Giuseppe Floridia, Hiroshi Takase
In this paper we prove an observability inequality for a degenerate transport equation. First we introduce a local in time Carleman estimate for the degenerate equation, then we apply it to obtain a global in time observability inequality by using also an energy estimate.
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We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.
Observability Inequalities and Measurable Sets
本文给出了$Ω\times(0,T)$上热方程的两个能观性不等式.在第一种情况下，观察来自$Ω\times(0,T)$中的正测度子集，而在第二种情况下，观察来自$\partialΩ\times(0,T)$中的正表面测度子集。证明了当$Ω$是有界Lipschitz局部星形区域时的Lebeau-Robbiano谱不等式.给出了上述能观性不等式的一些应用。
In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.
Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials
该研究将自由 Schrödinger 方程的能观性与唯一延拓不等式推广到半轴上的反平方势情形，并利用解表示、Hankel 变换不确定性原理及紧支集插值不等式完成证明。
Shape-Design Approximation for a Class of Degenerate Hyperbolic Equations with a Degenerate Boundary Point and Its Application to Observability
本文研究了退化双曲方程的形状设计近似方法，通过移除退化点的小邻域，得到非退化问题并证明其收敛性，最终推导出观测不等式。
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their origina…
Shape Design for a Class of Degenerate Parabolic Equations with Boundary Point Degeneracy and Its Application to Boundary Observability
本文研究退化抛物方程的边界点退化问题，通过形状设计方法分析边界观测问题，提出高维抛物方程的形状设计方法，建立退化方程的观测不等式。
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{ö}dinger equation to the magnetic Schr{ö}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{ö}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on loga…
Shape Design for Degenerate Parabolic Equations with Degenerate Boundaries and Its Application to Boundary Observability
本文研究了退化抛物方程在退化边界条件下的well-posedness，提出了一种形状设计问题，通过一系列均匀抛物方程近似退化抛物方程，并推导了与退化抛物方程相关的边界观测不等式。
Observability Inequalities and the Logvinenko--Sereda Theorem for the Dunkl Transform
本文针对Dunkl热方程建立了可观测性、Hölder型插值及谱不等式的等价性，并给出了Bernstein不等式与谱常数为$e^{C(1+N)}$的Logvinenko--Sereda定理。
Observability inequalities on lattice points are established for non-negative solutions of the heat equation with potentials in the whole space. As applications, some controllability results of heat equations are derived by the above-mentioned observability inequalities.
We develop a new abstract derivation of the observability inequalities at two points in time for Schrödinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schrödinger equation.
Carleman estimates and observability inequalities for parabolic equations with interior degeneracy
我们考虑一个抛物型问题的退化的空间域的内部，我们专注于可控性的结果，通过Carleman估计相关的伴随问题。本文的新颖之处在于，退化也是在内部的控制区域，使以前的结果可以适应这种情况。
Observability inequality, the interpolation inequality and the spectral inequality for the degenerate parabolic equation in R
研究ℝ上退化抛物方程的能观性、Hölder型插值与谱不等式关系，证明在尺度L上具γ-厚度的可测集满足能观性不等式。
We consider the transport equation $\ppp_t u(x,t) + H(t)\cdot \nabla u(x,t) = 0$ in $\OOO\times(0,T),$ where $T>0$ and $\OOO\subset \R^d $ is a bounded domain with smooth boundary $\ppp\OOO$. First, we prove a Carleman estimate for solutions of finite energy with piecewise continuous weight functions. Then, under a further condition which guarantees that the orbits of $H$ intersect $\ppp\OOO$, we prove an energy estimate which in turn yields an observability inequality. Our results are motivated by applications to inverse problems.
In this paper, we study quantitative spatial analytic bounds and unique continuation inequalities of solutions for fractional heat equations with an analytic lower order term on the whole space. At first, we show that the solution has a uniform positive analytic radius for all time, and the solution enjoys a log-type ultra-analytic bound if the coefficient is ultra-analytic. Second, we prove a Hölder type interpolation inequality on a thick set, with an explicit dependence on the analytic radius of coefficient. Finally, by the telescoping series method, we establish an observability inequality…
In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat…
Boundary Sidewise Observability of the Wave Equation
针对有界域上带非齐次边界Dirichlet数据或边界子集支撑源的波动方程，研究远离源支撑的边界测量反演源的问题。证明可观测性不等式需损失无穷多导数，并在源满足拟微分条件及微局部几何条件下建立Sobolev范数可观测性不等式，方法基于微局部缺陷测度。
Observability inequalities for heat equations with potentials
本文研究时变 Lipschitz 位势热方程的能观性不等式，利用位势的空间与时间正则性改进 Carleman 常数，并导出零能控性的定量正则控制及一维最优常数。
Application of uncertainty principles for decaying densities to the observability of the Schrödinger equation
本文利用适应于衰减密度的不确定性原理，研究了Schrödinger方程在特定可测集上的可观测性不等式。
Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators
设${X_j},{Y_j}(j = 1, \cdot \cdot \cdot,n)$是满足Hörmander条件的向量场，${Δ_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$是满足Hörmander条件的向量场.本文建立了退化椭圆型偏微分算子${Δ_L}$和$Δ_L^2$的Dirichlet特征值不等式.这些不等式将Laplacian的Dirichlet特征值的Yang不等式推广到这里，并且不等式的形式比Yang不等式更一般。为了得到它们，我们给出了Chebyshev不等式的一个推广。
We obtain Caccioppoli--type estimates for nontrivial and nonnegative solutions to the anticoercive partial differential inequalities of elliptic type involving degenerated $p$--Laplacian: $-Δ_{p,a} u:= -\mathrm{div}(a(x)|\na u|^{p-2}\na u)\ge b(x)Φ(u)$, where $u$ is defined in a domain $Ω$. Using Caccioppoli--type estimates, we obtain several variants of Hardy--type inequalities in weighted Sobolev spaces.
A Class of Degenerate Hyperbolic Equations with Neumann Boundary Conditions and Its Application to Observability
本文研究了一类退化双曲方程的混合观测不等式，通过边界和内部观测结合，证明了在特定参数范围内解的可控性，方法结合了加权框架、正则性提升和乘子论证。
Free vibrations of axially moving strings: Energy estimates and boundary observability
本文研究了轴向运动弦的小振动，通过级数公式和显式系数求解波动方程，定义了时间守恒的能量表达式，并建立了具有显式常数的边界能观性不等式。
Hardy Type Inequalities Related to Degenerate Elliptic Differential Operators
证明了与拟线性二阶退化椭圆型微分算子L_p（u）：=- \nabla_L^*（\abs{\nabla_L u}^{p-2} \nabla_L u）有关的几个Hardy型不等式.如果k是一个正的权值，使得-L_p k>= 0，则Hardy型不等式c \int_Ω \frac{\abs u^p}{ϕ^p} \abs{\nabla_L ϕ}^p d \le \int_Ω \abs{\nabla_L u}^p d成立。我们发现一个明确的价值观的常数，其中，在大多数情况下，结果最佳。作为特例，我们得到了Carnot群上次椭圆算子的Hardy不等式.
Hidden Boundary Trace Regularity and an Observability Estimate with Interior Remainder for Boundary-Degenerate Hyperbolic Equations
本文研究了二维退化双曲方程的隐边界迹正则性，建立了加权Sobolev空间中的well-posedness，并证明了法向导数的L^2迹估计。通过适应各向异性退化的Carleman权重，推导出具有低阶内部余项的长时间观测估计，同时识别了临界阈值处的框架级障碍。
Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
本文研究d维环面上拉普拉斯特征函数关于任意Borel测度的迹不等式与可观测性不等式，基于测度可积性、正则性及支撑集维数给出充分与必要条件，将Zygmund和Bourgain–Rudnick经典结果推广到高维，并应用于量子极限约束与Schrödinger方程控制理论。
We establish new Carleman estimates for the wave equation, which we then apply to derive novel observability inequalities for a general class of linear wave equations. The main features of these inequalities are that (a) they apply to a fully general class of time-dependent domains, with timelike moving boundaries, (b) they apply to linear wave equations in any spatial dimension and with general time-dependent lower-order coefficients, and (c) they allow for significantly smaller time-dependent regions of observations than allowed from existing Carleman estimate methods. As a standard applicat…
Spectral inequalities for Schrödinger equations with various potentials
本文研究了全空间中薛定谔算子在幂增长或无穷远处连续消失等不同势下的谱不等式，证明其定量依赖于传感器集合密度、势增长率及谱，并将其应用于获取热方程的定量能观性不等式。
Boundary controllability for degenerate/singular hyperbolic equations in nondivergence form with drift
该研究针对带漂移项的非散度型退化/奇异波动方程的零能控问题，借助能量方法与边界能观性，给出了局域于非退化边界点的控制下边界能控性的相关条件。
Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly…
Degenerate free discontinuity problems and spectral inequalities in quantitative form
本文引入一个几何项带负号的新退化泛函，通过研究其与跳跃集相互作用的障碍问题，解决了带有 Robin 边界条件的 Laplace 算子半线性问题特征值相关的 Sobolev-Poincaré 不等式最佳常数定量估计问题。
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Ω= (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $ω=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $ω= (a,b) \times ω_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $ω$ and the boundary of $Ω$: $$T_{\min} \geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our…
We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is known that the classical Sobolev inequality is sufficient and almost necessary for the Dirichlet problem to be solvable with a quantitative bound on the solution [11]. When the degeneracy is of infinite type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7]. In this paper we investigate this connection further and reduce the gap betwe…
Fractal Remez inequality on the sphere and observability of the heat equation
该研究在单位球面上建立分形雷梅兹不等式，基于此得到球面上热方程的精确可观测性不等式并改进已有结果，还证明了带超二次势的热方程的低维可观测性不等式。
We present several characterizations, via some weak observability inequalities, on the complete stabilizability for a control system $[A,B]$, i.e., $y'(t)=Ay(t)+Bu(t)$, $t\geq 0$, where $A$ generates a $C_0$-semigroup on a Hilbert space $X$ and $B$ is a linear and bounded operator from another Hilbert space $U$ to $X$. We then extend the aforementioned characterizations in two directions: first, the control operator $B$ is unbounded; second, the control system is time-periodic. We also give some sufficient conditions, from the perspective of the spectral projections, to ensure the weak observa…
We prove an equivalence between weighted Poincare inequalities and the existence of weak solutions to a Neumann problem related to a degenerate p- Laplacian. The Poincare inequalities are formulated in the context of degenerate Sobolev spaces defined in terms of a quadratic form, and the associated matrix is the source of the degeneracy in the p-Laplacian.
We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic…
This article is devoted to the study of the behavior of generalized entire solutions for a wide class of quasilinear degenerate inequalities modeled on the following prototype with p-Laplacian in the main part \begin{equation*} {\underset{m}{\overset{i=1}{\sum}}} X_i^*(|\mathbf{X}u|^{p-2} X_i u)\geq |u|^{q-2}u, \ \ x\in {\mathbb{R}}^{n},\ q>1,\ p>1, \end{equation*} where ${\mathbb{R}}^{n}$ is a Carnot-Carathéodory metric space, generated by the system of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes the adjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$…
We extend the results of [5], where we proved an equivalence between weighted Poincaré inequalities and the existence of weak solutions to a family of Neumann problems related to a degenerate $p$-Laplacian. Here we prove a similar equivalence between Poincaré inequalities in variable exponent spaces and solutions to a degenerate $p(x)$-Laplacian, a non-linear elliptic equation with nonstandard growth conditions.
We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.
We prove Poincaré and Sobolev inequalities in matrix A${}_p$ weighted spaces. We then use these Poincaré inequalities to prove existence and regularity results for degenerate systems of elliptic equations whose degeneracy is governed by a matrix A${}_p$ weight. Such results parallel earlier results by Fabes, Kenig, and Serapioni for a single degenerate equation governed by a scalar A${}_p$ weight. In addition, we prove Cacciopoli and reverse Hölder inequalities for weak solutions of the degenerate systems. As a means to prove the Poincaré inequalities we prove that the Riesz potential and frac…
Hausdorff type Time-Trace Observability for Airy Equations on the Line and Point Observability on the Torus
研究实直线和环面上Airy方程的可观性，通过引入新思想，证明了从Hausdorff厚集的时间迹可观性不等式、周期集观测的可观性不等式，给出环面上有限点可观性充要条件及相关推论，得到不同条件下的可观性结果。
Observability of wave and plate equations with rough coefficients and interfaces: a multiplier approach
本文采用乘子方法，通过正则化过程推导主部含粗糙系数及界面的波动方程与板方程的可观测性，避免分析奇异集合附近解的正则性，建立了含界面情形的新可观测性结果。
The paper deals with the controllability of a degenerate beam equation. In particular, we assume that the left end of the beam is fixed, while a suitable control $f$ acts on the right end of it. As a first step we prove the existence of a solution for the homogeneous problem, then we prove some estimates on its energy. Thanks to them we prove an observability inequality and, using the notion of solution by transposition, we prove that the initial problem is null controllable.
In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introduci…
Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations
将Bôcher关于Laplace方程的一个定理推广到一类共形不变的完全非线性退化椭圆型方程.我们还证明了局部Lipschitz黏性解的Harnack不等式和连续径向对称黏性解的分类。
Moving localized observations and Ces{à}ro asymptotic observability for conservative PDEs
本文提出一种确定性长时机制，通过移动局部观测获得Cesàro渐近可观察性不等式，利用凸化替代全观测，并通过切换实现移动观测器，最终适用于多种PDEs的内部观测和边界模型。
Nonlocal equations with degenerate weights
该研究引入带退化权的分数阶加权Sobolev空间，推导其嵌入与Poincaré不等式，证明对应非局部积分微分方程解的正则性，将经典结果推广到非线性非局部场景。
Rotating waves in nonlinear media and critical degenerate Sobolev inequalities
研究了单位球中带Dirichlet边界条件的非线性波动方程的旋转波解，通过分析相关临界退化Sobolev不等式及极值函数，推导基态解存在的充要条件，并探讨其非径向性及方法对更一般区域的推广。
Observability inequality, log-type Hausdorff content and heat equations
本文研究有界域与全空间热方程的能观测性不等式，以对数型 Hausdorff 容度刻画观测集，并借助谱不等式、Remez 不等式及容度工具证明其成立条件与一维最优尺度。
Nonlocal Harnack inequalities
本文提出并证明了一个非局部的、可能退化的积分-微分算子极小元的一般Harnack不等式，其模型是分数p-Laplacian算子.
Stochastic variational inequalities and regularity for degenerate stochastic partial differential equations
研究了退化拟线性SPDE解的正则性和特征。我们的结果是双重的：首先，我们证明了某些退化，拟线性SPDE驱动Lipschitz连续噪声的解决方案的规律性结果。特别是，这提供了一个表征的解决方案，这样的SPDE（广义）强解。其次，对于一维随机中曲率流与正常的噪音，我们适应随机变分不等式的概念，以前得到的解决方案的表征仅在限制意义上。这解决了[Es-Sarhir，von Renesse; SIAM，2012]中的一个未解决的问题，并锐化了[Es-Sarhir，von Renesse，Stannat; NoDEA，2012]中获得的正则性。
Global regularity estimates for $p(x)$-Laplacian variational inequalities with singular or degenerate matrix-valued weights
本文研究了带有奇异或退化矩阵权的 $p(x)$-拉普拉斯变分不等式的全局正则性，通过构造性水平集方法，得出了加权 Calderón-Zygmund 和 Orlicz 型估计，并展示了结果的最优缩放参数特性。
Normal trace inequalities and decay of solutions to the nonlinear Maxwell system with absorbing boundary
本文研究了具有吸收边界条件的非线性Maxwell系统，证明了小数据解的全局存在性和指数衰减性，并通过迹估计和正则性分析改进了线性自治情况下的结果。
Boundary Spectral Inequalities from Measurable Sets on $W^{2,\infty}$ manifolds and their applications
该文在$W^{2,\infty}$流形上建立Dirichlet与Neumann特征函数的定量边界谱不等式，进而证明热方程解的边界可观测性及由边界法向导数唯一确定初态，并给出最优阶对数稳定性估计。
Liouville theorems for nonlocal Lane--Emden inequalities with degenerate kernels
本文通过测试函数方法和二进制分解，证明了非局部算子下兰恩-埃门德不等式的勒维叶型定理，仅在特定条件下存在平凡解。
Harnack Inequalities for Degenerate Diffusions
我们研究了种群遗传学中一类退化扩散算子的各种概率和分析性质，即所谓的广义Kimura扩散算子。我们的主要结果是具有奇异低阶系数的退化抛物型方程弱解的随机表示，以及Kimura抛物型方程非负解的尺度不变Harnack不等式的证明.我们建立的解的随机表示是Feynman-Kac公式中关于扩散矩阵的退化性、漂移系数的有界性和弱解的先验正则性等假设的经典结果的相当大的推广。
Quantitative observability for the Schrödinger equation with an anharmonic oscillator
本文针对具有非谐振子势的薛定谔方程，构造了短时间区间上的定量可观测性不等式，给出观测常数的显式表达，并通过谱不等式、唯一紧性论证及 Szegö 极限定理刻画了可观测集合的几何特征。
We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $Ω\subset\mathbb{R}^n$, let $ρ$ be a quasi-metric on $Ω$, and let $Q$ be an $n\times n$ semi-definite matrix function defined on $Ω$. For an open set $Θ\SubsetΩ$, we give sufficient conditions to show that if the local weak Sobolev inequality % \[ \Big(\fint_B |f|^{pσ}dx\Big)^\frac{1}{pσ} \leq C\Big[ r(B)\fint_B |\sqrt{Q}\nabla f|^pdx + \fint_B |f|^pdx\Big]^\frac{1}{p} \] holds for some $σ>1$, all balls $B\subset Θ$, and function…
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