A parabolic flow toward solutions of the optimal transportation problem on domains with boundary
Jun Kitagawa
我们考虑一个抛物线版本的质量运输问题，并表明，它收敛到一个解决方案的原始质量运输问题在适当的条件下的费用函数，初始和目标域。
We consider a parabolic version of the mass transport problem, and show that it converges to a solution of the original mass transport problem under suitable conditions on the cost function, and initial and target domains.
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This paper introduces a family of transportation costs between non-negative measures. This family is used to obtain parabolic and reaction-diffusion equations with drift, subject to Dirichlet boundary condition, as the gradient flow of the entropy functional $\int_Ωρ\logρ+Vρ+1\hspace{1mm}dx$. In 2010, Alessio Figalli and Nicola Gigli introduced a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in their work. The costs in this paper allow the drift and the boundary co…
In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).
We develop an $\e$-regularity theory at the boundary for a general class of Monge-Ampère type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between Hölder densities supported on $C^2$ uniformly convex domains are $C^{1,α}$ up to the boundary, provided that the cost function is a sufficient small perturbation of the quadratic cost $-x\cdot y$.
We show a result of maximal regularity in spaces of Hölder continuous function, concerning linear parabolic systems, with dynamic or Wentzell boundary conditions, with an elliptic diffusion term on the boundary.
In this paper we consider an initial boundary value problem for a semilinear parabolic equation with nonlinear nonlocal boundary condition. We prove comparison principle, the existence theorem of a local solution and study the problem of uniqueness and nonuniqueness
Regularity of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
考虑一维抛物型自由边界值问题，在自由边界上具有非局部（积分-微分）条件。得到了自由边界的$C^m$正则性结果.特别地，给出了自由边界无穷可微的一个充要条件。
We obtain up to a flat boundary regularity results in parabolic Hölder spaces for viscosity solutions of fully nonlinear parabolic equations with oblique boundary conditions.
We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.
We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic Hölder spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.
In this paper we consider an initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal Neumann boundary condition. We prove comparison principle, the existence theorem of a local solution and study the problem of uniqueness and nonuniqueness.
Initial-Boundary Value Problems for Parabolic Equations
证明了p-Laplacian演化模型下抛物型方程非齐次初边值问题弱解的存在唯一性.
In this article we establish the exact growth of the solution to the singular quasilinear p-parabolic free boundary problem in non-divergence form near the free boundary from which follows its porosity.
In this paper we establish the exact growth of the solution of the singular quasilinear p-parabolic obstacle problem near the free boundary from which we deduce its porosity.
We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.
Boundary behavior of nonnegative solutions of fully nonlinear parabolic equations
研究了完全非线性抛物型Pucci极值算子粘性非负解的边界表现.我们建立了$C^{1，1}圆柱上的局部和整体比较定理，以及一个向后的Harnack不等式.
Parabolic Boundary Harnack Principles in Domains with Thin Lipschitz Complement
本文证明了热方程在薄抛物Lipschitz集的补集上非负解的前向和后向抛物边界Harnack原理，这些薄抛物Lipschitz集作为$\mathbb{R}^{n-2}\times\mathbb{R}$上抛物Lipschitz函数$f$的子图$E=\{(x,t): x_{n-1}\leq f(x'',t),x_n=0\}\subset \mathbb{R}^{n-1}\times\mathbb{R} $给出.我们的动机是应用到抛物自由边界问题薄（即余维2）自由边界。特别地，在本文的最后，我们展示了如何证明空间$C^{1,α}$正则性的自由边界的抛物Signorini问题。
$C^{2,α}$ regularity of free boundaries in optimal transportation
本文通过建立自由边界的几何估计并研究Monge–Ampère方程第二边值问题，证明最优运输中势函数与自由边界均具有$C^{2,\alpha}$正则性，解决了Caffarelli–McCann公开问题。
We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in $C^1$ and $C^{k,α}$ domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary. As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.
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.
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.
We consider initial boundary value problem for uniformly 2-parabolic differential operator of second order in cylinder domain in ${\mathbb R}^n $ with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo-Galerkin method we prove that problem has unique solution in special Bochner space.
In this paper we consider initial boundary value problem for nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition and nonnegative initial datum. We prove comparison principle, global existence and blow-up of solutions.
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.
Symmetrizers and Continuity of Stable Subspaces for Parabolic-Hyperbolic Boundary Value Problems
本文证明了抛物-双曲边值问题的稳定子空间对参数极限值的连续性。分析的基础上进行的建设在以前的文件Kreiss'型对称化。
In this paper, a new type of comparison theorem is presented for some initial-boundary value problems of second order nonlinear parabolic systems with nonlinear boundary conditions. This comparison theorem has an advantage over the classical ones, since this makes it possible to compare two solutions satisfying different types of boundary conditions. Some applications are given in the last section, where the existence of blow-up solutions is shown for some nonlinear parabolic equations and systems with nonlinear boundary conditions.
Large-scale boundary estimates of parabolic homogenization over rough boundaries
本文针对粗糙边界上具有快速振荡周期系数的二阶抛物系统或方程，通过两次近似的定量方法获得了大尺度边界估计。
We present weighted Sobolev spaces and prove a trace theorem for the spaces. As an application, we discuss non-zero boundary value problems for parabolic equations. The weighted parabolic Sobolev spaces we consider are designed, in particular, for the regularity theory of stochastic partial differential equations on bounded domains.
In this paper, we prove $\mathcal{H}^{2+α}$ regularity for viscosity solutions to non-convex fully nonlinear parabolic equations near the boundary. This constitutes the parabolic counterpart of a similar $C^{2, α}$ regularity result due to Silvestre and Sirakov proved in [15] for solutions to non-convex fully nonlinear elliptic equations.
Maximal parabolic regularity for divergence operators including mixed boundary conditions
证明了散度型椭圆二阶算子$A$在分布空间上满足极大抛物正则性，即使其下的区域是高度非光滑的，$A$的系数也是不连续的，并且$A$是用混合边界条件补的.应用于非光滑数据的拟线性抛物型方程。
Blow-up of solutions for a semilinear parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition
研究含非线性记忆与吸收项的半线性抛物方程在非线性非局部边界条件下的初边值问题，通过理论分析证明了解的整体存在性与有限时刻爆破。
Boundary value problems for parabolic operators in a time-varying domain
证明了具有边界爆破低阶系数的二阶线性一致抛物型算子的散度型和非散度型Dirichlet边值问题解的存在唯一性.该区域可能是时变的，非光滑的，并满足外测度条件。
Global existence of classical solutions for a nonlocal one dimensional parabolic free boundary problem
本文研究了一维抛物型自由边界值问题在自由边界上具有非局部（积分-微分）条件。在充分光滑的初边值条件和相容性条件下，我们证明了经典解的整体存在唯一性。我们的方法是基于一个等价的系统的非线性积分方程的分析。
We study existence and uniqueness of solutions to a class of nonlinear degenerate parabolic equations, in bounded domains. We show that there exists a unique solution which satisfies possibly inhomogeneous Dirichlet boundary conditions. To this purpose some barrier functions are properly introduced and used.
On the second boundary value problem for Monge-Ampere type equations and optimal transportation
本文研究Monge-Ampere方程第二边值问题整体光滑解的存在性及其在最优运输中势函数正则性的应用.费用函数满足条件A3的一个弱形式，在此条件下，我们在最近的一篇文章中证明了内部正则性。因此，它们包括Caffarelli和Urbas的二次费用函数情况以及早期工作中的各种示例。该方法是通过推导解决方案的二阶导数的全球估计。
Regularity of the free boundary in an unstable parabolic problem
本文研究燃烧模型引发的不稳定抛物问题，在热量向前推进的物理情形下证明了自由边界具有 $C^{1,α/2}$ 超曲面正则性。
In this paper we study the optimality condition for the Venttsel boundary control of a parabolic equation, that is, the state of the dynamic system is governed by a parabolic equation together with an initial condition while the control is applied to the system via the Venttsel boundary condition. A first order necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained problem.
The global solvability of initial-boundary value problem for nondiagonal parabolic systems
本文研究拟线性非对角抛物型方程组。我们假设主椭圆算子，这是抛物系统的一部分，具有发散结构。在一定的条件下证明了该方程的经典解的适定性，并证明了该解在时间上整体存在。
In this paper, we prove boundary pointwise $C^{k,α}$ regularity for any $k\geq 1$ for fully nonlinear parabolic equations. As an application, we give a direct and short proof of the higher regularity of the free boundaries in obstacle-type problems.
We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.
We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove Hölder continuity of the solution in space and time.
We study solutions to a variational equation that models heat control on the boundary. This problem can be thought of as the two phase parabolic Signorini problem. We show that when the solution has a sign on the boundary, the study of the free boundary can be reduced to the study of the free boundary in the parabolic Signorini problem. We show that locally there is always a sign on the boundary; therefore, the optimal regularity of the solutions as well as the study of the free boundary is reduced to the one phase parabolic Signorini problem.
This paper establishes sharp local regularity estimates for viscosity solutions of fully nonlinear parabolic free boundary problems with singular absorption terms. The main difficulties are due to the blow-up of the source along the free boundary and the lack of a variational structure. The proof combines the power of the Ishii-Lions method with intrinsically parabolic oscillation estimates. The results are new, even for second-order linear operators in nondivergence form.
In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,α}$ regularity of the free boundary for optimal partial transport between convex domains for densities $f, g$ bounded from below and above. When $f, g \in C^α$, and $\partialΩ, \partialΩ^*\in C^{1,1}$ are far apart, by adopting our recent results on boundary regularity of Monge-Ampère equations \cite{CLW1}, our second result shows that the free boundaries are $C^{2,α}$. As an application, in the last we also obtain these regularities of the free…
A parabolic inverse problem with mixed boundary data. Stability estimates for the unknown boundary and impedance
我们考虑的问题，确定一个不可访问的部分，通过热测量的导体的边界。我们研究的腐蚀问题，其中规定的罗宾型条件的损坏部分，我们证明对数稳定性估计。
Elliptic and parabolic reguarity for second order divergence operators with mixed boundary conditions
我们研究非Lipschitz域上包含混合边界条件的二阶方程和系统。关键的结果是插值合适的函数空间。在此基础上，利用Sneiberg同构定理，得到了椭圆正则性和抛物正则性结果.
This paper studies the regularity of the free boundary for viscosity solutions to a parabolic Bernoulli-type free boundary problem with variable coefficients. The main result is that Lipschitz free boundaries are $C^1$ with a normal vector that varies with a Hölder modulus of continuity in both space and in time. As a consequence, the viscosity solution satisfies the free boundary problem in a classical sense.
A drift approximation for parabolic PDEs with oblique boundary data
本文研究了有界区域中具有零斜边界数据的拟线性抛物型偏微分方程的解。我们的主要结果指出，解决方案可以近似为一个偏微分方程的解决方案在整个空间与惩罚漂移项。收敛性是局部一致的，并得到了最优的误差估计。
Parabolic power concavity and parabolic boundary value problems
研究了抛物型方程边值问题\begin{equation} \tag{$P$} \left\{\begin{array}{ll} \partial_t u=Δu +f(x,t,u,\nabla u) & \mbox{in}\quadΩ\times(0,\infty),\vspace{3pt}\\ u(x,t)=0 & \mbox{on}\quad\partial Ω\times(0,\infty),\vspace{3pt}\\ u(x,0)=0 & \mbox{in}\quadΩ, \end{array} \right. \end{equation}解的幂级数性质，其中$Ω$是${\bf R}^n$中的有界凸区域，$f$是$Ω\times(0,\infty)\times{\bf R}\times{\bf R}^n$中的非负连续函数.给出了在$\barΩ\times[0,\infty)$中$(P)$的解是抛物线幂凹的一个充分条件.
New unexpected limit operators for homogenizing optimal control parabolic problems with dynamic reaction flow on the boundary of critically scaled particles
本文研究了具有动态边界条件的抛物型最优控制问题的均匀化，通过取极限发现临界尺度导致出现了新的意外项。
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to degenerate parabolic equations of $p$-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity.
A nonlinear divergence parabolic equation with dynamic boundary conditions of Wentzell type is studied. The existence and uniqueness of a strong solution is obtained as the limit of a finite difference scheme, in the time dependent case and via a semigroup approach in the time-invariant case.
We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.
In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Two theorems on strong solvability and the existence of eigenvalues of the considered problem have been proved.
Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem
我们给出了一个全面的治疗抛物Signorini问题的基础上推广的Almgren的单调性的频率。其中包括解的最优正则性证明、自由边界点的分类、正则集的正则性和奇异集的结构。
Boundary Hölder gradient estimates for parabolic $p$-Laplace type equations
本文研究了抛物 $p$-Laplace 型方程粘性解的边界正则性，获得了边界逐点和全局 $C^{1,α}$ 正则性。
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。
We investigate a general nonhomogeneous parabolic initial-boundary value problem in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem.
Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
本文针对完全非线性退化或奇异抛物型方程，利用紧性方法、障碍函数构造、正则化技术及小扰动解的边界正则性，建立了统一的边界Hölder梯度估计，推广了已有结果。
Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
研究一类退化奇异抛物型方程非负解的边界表现，其原型是抛物型$p$ -Laplacian算子.假设这样的解决方案连续消失的Lipschitz圆柱的横向部分$S_T$的一些杰出的部分，我们证明了Carleson型估计，并推导出一些额外的假设下的方程或域的后果。然后，我们证明了类似的估计一类退化/奇异抛物型方程的非负解的多孔介质类型。
We study the generation property for a fourth order operator in divergence or in non divergence form with suitable Neumann boundary conditions. As a consequence we obtain the well posedness for the parabolic equations governed by these operators. The novelty of this paper is that the operators depend on a function $a: [0,1] \rightarrow \R_+$ that degenerates somewhere in the interval.
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