Guy Barles
We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities.
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In this work, we study some properties of the viscosity solutions to a degenerate parabolic equation involving the non-homogeneous infinity-Laplacian.
This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results apply to the Pucci operator, the normalized $q$-Laplacians with $1<q\leq\infty$, the Finsler Laplacian and more general quasilinear operators.
On equivalence of entropy and viscosity solutions to degenerate parabolic equations and applications
该研究证明了各向异性退化抛物-双曲方程与退化拟线性Hamilton-Jacobi方程的熵解和粘性解的等价性，并应用该结论得到周期情形下两类方程对应解的大时间行为。
We study Phragmén-Lindelöf properties of viscosity solutions to a class of doubly nonlinear parabolic equations in $\mathbb{R}^n\times (0,T)$. We also include an application to some doubly nonlinear equations.
We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in $BV$, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.
We study asymptotic decay rates of viscosity solutions to some doubly nonlinear parabolic equations, including Trudinger's equation. We also prove a Phragmén-Lindelöf type result and show its optimality.
The global equi-continuity estimate on $L^p$-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of $L^p$-viscosity solutions is established via an approximation of given data. The local Hölder continuity estimate on the space derivative of $L^p$-viscosity solutions is shown when the obstacles belong to $C^{1, β}$, and $p>n+2$.
In this article we present several results concerning uniqueness of $C$-viscosity and $L_{p}$-viscosity solutions for fully nonlinear parabolic equations. In case of the Isaacs equations we allow lower order terms to have just measurable bounded coefficients. Higher-order coefficients are assumed to be Hölder continuous in $x$ with exponent slightly less than $1/2$. This case is treated by using stability of maximal and minimal $L_{p}$-viscosity solutions.
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequali…
We study the relationship of viscosity and weak solutions to the equation \[ \smash{\partial_{t}u-Δ_{p}u=f(Du)} \] where $p>1$ and $f\in C(\mathbb{R}^{N})$ satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $p\geq2$.
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDE problem involving sublinear operators. This is done through a dynamic programming principle derived from [8]. The formula can be seen as a nonlinear extension of the Feynman--Kac formula and is based on the backward stochastic differential equations theory.
We provide a representation formula for viscosity solutions to a class of nonlinear second order parabolic PDEs given as a sup--envelope function. This is done through a dynamic programming principle derived from Denis, Hu, Peng (2010). The formula can be seen as a nonlinear extension of the Feynman--Kac formula and is based on the backward stochastic differential equations theory.
We study Phragmén-Lindelöf properties for viscosity solutions to a class of nonlinear parabolic equations of the type $H(Du, D^2u+Z(u)Du\otimes Du)+χ(t)|Du|^σ-u_t=0$ under a certain boundedness condition on $H$. We also state results for positive solutions to a class of doubly nonlinear equation $H(Du, D^2u)-f(u)u_t=0$.
We investigate the $C^{1+α}$-regularity of solutions of parabolic equations $\partial_{t}v+H(v,Dv,D^{2}v,t,x)=0$. Our main result says that under rather general assumptions there exist $C$-viscosity and $L_{p}$-viscosity solutions which are in $C^{1+α}_{loc}$. We allow $H$ to be just measurable in $t$ and for its principal part to have sufficiently small discontinuities as a function of~$x$. No Lipschitz continuity of $H$ with respect to $v,Dv$ is required.
Nonnegative Viscosity and Weak Solutions to Trudinger's Parabolic Equation
本文研究Trudinger抛物方程，证明对于非负函数，黏性解与弱解等价。
In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold $M$ has nonnegative sectional curvature, we get the finest results. If one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, the assumptions on curvature can be thrown away.
We show how a theorem about the solvability in $W^{1,2}_{\infty}$ of special parabolic Isaacs equations can be used to obtain the existence and uniqueness of viscosity solutions of general uniformly nondegenerate parabolic Isaacs equations. We apply it also to establish the $C^{1+χ}$ regularity of viscosity solutions and show that finite-difference approximations have an algebraic rate of convergence. The main coefficients of the Isaacs equations are supposed to be in $C^γ$ with respect to the spatial variables with $γ$ slightly less than $1/2$.
We study a version of the strong minimum principle, and large time asymptotics of positive viscosity solutions to classes of doubly nonlinear parabolic equations of the form $$ H(Du,D^2u)-u^{k-1}u_t=0,\;\;k\geq 1,\quad\mbox{in $Ω\times [0,T)$},$$ where $Ω\subset \mathbb{R}^n$ is a bounded domain and $0<T\leq \infty$. The spatial operator $H$ is homogeneous with power $k$.
We introduce the notion of δ-viscosity solutions for fully nonlinear uniformly parabolic PDE on bounded domains. We prove that δ-viscosity solutions are uniformly close to the actual viscosity solution. As a consequence we obtain an error estimate for implicit monotone finite difference approximations of uniformly parabolic PDE.
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
In this work, we show existence and uniqueness of positive solutions of $H(Du, D^2u)+χ(t)|Du|^Γ-f(u)u_t=$ in $Ω\times(0, T)$ and $u=h$ on its parabolic boundary. The operator $H$ satisfies certain homogeneity conditions, $Γ>0$ and depends on the degree of homogeneity of $H$, $f>0$, increasing and meets a concavity condition. We also consider the case $f\equiv 1$ and prove existence of solutions without sign restrictions.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely solvable in the viscosity sense. Moreover, some spectral properties of the elliptic operator are investigated, proving existence and simplicity of the first eigenvalue. Eventually, parabolic solutions are proven to converge to the corresponding limiting elliptic solution in the long-time limit.
In this paper, we show the existence and uniqueness of viscosity solution to the Cauchy-Dirichlet problem for a class of fully nonlinear parabolic equations. This extends recent results of Eyssidieux-Guedj-Zeriahi.
We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates proved in [S.N. Armstrong and H.V. Tran, Viscosity solutions of general viscous Hamilton-Jacobi equations, Math. Ann., 361 (2015)], as well as on viscosity solution arguments.
We introduce a notion of viscosity solutions for a general class of elliptic-parabolic phase transition problems. These include the Richards equation, which is a classical model in filtration theory. Existence and uniqueness results are proved via the comparison principle. In particular, we show existence and stability properties of maximal and minimal viscosity solutions for a general class of initial data. These results are new even in the linear case, where we also show that viscosity solutions coincide with the regular weak solutions introduced in [Alt&Luckhaus 1983].
Boundary Regularity for viscosity solutions of Fully nonlinear degenerate/singular parabolic equations
本文研究了完全非线性退化/奇异抛物型方程粘性解的边界正则性，证明了边界 $C^{1,α}$ 和 $C^{2,α}$-正则性，并导出了全局正则性结果及 Cauchy-Dirichlet 问题的可解性。
In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of th…
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.
This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Hölder continuous.
We derive two equivalent definitions of the viscosity solutions to the homogeneous sub-p- Laplace parabolic equations on the Heisenberg group, and characterize the viscosity solutions in terms of an asymptotic mean value formula. Moreover, we construct an example to show that these formulae do not hold in non-asymptotic sense.
Using a generalized assumption of Osgood type, we prove a comparison result between viscosity sub and supersolutions of fully nonlinear, possibly strongly degenerate, parabolic equations under rather generale assumtpions. The principle allows to consider dependence on the gradient witha quadratic growth and also on complex zeroth-order terms. The result it applies in particular to a differential model for pricing Mortgage-Backed Securities.
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.
In this paper we study linear parabolic equations on a finite oriented starshaped network; the equations are coupled by transmission conditions set at the inner node, which do not impose continuity on the unknown. We consider this problem as a parabolic approximation of a set of first order linear transport equations on the network and we prove that, when the diffusion coefficient vanishes, the family of solutions converges to the unique solution to the first order equations and satisfies suitable transmission conditions at the inner node, which are determined by the parameters appearing in th…
Gradient regularity for viscosity solutions to quasilinear parabolic equations with mixed singular-degenerate structure
本文研究了一类具有混合退化-奇异结构的拟线性抛物型方程粘性解的正则性，通过Jensen-Ishii方法和内在尺度技术，证明了解的Lipschitz正则性和梯度的Hölder连续性。
We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang, our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Souganidis in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof in Ekren, Touzi & Zhang, but requires an $L^p-$type of continuity (with respect to t…
This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.
We study and compare two concepts for weak solutions to semilinear parabolic path-dependent partial differential equations (PPDEs). The first is that of mild solutions as it appears, e.g., in the log-Laplace functionals of historical superprocesses. The aim of this paper is to show that mild solutions are also solutions in a viscosity sense. This result is motivated by the fact that mild solutions can provide value functions and optimal strategies for problems of stochastic optimal control. Since unique mild solutions exist under weak conditions, we obtain as a corollary a general existence re…
The parabolic normalized p-Laplace equation is studied. We prove that a viscosity solution has a time derivative in the sense of Sobolev belonging locally to $L^2$.
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 study the boundary behavior of viscosity nonnegative solutions of fully nonlinear parabolic Pucci extremal operators. We establish local and global comparison theorems in $C^{1,1} cylinders, along with a backward Harnack inequality.
For a parabolic equation associated to a uniformly elliptic operator, we obtain a $W^{3, \varepsilon}$ estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic $W^{2,\varepsilon}$ estimates with a comparison principle argument. As an application, we show, assuming the operator is $C^1$, that a viscosity solution is $C^{2,α}$ on the complement of a closed set of Hausdorff dimension $\varepsilon$ less than that of the ambient space, where the constant $\varepsilon>0$ depends only on the dime…
Time derivative estimates for normalized parabolic $p$-Laplace equations
本文建立了归一化抛物型$p$-Laplace方程有界粘性解时间导数的局部有界性，并通过构造反例表明$p=1$时解在时间上非Hölder连续。
We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosi…
We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub- and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.
This paper is concerned with the study of the Strong Maximum Principle for semicontinuous viscosity solutions of fully nonlinear, second-order parabolic integro-differential equations. We study separately the propagation of maxima in the horizontal component of the domain and the local vertical propagation in simply connected sets of the domain. We give two types of results for horizontal propagation of maxima: one is the natural extension of the classical results of local propagation of maxima and the other comes from the structure of the nonlocal operator. As an application, we use the Stron…
On the Viscosity Solutions of Parabolic p-Laplacian Equations with Capillary-Type Boundary Conditions
本文研究了具有毛细型边界条件的奇异/退化抛物p-拉普拉斯方程粘性解的适定性及大时间渐进行为，通过最大原理和近似过程结合椭圆特征值问题，证明了解的存在性、唯一性和渐进行为，同时讨论了非凸域的扩展。
In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge-Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of Dynamic Programming Principles for certain two-player, zero-sum games.
The paper is concerned with a class of mathematical models for polymeric fluids, which involves the coupling of the Navier-Stokes equations for a viscous, incompressible, constant-density fluid with a parabolic-hyperbolic integro-differential equation describing the evolution of the polymer distribution function in the solvent, and a parabolic integro-differential equation for the evolution of the monomer density function in the solvent. The viscosity coefficient appearing in the balance of linear momentum equation in the Navier-Stokes system includes dependence on the shear-rate as well as on…
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.
We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data, via comparison with one-dimensional solutions. This work extends multiple results of Andrews and Clutterbuck for quasilinear equations to fully nonlinear equations.
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.
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.
Li-Yau and Harnack estimates for nonlinear parabolic equations
本文为多类非线性抛物方程建立全局Harnack不等式，包括一致抛物流、加权p-Laplacian热流及各向同性扩散，利用极大值原理和加权测地线变分，并在粘性意义下推导Li-Yau不等式。
We study unbounded (viscosity) supersolutions of the Evolutionary p-Laplace Equation in the slow diffusion case. The supersolutions fall into two widely different classes, depending on whether they are locally summable to the power p-2 or not. Also the Porous Medium Equation is studied.
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.
We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known a…
In this paper we deal with a second order nonlinear evolution inclusion, with a nonmonotone, noncoercive viscosity term. Using a parabolic regularization (approximation) of the problem and {\it a priori} bounds that permit passing to the limit, we prove that the problem has a solution.
In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:\[\{[c]{l}\dfrac{\partial u}{\partial t}(t,x)+\mathcal{L}_tu(t,x)+f(t,x,u(t,x))\in\partialϕ(u(t,x)),\quad t\in[0,T),x\in\mathbb{R}^d, u(T,x)=h(x),\quad x\in\mathbb{R}^d,\] where $\partialϕ$ is the subdifferential operator of the proper convex lower semicontinuous function $ϕ:\mathbb{R}^k\to (-\infty,+\infty]$ and $\mathcal{L}_t$ is a second differential operator given by $\mathcal{L}_tv_i(x)={1/2}\operatorname {Tr}[σ(t,x)σ^*(t,x)\mathrm…
In this paper we are concerned with rate of convergence of parabolic systems with large diffusion. We will exhibit the exact moment that spatial homogenization occurs and estimate the continuity of attractors by a rate of convergence. We will show an example where our estimate is optimal.
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