Agnid Banerjee, Federico Buseghin, Nicola Garofalo
In this paper we establish the optimal interior regularity and the $C^{1,γ}$ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.
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Cagnetti, Gomes, Mitake and Tran (2013) introduced a new idea to study the large time behavior for degenerate viscous Hamilton--Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton--Jacobi equations. We establish the convergence result under rather general assumptions.
We prove the natural weighted Calderón and Zygmund estimates for solutions to elliptic and parabolic obstacle problems in nondivergence form with discontinuous coefficients and irregular obstacles. We also obtain Morrey regularity results for the Hessian of the solutions and Hölder continuity of the gradient of the solutions.
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.
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。
We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $|y|^a$ for $a \in (-1,1)$. Such problem arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($a=0$).
Riesz potential estimates for non-linear elliptic obstacle problems
研究涉及测度数据的非线性椭圆障碍问题，核心方法是根据里斯势建立解的逐点梯度估计，主要贡献是得到相关估计结果。
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.
Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of the model is exploited to construct these solutions as minimizers of a suitably regularized energy, which allows us to weaken considerably the assumptions on the model used in a previous article.
We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on the gradient: \[ \min \left\{f-|Du|^γF(D^2u),u-ϕ\right\} = 0 \quad\textrm{ in }\quad Ω. \] We obtain existence of solutions and prove sharp regularity estimates along the free boundary points, namely $\partial\{u>ϕ\} \cap Ω$. In particular, for the homogeneous case ($f\equiv0$) we get that solutions are $C^{1,1}$ at free boundary points, in the sense that they detach from the obstacle in a quadratic fashion, thus beating the optimal regularity allowed for such degenerate operators. We also pr…
The aim of this paper is twofold: to prove, for L^1-data, the existence and uniqueness of an entropy solution to the obstacle problem for nonlinear elliptic equations with variable growth, and to show some convergence and stability properties of the corresponding coincidence set. The latter follow from extending the Lewy--Stampacchia inequalities to the general framework of L^1.
We deal with a global Calderón-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p \le 2-\frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.
Singular elliptic measure data problems with irregular obstacles
本文研究带测度数据和不规则障碍的椭圆问题，在强奇异范围$1<p\le2-1/n$内建立新的比较估计，证明梯度和零阶位势的内蕴估计。
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 1963, Littman, Stampacchia, and Weinberger proved a mean value theorem for elliptic operators in divergence form with bounded measurable coefficients. In the Fermi lectures in 1998, Caffarelli stated a much simpler mean value theorem for the same situation, but did not include the details of the proof. We show all of the nontrivial details needed to prove the formula stated by Caffarelli, and in the course of showing these details we establish some of the basic facts about the obstacle problem for general elliptic divergence form operators, in particular, we show a basic quadratic nondegene…
Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schrödinger term: $-Δ_p u + \mathbb{V}|u|^{p-2}u$ with bound constraints $ψ_1 \le u \le ψ_2$ in non-smooth domains. This problem has its own interest in mathematics, engineering, physics and other branches of science. Our approach makes a novel connection between the study of Calderón-Zygmund theory for nonlinear Schrödinger type equations and variational inequalities for double obstacle problems.
Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth
研究具有广义Orlicz增长的椭圆障碍问题，通过建立解在Musielak-Orlicz空间的存在性，利用非线性Wolff势导出解的梯度估计，得到解的$C^{1,\alpha}$正则性结果。
In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.
We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper "The regularity of free boundaries in higher dimensions."
Gradient potential estimates for elliptic double obstacle problems with Orlicz growth
该研究针对含测度数据的 Orlicz 增长椭圆型双障碍问题，利用分数次极大算子与非线性 Wolff 位势建立解及其梯度的逐点和振荡估计，证明解具有 $C^{1,α}$ 正则性。
We study the regularity of the solution of the double obstacle problem form for fully non linear parabolic and elliptic operators. We show that when the obstacles are sufficiently regular the solution is $C^{1,α}$ in the interior for both the parabolic and the elliptic cases.
We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.
We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators; more specifically, we consider viscosity solutions of \[ |D u|^γF(x, D^2u) = f(x)χ_{\{u>ϕ\}} \textrm{ in } B_1 \] with $γ>0$, $ϕ\in C^{1, α}(B_1)$ for some $α\in(0,1]$ and $f\in L^\infty(B_1)$ constrained to satisfy \[ u\geq ϕ\textrm{ in } B_1 \] and prove that they are $C^{1,β}(B_{1/2})$ (and in particular along free boundary points) where $β=\min\left\{α, \frac{1}{γ+1}\right\}$. Moreover, we achieve such a feature by using a recently developed geometric appro…
The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity many solutions for the problems.
In this paper we study the existence of solutions of thedegererate elliptic system.
We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where $\nabla u \in K$, where $K\subset \mathbb{R}^N$ is a Borel set containing the origin.
The aim of this paper is to study the obstacle problem with an elliptic operator having degenerate coercivity. We prove the existence of an entropy solution to the obstacle problem under the assumption of $L^{1}-$summability on the data. Meanwhile, we prove that every entropy solution belongs to some Sobolev space $W^{1,q}(Ω)$.
Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an interpretation of global irregularity in terms of the associated heat semigroup is given.
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。
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。
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。
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
An elliptic equation div(F(Du)) = f whose ellipticity strongly degenerates for small values of Du (say, F = 0 on B(0,1)) is considered. The aim is to prove regularity for F(Du). The paper proves a continuity result in dimension two and presents some applications.
Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation
该研究聚焦于一类带临界增长与对数扰动的非线性退化椭圆算子的Dirichlet问题，探究其弱解的存在性。
In this paper we study the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called \emph{regular points} in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator $(\partial_t - Δ_x)^s$ for $s \in (0,1)$. Our regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. Our approach is based on first establishing the boundedness of the time-derivat…
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.
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。
In this article we study an elliptic problem with degenerate coercivity. We will show that the presence of some lower order terms has a regularizing effect on the solutions.
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.
We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.
In this paper we develop a detailed study on maximum and comparison principles for a degenerate elliptic system. Explicit lower bounds for principal eigenvalues of this system in terms of the measure of $Ω$ are also proved.
We derive a comparison principle for a degenerate elliptic partial differential equation without boundary conditions which arises naturally in optimal learning strategies. Our argument is direct and exploits the degeneracy of the differential operator to construct (logarithmically) diverging barriers.
Free boundary regularity in obstacle problems with a degenerate forcing term
本文研究退化强迫项下的障碍问题，通过epiperimetric不等式证明自由边界正则性，揭示解的弱方向单调性及原点正则性条件。
Gradient regularity for a class of elliptic obstacle problems
该研究针对带障碍约束的非自治积分泛函局部极小值问题，在偏映射系数满足合适Sobolev正则性的条件下，证明了局部极小值的更高可微性与Lipschitz连续性，推进了椭圆障碍问题的梯度正则性研究。
We show an existence of a weak solution of a degenerate and/or singular semilinear elliptic boundary value (nonhomogeneous) problem lying between a given weak subsolution and a given weak supersolution. It has been applied for an existence result of large solution to a similar problem.
This paper is concerned with semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\mathbb R^n$. This class of equations includes Bellman equations containing operators of Lévy-Itô type. Hölder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.
We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by Richardson's method.
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the…
The weak maximum principle for solutions of degenerate elliptic equations with lower order terms
本文研究退化椭圆方程含低阶项解的弱最大原理，基于作者与他人最近证明的存在性结果提出新证明。
We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Ampère equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.
We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.
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 $C^{1, γ}$.
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.
This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.
Failure of Calderón-Zygmund estimates for degenerate elliptic PDEs with $A_p$-weights when $p > 2$
本文通过凸积分技术展示，在A_p权重(p>2)下，退化线性椭圆PDE的加权Calderón-Zygmund估计失效。
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。
In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form $\nabla^2 ψ+ L(x,\nabla ψ)$, including the conformal hessian operator.
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.
This paper studies the Sobolev regularity of weak solution of degenerate elliptic equations in divergence form $\text{div}[\mathbf{A}(X) \nabla u] = \text{div}[\mathbf{F}(X)]$, where $X = (x,y) \in \mathbb{R}^{n} \times \mathbb{R}$ . The coefficient matrix $\mathbf{A}(X)$ is a symmetric, measurable $(n+1) \times (n+1)$ matrix, and it could be degenerate or singular in the one dimensional $y$-variable as a weight function in the Muckenhoupt class $A_2$ of weights. Our results give weighted Sobolev regularity estimates of Calderón-Zygmund type for weak solutions of this class of singular, degene…
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.
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