Alberto Favaron
We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.
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This paper studies Hölder regularity property of bounded weak solutions to a class of strongly coupled degenerate parabolic systems.
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.
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。
We establish Holder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are obtained. These problems have a numeros applications which occur in fluid mechanics, environmental engineering and atmospheric dispersion of pollutants.
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。
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized solutions.
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。
Sobolev-type regularity results are proved for solutions to a class of second order elliptic equations with a singular or degenerate weight, under non-homogeneous Neumann conditions. As an application a Pohozaev-type identity for weak solutions is derived.
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
This paper deals with a class of nonlinear anisotropic parabolic equations with degenerate coercivity. Using the anisotropic Gagliardo-Nirenberg-type inequality, we prove some existence and regularity results for the solutions under the framework of anisotropic Sobolev spaces, which generalize the previous results of [10,18,23].
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.
Gradient regularity for fully nonlinear equations with degenerate coefficients
针对在超曲面上退化的完全非线性方程，推导了粘性解的$C^{1,α}$正则性估计。
In this paper we prove regularity results for a class of nonlinear degenerate elliptic equations of the form $\displaystyle -\operatorname{div}(A(|\nabla u|)\nabla u)+B\left( |\nabla u|\right) =f(u)$; in particular, we investigate the second order regularity of the solutions. As a consequence of these results, we obtain symmetry and monotonicity properties of positive solutions for this class of degenerate problems in convex symmetric domains via a suitable adaption of the celebrated moving plane method of Alexandrov-Serrin.
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.
A short note on maximal regularity for the heat kernel in Besov spaces and a degenerate 3D Keller-Segel system
本文证明在临界Lorentz空间中具有小初值的退化Keller-Segel方程的整体存在性。
The regularity of the solutions to the Muskat equation: the degenerate regularity near the turnover points
本文研究了Muskat问题解在翻转点附近的解析性，证明在满足额外条件时，足够光滑的解在翻转点处退化的区域内具有解析性，补充了已有研究结果。
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;…
Regularity of Solutions to a Class of Degenerate Cross Diffusion Systems of $m$ Equations
本文针对 $N$ 维区域上含 $m$ 个方程的退化交叉扩散系统，研究弱解正则性及经典解的全局存在性。
In this note, we prove $\mathcal{C}^{1,γ}$ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of \cite{BDL1} concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
We study the Cauchy-Dirichlet problem associated to a phase transition modeled upon the degenerate two-phase Stefan problem. We prove that weak solutions are continuous up to the parabolic boundary and quantify the continuity by deriving a modulus. As a byproduct, these a priori regularity results are used to prove the existence of a so-called physical solution.
In this paper we prove Holder regularity of the derivative of radial solutions to fully nonlinear equations when the operator is hessian, homogenous of degree 1 in the Hessian, homogenous of some degree $α>-1$ in the gradient and which is elliptic when the gradient is not null.
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
We establish the local Lipschitz continuity and the higher differentiability of vector-valued local minimizers of a class of energy integrals of the Calculus of Variations. The main novelty is that we deal with possibly degenerate energy densities with respect to the x-variable.
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 a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p-Poisson equation, for the porous medium equation and for the doubly nonlinear equation.
We prove that every weak solution to a certain class of infinitely degenerate quasilinear equations is continuous. An essential feature of the operators we consider is that their Fefferman-Phong associated metric may be non doubling with respect to Lebesgue measure.
Partial Regularity for Harmonic Maps into Spheres at a Singular or Degenerate Free Boundary
本文证明了在域度量沿自由边界以$d^α$速率退化或奇异时，具有自由边界数据的弱平稳调和映照的部分正则性。
Gradient regularity for degenerate fully nonlinear free transmission problems with Hamiltonian terms
研究含哈密顿项的退化完全非线性自由传输问题，通过置于粘性不等式背景建立梯度局部赫尔德正则性，基于新估计与局部分析获逐点\(C^{1,\alpha}\)正则性。
Nonlocal degenerate Isaacs operators: Hölder regularity
本文研究非局部退化伊萨克算子的霍尔德正则性，证明在s接近1时解的霍尔德连续性，并应用得到利奥维勒定理。
In this paper we prove Holder regularity of the gradient for solutions of Dirichlet problem associate to degenerate elliptic equations, extending the recent result of Imbert and Silvestre. Indeed we obtain regularity up to the boundary and when the equation has lower order terms.The proof follows their scheme but requires new tools and new ideas. In particular we give some a priori Lipschitz and Hölder estimates in the presence of boundary condition on one part of the boundary.
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。
In this paper we establish some regularity results concerning the behavior of weak solutions and very weak solutions of the degenerate wave equation near the boundary. For the nondegenerate case, the correponding results were originally obtained by Fabre and Puel (J. of Diff. Eq. 106, 1993). This kind of results is closely related to the exact boundary controllability for the wave equation as the limit of internal controllability.
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some degenerate-singular Monge-Ampère type equations and prove the sharp boundary estimates for convex viscosity solutions of the problem. As a result, we obtain the optimal global Hölder regularity of the convex viscosity solutions.
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior $C^{1, α}$ regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.
Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and Hénon-type equations
本文研究退化或奇异全非线性死核系统与强吸收项的耦合问题，提出粘性解在自由边界处的正则性提升、非退化性、自由边界测度估计、Liouville型结果及解的发散行为，同时推导了退化权重和强吸收下Hénon型方程粘性解的精确正则性估计。
We prove local boundedness, Harnack's inequality and local regularity for weak solutions of quasilinear degenerate elliptic equations in divergence form with Rough coefficients. Degeneracy is encoded by a non-negative, symmetric, measurable matrix valued function Q(x) and two suitable non-negative weight functions. We setup an axiomatic approach in terms of suitable geometric conditions and local Sobolev-Poincaré inequalities. Data integrability is close to L1 and is exploited in terms of a suitable Stummel-Kato class that in some cases is necessary for local regularity.
This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform Hölder regularity in terms of $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence free drift.
We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.
In this paper, we establish $C^{1, α}$ regularity upto the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior $C^{1, α}$ regularity result established in [21] for equations with similar structural assumptions. The proof of our main result is achieved via compactness arguments combined with new boundary Hölder estimates for equations which are uniformly elliptic when the gradient is either small or large.
We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial Ω} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.
In the present paper, a class of fully non-linear elliptic equations are considered, which are degenerate as the gradient becomes small. Hölder estimates obtained by the first author (2011) are combined with new Lipschitz estimates obtained through the Ishii-Lions method in order to get $C^{1,α}$ estimates for solutions of these equations.
C^{1,α} regularity for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions
本文研究了一类具有斜边界条件的奇异/退化全非线性椭圆方程的粘性解的全局 $C^{1, α}$ 正则性，扩展了 BKO25 的结果至更广的方程类，包括奇异情形。
We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary, and assuming local boundedness, we prove continuity. On globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th o…
Optimal regularity for degenerate Kolmogorov equations with rough coefficients
针对满足抛物型Hörmander条件、系数时间可测空间Hölder连续的退化方程，通过广义强解概念证明基本解存在性、最优Hölder正则性及高斯估计，为Langevin型扩散相关倒向Kolmogorov方程研究提供关键支撑。
Uniform regularity for degenerate elliptic equations in perforated domains
针对穿孔区域中快速振荡系数的退化椭圆方程，建立了定量收敛速率，并获得了加权的Lipschitz和$W^{1,p}$一致估计。
In this manuscript we establish local Hölder regularity estimates for bounded solutions of a certain class of doubly degenerate evolution PDEs. By making use of intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In such a scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence for our findings and approach, we address a Liouville type resu…
We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in \cite{MonsaingonNovikovRoquejoffre}. We set up a finite difference scheme allowing to compute approximate solutions and capture the free boundaries, and we carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, we prove the Lipschitz regularity of the free boundaries. Our simulations indicate that this regularity is optimal, and the free boundaries s…
On the regularity and existence of weak solutions for a class of degenerate singular elliptic problem
针对权重在原点附近可消失或爆炸、且奇异性可在区域内变化的退化奇异椭圆问题，本文给出权重、奇异指数和源函数的充分条件，证明弱解的正则性与存在性。
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Ω$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Hölder regularity up to the boundary, that is, $u/d^s\in C^α(\overlineΩ)$ for some $α\in(0,1)$, $d$ being the distance from the boundary.
We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.
We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,α}$ regularity of viscosity solutions to such equations.
We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1<p<2$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points.
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.
Cauchy problem for singular-degenerate porous medium type equations: well-posedness and Sobolev regularity
本文研究扩散系数兼具多孔介质型退化与奇异性的拟线性反应扩散方程柯西问题，利用m-增生算子理论、动力学公式和Fourier分析技术证明了其适定性与Sobolev正则性。
Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations
本文研究一类退化或奇异完全非线性积分微分方程解的正则性，分别在退化和奇异情形下建立了边界正则性、Schauder型更高正则性及梯度Hölder连续性，结果在a(x)≡0时也具新颖性，并可推广至退化非局部归一化p-Laplacian方程。
Vecer derived a degenerate parabolic equation with a boundary condition characterizing the price of Asian options with generally sampled average. It is well understood that there exists a unique probabilistic solution to such a problem but it remained unclear whether the probabilistic solution is a classical solution. We prove that the probabilistic solutions to Vecer's PDE are regular.
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