Tonia Ricciardi
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.
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Higher order H{ö}lder approximation by solutions of second order elliptic equations
本文研究了由二阶椭圆方程解实现的高阶Hölder逼近问题，证明了在特定正则性条件下函数的全局逼近存在性，并探讨了其导数性质。
Uniform bound for the volume of the solutions of elliptic equations
本文研究了具有Hölder权重或边界奇性及Dirichlet条件的变分问题，证明了解在解析域上体积的有界性。
Improved Regularity for Nonlocal Hamilton--Jacobi Equations with Superlinear Hamiltonians
研究一类由x依赖积分微分算子和强制超线性哈密顿量驱动的非局部Hamilton-Jacobi方程的粘性解正则性，在一般结构假设下建立Hölder正则性，并在弱椭圆性条件下证明全局Lipschitz连续性，通过反例说明假设的尖锐性。
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.
We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable H{ö}lder coefficients when the underlying domain is bounded and has a C 1+$κ$-boundary for some $κ$ > 0. We also prove a number of other results such as gradient estimates for heat kernels and Green functions G of elliptic operators with possibly complex-valued coefficients. We establish H{ö}lder continuity of $\nabla$ x $\nabla$ y G up to the boundary. These results are used to prove L p-estimates for commutators of Dirichlet-to-Neumann operators on the boundary of C 1+$κ$-domains. Such estima…
Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is uniformly elliptic, H$\ddot o$lder continuous and satisfies the natural structure conditions of fractional order, but without the concavity assumption of $F$. These assumptions are weaker and the result is stronger than that of Caffarelli and Wang[1], Chen[2].
A Classical Elliptic Regularity Approach to Almost Harmonic Maps and Related Systems
提出基于Campanato型离散迭代和Caccioppoli估计的统一抽象框架，证明在稳定于缩放且满足离散振荡衰减公理的容许对中，映射自动局部Hölder连续，并达到最优指数，避免了现有理论中的对偶、不等式和移动标架等技术。
$C^{1,α}$ regularity for degenerate fully nonlinear elliptic equations with oblique boundary conditions on $C^1$ domains
本文在C^1域上研究带斜边界条件的退化完全非线性椭圆方程，通过先建立几乎C^1平坦域中的一致边界Hölder估计，再利用扰动与紧性方法证明了粘性解直到边界的尖锐C^{1,α}正则性。
Liouville theorems and optimal regularity in elliptic equations
该研究建立了含可测系数椭圆偏微分方程解的最优正则性与无穷远刘维尔性质的关联，通过单调性公式、填洞技巧等方法分别处理二维与高维情形，探讨了增长性与正则性指数的相互作用。
Multi-wave inverse problems are indirect imaging methods using the interaction of two different imaging modalities. One brings spatial accuracy, and the other contrast sensitivity. The inversion method typically involve two steps. The first step is devoted to accessing internal datum of quantities related to the unknown parameters being observed. The second step involves recovering the parameters themselves from the internal data. To perform that inversion, a typical requirement is that the Jacobian of fields involved does not vanish. A number of authors have considered this problem in the pas…
We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.
We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{ö}lder estimates and the Harnack inequality for uniformly elliptic or parabolic equations with rough coefficients in divergence form. The theory of hypoellipticity of H{ö}rmander shows, under general "bracket" conditions, the regularity of solutions to partial differential equations combining first and second order derivative operators when ellipticity fails in some directions. We discuss recent exten…
We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{ö}lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic…
In this paper, we prove some isoperimetric inequalities and give a sharp bound for the positive solution of sublinear elliptic equations.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
The sharp growth rate in nonuniformly elliptic Schauder theory
该研究探讨非一致椭圆Schauder理论，得出在椭圆率增长的最优假设下，Schauder估计对非一致椭圆问题成立的结论。
Boundary regularity and Hopf lemma for nondegenerate stable operators
本文针对满足外部C^{1,dini}性质的有界开集，证明了非退化稳定积分微分算子方程解的尖锐边界Hölder正则性与Hopf型边界引理，并展示了估计在s→1-时的稳健性，从而恢复二阶方程经典结果。
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.
Generalized Morrey-Campanato estimates for elliptic equations with coefficients of integrable oscillation
针对低正则性系数和源项的散度型椭圆方程，引入广义Morrey和Campanato空间，建立弱解梯度的正则性估计，并恢复经典Hölder、Lebesgue估计及分数阶Sobolev正则性结果。
We consider a second-order selfadjoint elliptic operator with an anisotropic diffusion matrix having a jump across a smooth hypersurface. We prove the existence of a weight-function such that a Carleman estimate holds true. We moreover prove that the conditions imposed on the weight function are necessary.
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;…
Sharp gradient estimates for a class of singular/degenerate fully nonlinear elliptic equations with oblique boundary conditions and Hamiltonian terms
研究一类具斜边界条件和哈密顿项的奇异/退化完全非线性椭圆方程，在一般条件下，于统一框架内建立直至边界的精确 \(C^{1,\alpha}\) 正则性。
We show, using symmetrization techniques, that it is possible to prove a comparison principle (we are mainly focused on $L^1$ comparison) between solutions to an elliptic partial differential equation on a smooth bounded set $Ω$ with a rather general boundary condition, and solutions to a suitable related problem defined on a ball having the same volume as $Ω$. This includes for instance mixed problems where Dirichlet boundary conditions are prescribed on part of the boundary, while Robin boundary conditions are prescribed on its complement.
We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequ…
We prove a self-improving property for reverse H{ö}lder inequalities with non-local right hand side. We attempt to cover all the most important situations that one encounters when studying elliptic and parabolic partial differential equations as well as certain fractional equations. We also consider non-local extensions of A$\infty$ weights. We write our results in spaces of homogeneous type.
In this paper, we prove sharp estimates and existence results for anisotropic nonlinear elliptic problems with lower order terms depending on the gradient. Our prototype is: $ \left\{ \begin{array}{ll} -\mathcal Q_{p}u =[H(Du)]^{q}+f(x) &\text{in }Ω,\\ u=0&\text{on }\partialΩ. \end{array} \right. $ Here $Ω$ is a bounded open set of $\mathbb R^{N}$, $N\ge 2$, $0<p-1<q\le p<N$, and $\mathcal Q_{p}$ is the anisotropic operator $ \mathcal Q_{p} u ={\rm div}\left( [H(Du)]^{p-1}H_ξ(Du) \right)$, where $H$ is a suitable norm of $\mathbb R^{N}$. Moreover, $f$ belongs to an appropriate Marcinkiewicz sp…
Meyers exponent rules the first-order approach to second-order elliptic boundary value problems
本文研究散度形式二阶椭圆边值问题的一阶方法，通过算子值 Fourier 乘子证明了 Dirac 型算子预解式的全局 Lp 界与弱解梯度高阶可积性的最优指数一致。
In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $λ_{F}(p,Ω)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal P$-function method, it is possible to get several sharp estimates for $λ_{F}(p,Ω)$ in terms of several geometric quantities associated to the domain. The $\mathcal P$-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
We consider the equation $-Δu= |x|^α|u|^{p-1}u$ for any $α\geq 0$, either in $\mathbb R^2$ or in the unit ball $B$ of $\mathbb R^2$ centered at the origin with Dirichlet or Neumann boundary conditions. We give a sharp description of the asymptotic behavior as $p\rightarrow +\infty$ of all the radial solutions to these problems. In particular, we show that there is no uniform a priori bound (in $p$) for nodal solutions under Neumann or Dirichlet boundary conditions. This contrasts with the recently shown fact that positive solutions have uniform a priori bounds for $α=0$ and Dirichlet boundary …
Sharp Logarithmic Ultra-analyticity for Fractional and Nonlocal Elliptic Equations
本文通过傅里叶分析框架研究分数阶和非局部纯势方程，证明了解的对数超解析估计，并展示了指数和常数的尖锐性，同时建立了加权超解析尺度的传递原理。
We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-Δ)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under some general assumptions. Furthermore, in dimension 2, we can obtain essentially sharp lower bounds for some forth order elliptic operators, the sharpness is shown by constructing a Meshkov type example.
Sharp gradient bounds for uniformly elliptic PDE, eigenvalue asymptotics, and the Landis conjecture
本文针对一致椭圆方程的三个经典问题，提出尖锐梯度界并改进特征值渐近与Landis猜想，统一方法结合Harnack不等式、对偶与重标度，以反例证明最优性。
In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of \cite{BO}, the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above $p(x)$, see \eqref{energy_introduction} and \eqref{byun_ok_estimate}. Here, we bridge this gap to obtain the end point case of the estimates obtained in \cite{BO}, see \eqref{our_estimate}. In order to do …
In this survey we formulate our results on different forms of maximum principles for linear elliptic equations and systems. We start with necessary and sufficient conditions for validity of the classical maximum modulus principle for solutions of second order strongly elliptic systems. This principle holds under rather heavy restrictions on the coefficients of the systems, for instance, it fails for the Stokes and Lamé systems. Next, we turn to sharp constants in more general maximum principles due to S. Agmon and C. Miranda. We consider higher order elliptic equations, Stokes and Lamé systems…
In this article we establish sharp $C^{1,α}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying coefficients as a special case. The sharp exponent $α$ is asymptotically optimal and is determined by the Hölder regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.
Sharp Morrey regularity for an even order elliptic system
本文在系数满足极小正则性假设下，为 Rivière 型偶数阶椭圆系统建立了精确的 Morrey 正则性理论，进一步推广了现有的 L^p-正则性理论及二阶与四阶椭圆系统结果。
We establish sharp $W^{2,p}$ regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator $F$. By means of geometric tangential methods, we show that if the {\it recession} of the operator $F$ -- formally given by $F^*(M):=\infty^{-1} F(\infty M)$ -- is convex, then any viscosity solution to the original equation $F(D^2u) = f(x)$ is locally of class $W^{2,p}$, provided $f\in L^p$, $p>d$, with appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they a…
Sharp criteria for nonlocal elliptic inequalities on manifolds
本文研究了完备非紧黎曼流形上非局部椭圆不等式正解的存在性，在分数阶Laplacian的Green函数满足拟度量性质时获得了充分必要判据，并在Li-Yau高斯型热核估计下给出了显式条件。
In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators under strong absorption conditions. We establish improved geometric regularity along the free boundary, for a sharp value depending only on structural parameters. Non degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.
Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity
本文研究非线性项系数在有界开集内外取相反符号的半线性椭圆方程，利用Lyapunov-Schmidt约化方法证明了次临界情形下集中正解与节点解的存在性及正解的多重性，并在临界情形下证明了非平凡拓扑区域中爆破解的存在性。
Weakly elliptic damping gives sharp decay
本文证明弱椭圆阻尼在非函数演算情形下为抽象阻尼波半群提供尖锐能量衰减，并应用于线性化水波和Kelvin--Voigt阻尼。
We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.
The purpose of this note is to prove sharp Strichartz estimates with derivative losses for the non elliptic Schrodinger equation posed on the 2 dimensional torus.
We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.
Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-Δu=\frac{u^{2^\star(s)-1}}{|x|^s}-μu^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $μ> 0$. When $μ=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $μ>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that $\liminf_{x\to…
We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $Δu\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give conditions on $f$ and $g$ which guarantee the availability or the absence of positive solutions of such equations in $\R^N$. Our results considerably improve the existing ones and are sharp or close to sharp in the model cases. In particular, we completely characterize the solvability of such equations when $f$ and $g$ have power growth at infinity. We also derive a s…
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}.
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 consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
Sharp decay estimates for subsolutions of fully nonlinear equations in thin domains
该研究将薄区域完全非线性方程粘性下解的衰减估计问题转化为一维微分不等式求解，得到体现区域几何与方程椭圆性相互作用的尖锐估计，可应用于1<p≤2时p-次调和函数的衰减估计
Sharp Hessian estimates for fully nonlinear elliptic equations under relaxed convexity assumptions, oblique boundary conditions and applications
本文利用几何切向方法与 Recession 算子，在弱于凸性的结构假设及斜边界条件下建立完全非线性椭圆方程粘性解的全局 Hessian/BMO 估计，并推广至障碍型问题。
In this paper we prove the validity of Gibbons' conjecture for a coupled competing Gross-Pitaevskii system. We also provide sharp a priori bounds, regularity results and additional Liouville-type theorems.
Sharp estimates for fundamental frequencies of elliptic operators generated by asymmetric seminorms in low dimensions
本文建立了一维精确非对称 Poincare 不等式并计算最优常数与极值函数，基于此一维理论全面研究了正齐次各向异性设定下平面基本频率及相关谱优化问题。
Sublinear elliptic equations with a sharp change of sign in the nonlinearity
本文研究了具有非线性符号突变的亚线性椭圆方程，通过变分方法探讨了解的唯一性、多重性和支撑特性，并与两相 Serrin 类问题建立联系。
Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions
本文通过谱分辨接触几何机制，解决了平面及三维全非线性椭圆上解的尖锐Hessian可积性猜想，给出了最优指数并识别了剩余差距的谱配置。
We study radial symmetry of large solutions of the semi-linear elliptic problem Δu + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.
We use "generalized" version of total variation, coarea formulas, isoperimetric inequalities to obtain sharp estimates for solutions (and for their gradients) to anisotropic elliptic equations with a lower order term, comparing them with the solutions to the convex symmetrized ones.
Sharp and improved regularity estimates for weighted quasilinear elliptic equations of $p-$Laplacian type and applications
本文针对Hardy-Hénon型加权p-Laplacian拟线性椭圆方程，利用几何切向方法、振荡机制与紧性缩放技术，建立了仅依赖通用参数的尖锐改进正则性估计，并推广了$C^{p^{\prime}}$猜想相关结果，同时应用于Liouville型结论。
This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + ε}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results prese…
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