D. Del Santo, M. Prizzi
We find minimal regularity conditions on the coefficients of a parabolic operator, ensuring that no nontrivial solution tends to zero faster than any exponential.
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We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.
In this work, we shall study the nonlinear inverse problems of recovering the Robin coefficients in elliptic and parabolic systems of second order, and establish their local Lipschitz stabilities. Some local Lipschitz stability was derived for an elliptic inverse Robin problem. We shall first restructure the arguments in \cite{chou04} for the local Lipschitz stability so that the stability follows from three basic conditions for the elliptic inverse Robin problem. The new arguments are then generalized to help establish a novel local Lipschitz stability for parabolic inverse Robin problems.
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 …
As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.
We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.
We develop an improved version of the parabolic Lipschitz truncation, which allows qualitative control of the distributional time derivative and the preservation of zero boundary values. As a consequence, we establish a new caloric approximation lemma. We show functions. The distance is measured in terms of spatial gradients as well as almost uniformly in time. Both results are extended to the setting of Orlicz growth.
We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $Ω\times \R^+$, where $Ω$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.
In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [Math. Scand. ${\bf 8}$ (1960), 277--286] and some more recent results of Del Santo and Prizzi [J. Math. Pures Appl. ${\bf 84}$ (2005), 471--491; Ann. Mat. Pura Appl., to appear].
In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.
We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.
Parabolic Lipschitz truncation for multi-phase problems: the degenerate case
本文研究抛物型多相问题中的Lipschitz截断方法，基于Whitney分解和覆盖引理，并利用文献[24]中的比较方案区分各相，处理退化情形。
Reconstruction of the initial condition in parabolic equations with Log-Lipschitz coefficients
针对系数关于时间Log-Lipschitz连续、关于空间Lipschitz连续的抛物方程，结合条件稳定性结果与适定变分问题，实现了从终值粗糙测量数据出发的初值重构。
For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain $Ω$ we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hoelder continuous for sufficiently $L^p$-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions are well-posed on $\mathrm{C}(\barΩ)$.
Lipschitz truncation method for parabolic double-phase systems and applications
本文提出一种基于 Whitney 型覆盖和单位分解的 Lipschitz 截断方法，用于证明 $p$-Laplace 型抛物双相 Dirichlet 问题的能量估计与唯一性，并展示非齐次问题的存在性。
Invariant measures for stochastic parabolic-hyperbolic equations in the space of almost periodic functions: Lipschitz flux case
该研究在通量与粘性函数Lipschitz连续及非退化条件下，分析任意维随机退化抛物-双曲方程概周期解的适定性和长时间行为，证明Besicovitch概周期函数可分子空间中不变测度的存在唯一性。
Lipschitz regularity for the parabolic $(s,p)-$obstacle problem
研究退化范围2 < p < 2/(1 - s)内抛物型分数阶p - 拉普拉斯方程的障碍问题，证明粘性解在空间上局部Lipschitz连续、时间上Hölder连续，p > 1/(1 - s)时时间正则性提升为Lipschitz连续。
We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the $p$-Laplace operator for all $p \in (1,\infty)$, but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space $C(\barΩ)$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in…
We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefo…
Lipschitz regularity for parabolic fractional $p$-Laplace equations
研究抛物分数 $p$-拉普拉斯方程局部弱解的空间局部利普希茨连续性，证明在 $sp > p-1$ 时的均匀时间正则性，适用于对称、平移不变核，包括可能不连续的核。
We consider functions $u\in L^\infty(L^2)\cap L^p(W^{1,p})$ with $1<p<\infty$ on a time space domain. Solutions to non-linear evolutionary PDE's typically belong to these spaces. Many applications require a Lipschitz approximation $u_λ$ of $u$ which coincides with $u$ on a large set. For problems arising in fluid mechanics one needs to work with solenoidal (divergence-free) functions. Thus, we construct a Lipschitz approximation, which is also solenoidal. As an application we revise the existence proof for non-stationary generalized Newtonian fluids in [DRW10]. Since ${\rm div} u_λ=0$, we are …
A Characterization of Solvability of the Parabolic $L^p$ Dirichlet Problem on Lipschitz Graph Domains Via Carleson Measure Estimates of Bounded Solutions
该研究通过有界解的Carleson测度估计，刻画了Lipschitz图区域上抛物型$L^p$Dirichlet问题的可解性，改进了需附加抛物一致可求长性等额外假设的现有结果。
In a Lipschitz cylinder, this paper is devoted to establish an almost sharp error estimate $O(\varepsilon\log_2(1/\varepsilon))$ in $L^2$-norm for parabolic systems of elasticity with initial-Dirichlet conditions, arising in the homogenization theory. To achieve the goal, with the parabolic distance function being a weight, we first developed some new weighted-type inequalities for the smoothing operator at scale $\varepsilon$ in terms of t-anisotropic Sobolev spaces, and then all the problems may be reduced to three kinds of estimate for the homogenized system, in which a weighted-type Caccio…
Big Pieces of Regular Parabolic (bi-)Lipschitz Images is Equivalent to Parabolic Uniform Rectifiability
该研究定义正则抛物（双）利普希茨像概念，将David-Semmes理论推广到抛物语境，证明抛物Ahlfors-David正则集的抛物一致可求长性等价于其具有n维时空的抛物（双）利普希茨像的大块。
The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies a non-degeneracy condition. This is the optimal regularity for this problem.
For the parabolic obstacle-problem-like equation $$Δu - \partial_t u = λ_+ χ_{\{u>0\}} - λ_- χ_{\{u<0\}} ,$$ where $λ_+$ and $λ_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary $\partial\{u>0\} \cup\partial\{u<0\}$ is in a neighborhood of each ``branch point'' the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper \cite{imrn} to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a c…
We shall study in this paper the Lipschitz type stabilities and convergence rates of Tikhonov regularization for the recovery of the radiativities in elliptic and parabolic systems with Dirichlet boundary conditions. The Lipschitz type stability estimates are derived. Due to the difficulty of the verification of the existing source conditions or nonlinearity conditions for the considered inverse radiativity problems in high dimensional spaces, some new variational source conditions are proposed. The conditions are rigorously verified in general dimensional spaces under the Lipschitz type stabi…
Continuity of attractors of parabolic equations with nonlinear boundary conditions and rapidly varying boundaries. The case of a Lipschitz deformation
本文研究了带非线性边界条件的抛物型方程在边界随参数$ε$趋于零时快速振荡变化下吸引子的连续性，通过引入变化域函数与Banach空间算子族的收敛性，证明了解收敛至含振荡因子的极限方程。
We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to infinity. We characterize velocity fields, so that positive solutions decay or lift-off at spatial infinity as time tends to infinity. This addresses the question of stability of the zero solution for decaying perturbations.
Lipschitz stability and reconstruction in inverse problems for semi-discrete parabolic operators
针对半离散抛物方程的反问题，本文利用受网格尺寸约束的新颖半离散Carleman估计，建立了恢复势函数的稳定性估计并提出了收敛的重构算法。
Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.
The fractional Lipschitz caloric capacity of Cantor sets
本文研究了康托集的分数Lipschitz热容量，通过分析空间梯度特性，得出了与解析和里斯容量类似的结果。
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as $η\to \infty$.
We derive an estimate for the minimal period of autonomous strongly damped hyperbolic problems. Our result corresponds to the works by Yorke, Busenberg et al. for ordinary differential equations as well as Robinson and Vidal-Lopez for parabolic problems. A general approach is developed for treating both hyperbolic and parabolic problems. An example of application to a class of beam equations is provided.
Boundary estimates for parabolic non-divergence equations in $C^1$ domains
本文研究了抛物非散度方程在$C^1$领域边界的正则性，通过显式连续模扩展了Hopf-Oleinik引理和Dini边界结果，并统一了Lipschitz领域与$C^{1-\varepsilon}$正则性的证明。
Lipschitz regularity for parabolic double phase equations with gradient nonlinearity
本文利用 Ishii-Lions 方法证明了一类具梯度非线性的抛物型双相方程粘性解的空间局部 Lipschitz 正则性与时间 Hölder 估计，并在适当条件下建立了粘性解与弱解的等价性。
We consider parabolic problems with non-Lipschitz nonlinearity in the different scales of Banach spaces and prove local-in-time existence theorem. New class of parabolic equations that have analytic solutions is obtained.
We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor $\mathcal A$ lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequ…
We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on all coefficients. Under the assumptions we show the Hölder continuity of the solution in the spatial component. Furthermore, additionally assuming the Dini continuity of the coefficient of the second order derivative, we have the better continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary…
We prove in this article that functions satisfying a dynamic programming principle have a local interior Lipschitz type regularity. This DPP is partly motivated by the connection to the normalized parabolic $p$-Laplace operator.
In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the …
In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in $(0,T) \times \R^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their $L^{\infty}$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (J. Funct. Anal. 2006) to get this type of…
In this paper we present an improvement of [Math. Ann. 345 (2009), 213--243], where the authors proved a result concerning continuous dependence for backward parabolic operators whose coefficients are Log-Lipschitz in $t$ and $C^2$ in $x$. The $C^2$ regularity with respect to $x$ had to be assumed for technical reasons. Here we remove this assumption, replacing it with Lipschitz-continuity. The main tools in the proof are Littlewood-Paley theory and Bony's paraproduct as well as a result of Coifman and Meyer [Astérisque 57, 1978, Th. 35].
In this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic …
Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph
本文证明抛物型Lipschitz图域上热量测度的$A_\infty$性质等价于边界抛物型一致可修正性，从而结合Lewis-Murray工作解决了热方程$L^p$ Dirichlet问题可解性的长期开放问题。
Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density
本文证明了二次抛物型微分方程在稀疏传感器集下的可观测性与零可控性，依据生成元奇异空间的不同情况调整传感器集条件，并利用耗散估计与不确定性关系进行证明。
In a cylindrical space-time domain with a convex, spatial base, we establish a local Lipschitz estimate for weak solutions to parabolic systems with Uhlenbeck structure up to the lateral boundary, provided homogeneous Dirichlet data are assumed on that part of the lateral boundary.
Parabolic boundary Harnack inequalities with right-hand side
本文用爆破技术建立允许非零右端项的抛物型边界 Harnack 不等式，并据此推出抛物障碍与 Signorini 问题中平坦 Lipschitz 自由边界的 $C^{1,\alpha}$ 正则性。
We study second order parabolic equations on Lipschitz domains subject to inhomogeneous Neumann (or, more generally, Robin) boundary conditions. We prove existence and uniqueness of weak solutions and their continuity up to the boundary of the parabolic cylinder. Under natural assumptions on the coefficients and the inhomogeneity we can also prove convergence to an equilibrium or asymptotic almost periodicity.
The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch du…
We study second order equations and systems on non-Lipschitz domains including mixed boundary conditions. The key result is interpolation for suitable function spaces. From this, elliptic and parabolic regularity results are deduced by means of Sneiberg's isomorphism theorem.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
In this paper we develop a potential theory for strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_{Y}-\partial_t, \] in unbounded domains of the form \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,y_m,t)\}, \] where $ψ$ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. Concerning $A=A(X,Y,t)$ we assume that $A$ is bounded, measurable, symme…
We prove regularity results such as interior Lipschitz regularity and boundary continuity for the Cauchy-Dirichlet problem associated to a class of parabolic equations inspired by the evolutionary $p$-Laplacian, but extending it at a wide scale. We employ a regularization technique of viscosity-type that we find interesting in itself.
In this note we consider the initial boundary value problem for the heat equation on cylinders based on Lipschitz domains with Besov data. We obtain a regularity exponent for the solution that improves the rate of convergence of nonlinear approximation methods.
We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. This result extends in some ways a classical result of Lions and Malgrange [12] and a recent result of the authors [10].
We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,t)\}. \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $ψ$ and $Ω$ we assume that $Ω$ is what we call an (unbounded) Lipschitz domain: $ψ$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euc…
Extrapolation of solvability of the parabolic $L^p$ Neumann problem on bounded Lipschitz cylinders
本文研究了抛物型 $L^p$ 霍尔德问题在有界利普希茨圆柱上的可解性外推问题，提出新的方法以解决该问题。
The purpose of this short note is to show how it is possible to combine existing results in the literature to get the unique continuation from sets of positive measure for time dependent parabolic equations with Lipschitz principal part and bounded lower order terms, result which was known in the case of analytic coefficients in [3].
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