with a lower bound of sectional curvature using the ABP method.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
Nonuniform ellipticity in variational problems and regularity
本文概述了非均匀椭圆问题正则性理论的近期进展，重点关注变分性质的问题。
We prove a $C^{1,α}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.
We prove a Meyers type regularity estimate for approximate solutions of second order elliptic equations obtained by Galerkin methods. The proofs rely on interpolation results for Sobolev spaces on graphs. Estimates for second order elliptic operators on rather general graphs are also obtained.
We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally $C^{1,γ}$-regular.
In this paper, we study the regularity of weak solutions to a class of nonlocal elliptic equations in Bessel potential spaces $H^{s,p}$. Our main results can be seen as an extension of the well-known $W^{1,p}$ regularity theory for local second-order elliptic equations in divergence form to the nonlocal setting.
We establish sharp global regularity results for solutions to nonhomogeneous, nonunifomrly elliptic systems with zero boundary conditions. In particular, we obtain everywhere Lipschitz continuity under borderline Lorentz assumptions on the forcing term, thus positively settling the optimality issue raised in \cite{bdms}.
Regularity for solutions of non-uniformly elliptic equations in non-divergence form
本文证明了非散度型非一致椭圆方程的 Aleksandrov--Bakelman--Pucci 估计，并通过建立局部有界性和弱 Harnack 不等式研究了解的局部行为，采用对椭圆性施加可积性假设而非指定特定结构的方法。
Fine regularity for elliptic systems with discontinuous ingredients
本文研究了具有不连续系数的二阶线性椭圆系统强解在Morrey、BMO和Hölder空间中的正则性问题。
Second order regularity for solutions to anisotropic degenerate elliptic equations
本文研究退化各向异性椭圆方程解的正则性，建立了二阶估计并给出应力场正则性结果，且结果在欧几里得情形下亦为新。
We analyze fine properties of solutions to quasilinear elliptic equations with double phase structure and characterize, in the terms of intrinsic Hausdorff measures, the removable sets for Hölder continuous solutions.
A priori estimates for the weak solutions the Dirichlet problem for the uniformly higher-order elliptic equations in a smooth bounded domain $Ω\subset \Rn$ in generalized weighted Sobolev-Morrey spaces are obtained.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We establish partial regularity result for vector-valued solutions to second order elliptic system in divergence form. The coefficients safisfies Dini condition respect to $(x,u)$ with growth order lager than 2. We prove $C^1$-regularity of the solutions outside of singular set.
We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman's equations is of order at least $h^{2/3}$, where $h$ is the mesh size. The equations are considered in smooth bounded domains.
On the second-order regularity of widely degenerate elliptic systems
研究广泛退化椭圆型系统二阶正则性，通过证明临界集附近正则性及退化区域测度性质得出相关结论。
In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal $L_p$-$L_q$ regularity for these problems and give examples that our results are applicable. The Lopatinskii--Shapiro and the asymptotic Lopatinskii--Shapiro conditions are important to get boundedness of solution operators.
Elliptic problems with additional unknown distributions in boundary conditions are investigated in Besov and Sobolev-Triebel-Lizorkin spaces of low regularity, specifically of an arbitrary negative order. We find that the problems induce Fredholm bounded operators on appropriate pairs of these spaces.
We prove a Brezis--Kato regularity type results for solutions of the higher order nonlinear elliptic equation \[ L u = g(x,u)\qquad\text{in }Ω\] with an elliptic operator $L$ of $2m$ order with variable coefficients and a Carathéodory function $g:Ω\times \mathbb{C}\to\mathbb{C}$, where $Ω\subset\mathbb{R}^N$ is an open set with $N > 2m$.
We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is $C^{1,α}$ on the boundary of the domain, for an optimal and explicit $α$ given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian;…
We start in this paper a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.
In this paper, we consider the regularity of solutions to elliptic equation with Grushin's operator. By using the Feynman-Kac formula, we first get the expression of heat kernel, and then by using the properties of heat kernel, the optimal regularity of solutions will be obtained. The novelty of this paper is that the Grushin's operator is a degenerate operator.
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.
We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect to the $x$ and continuity with respect to $u$ and prove Hölder continuity of the solutions outside of singular sets.
This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.
In this note we prove an end-point regularity result on the Lp integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BMO regularity of second derivatives of solutions to elliptic equations.
High-order Regularity Theory for High-contrast Elliptic Homogenization
本文研究高对比椭圆正则化中的高阶正则性问题，提出高对比Caccioppoli不等式并证明全局高阶正则性结果。
Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions to non linear PDE.
In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^α$, $C^{1,α}$ and $C^{2,α}$ regularity. As byproducts, we also prove the A-B-P maximum principle, Harnack inequality, uniqueness and solvability of the equations.
Well-Posedness of Second-Order Uniformly Elliptic PDEs with Neumann Conditions
本文将 Nardi (2015) 的结果推广到带 Neumann 边界条件的散度型二阶一致椭圆方程，证明了解的存在唯一性，并给出 Schauder 估计。
In this paper we study a class of anisotropic equations with a lower order term whose coefficients lay in Marcinkiewicz spaces. We prove some regularity results for local solutions requiring any control on the norm of the coefficients.
Local and global $C^{1,β}$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type
本文研究了均匀椭圆$p$-拉普拉斯和Orlicz-拉普拉斯类型quasilinear方程的局部和全局$C^{1,β}$正则性，证明了在内部和边界上的梯度Hölder连续性。
We obtain an error estimate between viscosity solutions and δ-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is Lipschitz-continuous in space with linear growth in the Hessian. We also establish a rate of convergence for monotone and consistent finite difference approximation schemes for such equations.
We consider Fokker--Planck--Kolmogorov equations with unbounded coefficients and obtain upper estimates of solutions. We also obtain new estimates involving Lyapunov functions.
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.
We prove existence and regularity results for weak solutions of non linear elliptic systems with non variational structure satisfying $(p,q)$-growth conditions. In particular we are able to prove higher differentiability results under a dimension-free gap between $p$ and $q$.
We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and generalized Orlicz growth.
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 $C^{1,\text{Log-Lip}}$.
In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.
We consider nonlinear fourth order elliptic equations of double divergence type. We show that for a certain class of equations where the nonlinearity is in the Hessian, solutions that are C^{2,alpha} enjoy interior estimates on all derivatives.
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.
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.
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