Daniela De Silva, Ovidiu Savin
We investigate the Boundary Harnack Principle in Hölder domains of exponent $α>0$ by the analytical method developed in our previous work "A short proof of Boundary Harnack Principle".
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Hölder regularity and Harnack inequality for the logarithmic Laplacian
本文针对对数拉普拉斯算子，建立Schauder型估计，证明非齐次项为κ阶Hölder连续时内部解的正则性，并给出非负解的Harnack不等式。
Hölder Continuity and Harnack estimate for non-homogeneous parabolic equations
本文针对非散度型非齐次抛物型方程，建立了时间向前的内在Harnack不等式，由此导出解的Hölder连续性，并给出量化强最小值原理的全局Harnack型估计，为广义Harnack不等式提供了新证明。
Hölder and Harnack estimates for integro-differential operators with kernels of measure
本文针对具有测度核的椭圆型非局部方程，利用De Giorgi型方法建立弱解的Hölder与Harnack估计，推广了Dyda和Kassmann（2020）的已有结果。
This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hörmander vector fields. Adapting the iteration scheme of Jürgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show Hölder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parab…
Weak parabolic Harnack inequality and Hölder regularity for non-local Dirichlet forms
本文研究一般正则狄利克雷型，通过局部热核估计与增长引理建立弱抛物型哈纳克不等式的等价条件，并在取消上跳跃光滑性假设的情形下证明热函数与调和函数的赫尔德连续性，推广了已有理论。
In this paper we obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard $p(x)-$type growth. A model equation is the inhomogeneous $p(x)-$laplacian. Namely, \[ Δ_{p(x)}u:=\mbox{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)=f(x)\quad\mbox{in}\quadΩ\] for which we prove Harnack inequality when $f\in L^{q_0}(Ω)$ if $\max\{1,\frac N{p_{min}}\}<q_0\le \infty$. The constant in Harnack inequality depends on $u$ only through $\||u|^{p(x)}\|_{L^1(Ω)}^{p_{max}-p_{min}}$. Dependence of the constant on $u$ is known to be necessary in the case of variable $p(x)…
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form \begin{eqnarray} \text{div}\big(A(x,u,\nabla u)\big) = B(x,u,\nabla u)\text{ for }x\inΩ\nonumber \end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness …
James Serrin's fundamental contributions to the theory of quasilinear elliptic equations are well-known and widely appreciated. He also made less well-known contributions to the theory of quasilinear parabolic equations which we dicuss in this note. Jurgen Moser gave greatly simplified proofs of the De Giorgi-Nash regularity results for linear divergence structure elliptic and parabolic differential equations using an original iterative technique. Serrin extended Moser's techniques and applied them to the study of divergence structure quasilinear elliptic equations, and in collaboration with A…
Interior Harnack inequality and Hölder estimates for linearized Monge-Ampère equations in divergence form with drift
本文研究了带漂移项的散度形式线性化 Monge-Ampère 方程解的内部估计，证明了二维及高维（需附加 Hessian 可积性假设）情形下的内部 Harnack 不等式与 Hölder 估计，推广了 Le 的相关结果。
On Holder Continuity and Equivalent Formulation of Intrinsic Harnack Estimates for an Anisotropic Parabolic Degenerate Prototype Equation
本文利用内在Harnack估计证明各向异性退化抛物方程有界局部弱解的Hölder连续性，并给出这些估计在内在几何下的等价表述。
In this paper, we obtain the weak Harnack inequality and Hölder estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local Hölder estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off.
In 1957, De Giorgi [3] proved the Hölder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3].
Hölder estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2.
We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\timesΩ$$ where $L$ is a divergence form elliptic operator and $Ω\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $Ω$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem.
Nonlocal Harnack inequalities for nonlocal double phase equations I ; with positive bounded modulating coefficient with no Hölder condition
本文通过De Giorgi-Nash-Moser理论，证明了非局部双重相方程弱解的非局部Harnack不等式，并研究了其弱超解的局部有界性和弱Harnack不等式。
We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^γ \nabla\cdot (|x|^{-β} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb R}^N$. The weights $|x|^γ$ and $|x|^{-β}$, with $γ< N$ and $γ-2 < β\leq γ(N-2)/N$ can be both degenerate and singular and need not belong to the class $\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalit…
In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetr…
We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $Ω\cap B_1$, $u_1=u_2=0$ in $B_1\setminusΩ$, and $u_1,u_2\geq0$ in $\mathbb R^n$, then $u_1$ and $u_2$ are comparable in $B_{1/2}$. The result applies to arbitrary open sets $Ω$. When $Ω$ is Lipschitz, we show that the quotient $u_1/u_2$ is Hölder continuous up to the boundary in $B_{1/2}$.
On the Boundary Harnack Principle for operators with different lower order terms
本文在Lipschitz区域中，对具有相同主部但不同低阶项的两个线性一致椭圆型方程的解，建立了经典的边界Harnack原理。
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
We investigate the parabolic Boundary Harnack Principle for both divergence and non-divergence type operators by the analytical methods we developed in the elliptic context. Besides the classical case, we deal with less regular space-time domains, including slit domains.
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 study some geometric and potential theoretic properties of nodal domains of solutions to certain uniformly elliptic equations. In particular, we establish corkscrew conditions, Carleson type estimates and boundary Harnack inequalities on a class of nodal domains.
We prove a boundary Harnack principle in Lipschitz domains with small constant for fully nonlinear and $p$-Laplace type equations with a right hand side, as well as for the Laplace equation on nontangentially accessible domains under extra conditions. The approach is completely new and gives a systematic approach for proving similar results for a variety of equations and geometries.
We introduce a new boundary Harnack principle in Lipschitz domains for equations with right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.
We provide a general framework for the realization of powers or functions of suitable operators on Dirichlet spaces. The first contribution is to unify the available results dealing with specific geometries; a second one is to provide new results on rather general metric measured spaces that were not considered before and fall naturally in the theory of Dirichlet spaces. The main tool is using the approach based on subordination and semi-groups by Stinga and Torrea. Assuming more on the Dirichlet space, we derive several applications to PDEs such as Harnack and Boundary Harnack principles.
Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains
该研究针对Reifenberg平坦区域，证明了分数阶p-调和函数商的边界哈纳克不等式与赫尔德连续性，通过系列估计和变形方法将相关结果从能量类扩展到尾类。
The boundary Harnack principle on optimal domains
本文为满足特定几何条件的域证明了边界Harnack不等式，并探讨了其在形状优化和自由边界问题中的应用。
This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.
The $C^{1,α}$ boundary Harnack principle in a slit domain and its application to the Signorini problem
本文研究了具有变系数的Signorini问题自由边界$C^{2,α}$正则性，利用裂隙域中的$C^{1,α}$边界Harnack不等式，通过非标准退化椭圆方程的研究获得$C^{1,α}$Schauder估计。
We prove forward and backward parabolic boundary Harnack principles for nonnegative solutions of the heat equation in the complements of thin parabolic Lipschitz sets given as subgraphs $E=\{(x,t): x_{n-1}\leq f(x'',t),x_n=0\}\subset \mathbb{R}^{n-1}\times\mathbb{R} $ for parabolically Lipschitz functions $f$ on $\mathbb{R}^{n-2}\times\mathbb{R}$. We are motivated by applications to parabolic free boundary problems with thin (i.e co-dimension two) free boundaries. In particular, at the end of the paper we show how to prove the spatial $C^{1,α}$ regularity of the free boundary in the parabolic …
In this paper we consider a class of hypoelliptic second-order partial differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for $\mathcal{L}$. The involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy of the second order part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily …
We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes of the corresponding diffusions. In particular, we do not assume volume doubling property for the symmetric measure.
Differential Harnack inequalities and maximum principles of Morel-Oswald type for elliptic PDE in divergence form
本文为散度型一致椭圆方程正解建立Morel-Oswald型Hopf-Oleinik引理推广及新的微分Harnack估计，并给出最优全局结果。
We establish boundary estimates for non-negative solutions to the p-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental…
Higher order boundary Harnack principles in Dini type domains
本文研究Dini型区域中散度型椭圆方程的高阶边界Harnack原理，通过分析退化辅助方程，在双重及三重Dini条件下建立解及其比值的C^k估计。
A multi-point maximum principle to prove global Harnack inequalities for Schrödinger operators
本文提出多点最大值原理，直接证明黎曼流形上薛定谔方程正解的全局Harnack不等式，避免高阶导数，并给出二次势情形下的更优常数及Ornstein-Uhlenbeck算子的应用。
We consider the singular boundary-value problem Δu = f(u) in D; u|_dD= phi, where 1. D is a bounded C^2-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally Hölder continuous function such that f(u) -> 1 as u -> 0 at the rate u^{-α}, for some αin (0,1), 3. and phi is a positive continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.
We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{p-2}u\big)-Δ_p u=0,\quad p>1. \] The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain…
A multi-point maximum principle to prove the global Harnack inequality for the doubly nonlinear heat equation
本研究提出一种多点最大值原理的新方法，用于证明双重非线性热方程正粘性解满足的全局逐点Harnack不等式，该方法也可推导多孔介质方程等对应解的Harnack不等式。
Boundary Harnack principle on uniform domains
本文在满足标度不变椭圆Harnack不等式的空间上，证明了均匀领域上的标度不变边界Harnack原理，无需假设空间为测地空间，并通过最近的结果保证了格林函数的存在性。
We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain.
On higher order boundary Harnack and analyticity of free boundaries
本文建立非标准退化椭圆方程的 C^1,alpha Schauder 估计，借此给出高阶边界 Harnack 不等式的新证明，并应用于证明经典障碍问题中自由边界的解析性。
We study the Krylov-Safonov type Hölder estimate and Harnack inequality for viscosity solutions satisfying a uniformly parabolic equation only where the gradient with respect to the space-time variables is large.
Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
本文针对Kolmogorov方程，在可依赖高阶变量的渐近圆柱Lipschitz区域建立边界Harnack不等式，突破早期理论的对称性限制，通过多类估计与论证排除异常构型，得到正解的相关性质。
On the sharp Hölder exponent in the De Giorgi--Nash--Moser theory
本文构造了三维及以上空间中具有可测系数的均匀椭圆方程的解，其Hölder指数α=exp(-c_n K)，证明了除二维情形外，Bombieri-Giusti Harnack不等式得到的Hölder指数对椭圆率常数K的依赖是最优的。
We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.
We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In particular, Harnack's Principle and Minimum Principle are provided for $\mathcal{A}$-superharmonic functions and boundary Harnack inequality is proven for $\mathcal{A}$-harmonic functions.
The Boundary Harnack Principle and the 3G Principle in Fractal-Type Spaces
本文证明了一大类 Dirichlet 空间中有界内一致域上 Green 函数的广义 3G 原理，结果适用于高维分形及无明确 Hausdorff 维数的广义分形型空间，并探讨了其在 Schrödinger 算子上的应用。
Higher order boundary Harnack principle via degenerate equations
本文研究奇异/退化加权椭圆方程的高阶 Schauder 估计，并将其用于证明二阶椭圆方程解之比在节点集正则及奇异部分附近的高阶边界 Harnack 型正则性。
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=Δ_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}<p<2$. It is well-known that non-negative solutions behave for large times as $\mathcal{B}$, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of $\mathcal{B}$. This can be considered the nonlinear…
In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result for an inverse source problem…
This article is about the convex solution $u$ of the Monge--Ampère equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)| \leq C \operatorname{dist}(x,\partialΩ)^α$ is equivalent to (uniform) Hölder continuity with the same constant and exponent. Convex $α$-Hölder continuous functions are $W^{1,p}$ for $p < 1/(1{-}α)$. We prove Hölder continuity with the exponent $α=2/n$ for $n \geq 3$ and any $α\in (0,1)$ for $n=2$, provided that…
In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Székelyhidi], we prove that the set of Hölder $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible Hölder $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.
Hölder regularity of solutions of degenerate parabolic equations of general dimension
本文研究任意维数、系数有界可测的非散度型退化抛物方程，建立 ABP 估计、Harnack 不等式、Hölder 正则性与 Schauder 估计。
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the bou…
Shifted Composition I: Harnack and Reverse Transport Inequalities
提出移位复合规则这一信息论新原理，通过辅助移位界定随机过程分布间的散度，借此证明扩散过程的反向传输不等式并蕴含维度无关的 Harnack 不等式，同时连接了连续与离散时间方法。
Regularity results for quasiminima of a class of double phase problems
本文在度量测度空间框架下，采用基于De Giorgi方法、相分析与内蕴几何估计的变分方法，证明了p-Laplace型椭圆双相问题局部拟极小元的有界性、Hölder连续性与Harnack不等式等正则性结果。
Using a variational approach we study interior regularity for quasiminimizers of a $(p,q)$-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle, and Liouville's Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity a…
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