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The linear transport equation allows to advect level-set functions to represent moving sharp interfaces in multiphase flows as zero level-sets. A recent development in computational fluid dynamics is to modify the linear transport equation by introducing a nonlinear term to preserve certain geometrical features of the level-set function, where the zero level-set must stay invariant under the modification. The present work establishes mathematical justification for a specific class of modified level-set equations on a bounded domain, generated by a given smooth velocity field in the framework of the initial/boundary value problem of Hamilton–Jacobi equations. The first main result is the existence of smooth solutions defined in a time-global tubular neighborhood of the zero level-set, where an infinite iteration of the method of characteristics within a fixed small time interval is demonstrated; the smooth solution is shown to possess the desired geometrical feature. The second main result is the existence of time-global viscosity solutions defined in the whole domain, where standard Perron's method and the comparison principle are exploited. In the first and second main results, the zero level-set is shown to be identical with the original one. The third main result is that the viscosity solution coincides with the local-in-space smooth solution in a time-global tubular neighborhood of the zero level-set, where a new aspect of localized doubling the number of variables is utilized. [ABSTRACT FROM AUTHOR]

By exploiting the contact Hamiltonian dynamics (T* M × ℝ, Φt) around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the Φt-recurrent set, the strongly static set, the Aubry set, the Mañé set, and the Φt-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser (KAM) solutions and the existence of transitive orbits around the Aubry set. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions to the subelliptic p-Poisson equation as p → + ∞ in Carnot-Carathéodory spaces. In particular, introducing a suitable notion of differentiability, extend the celebrated result of Bhattacharya et al. (Rend Sem Mat Univ Politec Torino Fascicolo Speciale 47:15–68, 1989) and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the ∞ -Laplacian and the Eikonal equation. [ABSTRACT FROM AUTHOR]

We propose a new asymptotic expansion for the fractional p-Laplacian with precise computations of the errors. Our approximation is shown to hold in the whole range p ∈ (1 , ∞) and s ∈ (0 , 1) , with errors that do not degenerate as s → 1 - . These are super-quadratic for a wide range of p (better far from the zero gradient points), and optimal in most cases. One of the main ideas here is the fact that the singular part of the integral representation of the fractional p-Laplacian behaves like a local p-Laplacian with a weight correction. As a consequence of this, we also revisit a previous asymptotic expansion for the classical p-Laplacian, whose error orders were not known. Based on the previous result, we propose monotone finite difference approximations of the fractional p-Laplacian with explicit weights and we obtain the error estimates. Finally, we introduce explicit finite difference schemes for the associated parabolic problem in R d and show that it is stable, monotone and convergent in the context of viscosity solutions. An interesting feature is the fact that the stability condition improves with the regularity of the initial data. [ABSTRACT FROM AUTHOR]

In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron's method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]

We establish a convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs. Our proof is based on a novel convergence theorem for dynamic sublinear expectations and the stochastic representation of viscosity solutions as value functions. [ABSTRACT FROM AUTHOR]

In this paper, we state the equivalence between weak and viscosity solutions for non-homogeneous problems involving the anisotropic p (·) -Laplacian. The proof that viscosity solutions are weak solutions is performed by the inf-convolution technique. However, due to the anisotropic nature of the p (·) -Laplacian we adapt the definition of inf-convolution to the non-homogeneity of this operator. For the converse, we develop comparison principles for weak solutions. Since the locally Lipschitz assumption is crucial to get the viscosity-weak implication, we prove that a class of bounded viscosity solutions are indeed locally Lipschitz. [ABSTRACT FROM AUTHOR]

We show that a generic level set of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate. This extends the fundamental work of Evans and Spruck (J Geom Anal 5(1):79–116, 1995), which draws a similar connection between the viscosity solution and Brakke flows. [ABSTRACT FROM AUTHOR]

Dynamic programming equations for mean field control problems with a separable structure are Eikonal type equations on the Wasserstein space. Standard differentiation using linear derivatives yield a direct extension of the classical viscosity theory. We use Fourier representation of the Sobolev norms on the space of measures, together with the standard techniques from the finite dimensional theory to prove a comparison result among semi-continuous sub and super solutions, obtaining a unique characterization of the value function. [ABSTRACT FROM AUTHOR]

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k (0) = (n - 2 k) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan (λ i (H w)) = f k (x) such that v k - u k has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases f : R n ⊇ Ω → (- n π / 2 , n π / 2) . Our examples show it does not. [ABSTRACT FROM AUTHOR]

A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the rate O (ε 1 / 2) of convergence in periodic homogenization of forced graphical mean curvature flows in the laminated setting. We also discuss with an example that a faster rate cannot be obtained by utilizing Lipscthiz estimates. [ABSTRACT FROM AUTHOR]

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in C 2 , α (B R) ∩ C (B R ¯) for the inhomogeneous ∞ -Bilaplacian equation on a ball B R ⊂ R n : Δ ∞ 2 u : = (Δ u) 3 | D (Δ u) | 2 = f (x) with Navier Boundary conditions ( u = g ∈ C (∂ B R) , Δ u = 0 on ∂ B R ). We also prove that there exists a solution in C 1 , α (R n) for all α > 0 to the eigenvalue problem on R n : Δ ∞ 2 u = - λ u + f (x) whenever n ≥ 3 , λ < 0 , and f(x) is continuous, bounded, and supported on an annulus. [ABSTRACT FROM AUTHOR]

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term - div (| D u | p - 2 D u + a (x) | D u | q - 2 D u) = f (x , u , D u) , 1 < p ≤ q < ∞ , a (x) ≥ 0. We find some appropriate hypotheses on the coefficient a(x), the exponents p, q and the nonlinear term f to show that the viscosity solutions with a priori Lipschitz continuity are weak solutions of such equation by virtue of the inf (sup )-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary regularity for viscosity solutions of fully nonlinear elliptic equations. We use a unified, simple method to prove that if the domain Ω satisfies the exterior C 1 , Dini condition at x 0 ∈ ∂ Ω , the solution is Lipschitz continuous at x 0 ; if Ω satisfies the interior C 1 , Dini condition at x 0 , the Hopf lemma holds at x 0 . The key idea is that the curved boundaries are regarded as perturbations of a hyperplane. Moreover, we show that the C 1 , Dini conditions are optimal. [ABSTRACT FROM AUTHOR]

In the present article, we consider a double-phase eigenvalue problem with large exponents. Let λ(pn,qn)1{\lambda }_{\left({p}_{n},{q}_{n})}^{1} be the first eigenvalues and un{u}_{n} be the first eigenfunctions, normalized by ‖un‖ℋn=1\Vert {u}_{n}{\Vert }_{{{\mathcal{ {\mathcal H} }}}_{n}}=1. Under some assumptions on the exponents pn{p}_{n} and qn{q}_{n}, we show that λ(pn,qn)1{\lambda }_{\left({p}_{n},{q}_{n})}^{1} converges to Λ∞{\Lambda }_{\infty } and un{u}_{n} converges to u∞{u}_{\infty } uniformly in the space Cα(Ω){C}^{\alpha }\left(\Omega ), and u∞{u}_{\infty } is a nontrivial viscosity solution to a Dirichlet ∞\infty -Laplacian problem.

Pareto hull peeling is a discrete algorithm, generalizing convex hull peeling, for sorting points in Euclidean space. We prove that Pareto peeling of a random point set in two dimensions has a scaling limit described by a first-order Hamilton–Jacobi equation and give an explicit formula for the limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts with convex peeling, which converges to curvature flow. The proof involves direct geometric manipulations in the same spirit as Calder (Nonlinear Anal 141:88–108, 2016). [ABSTRACT FROM AUTHOR]

In this article, we develop a novel notion of viscosity solutions for first order Hamilton-Jacobi equations in proper CAT (0) spaces. The notion of viscosity is defined by taking test functions that are locally Lipschitz and can be represented as a difference of two semiconvex functions. Under mild assumptions on the Hamiltonian, we recover the main features of viscosity theory for both the stationary and the time-dependent cases in this setting: the comparison principle and Perron’s method. Finally, we show that this notion of viscosity coincides with the classical one in R N and we give several examples of Hamilton-Jacobi equations in more general CAT (0) spaces covered by this setting. [ABSTRACT FROM AUTHOR]

In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. Subsequently, we present consistent monotone schemes to approximate infinity ground states and higher eigenfunctions on grids. We prove that our method converges (up to a subsequence) to a viscosity solution of the eigenvalue problem, and perform numerical experiments which investigate theoretical conjectures and compute eigenfunctions on a variety of different domains. [ABSTRACT FROM AUTHOR]

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases. [ABSTRACT FROM AUTHOR]

We provide the Alexandroff–Bakelman–Pucci estimate and global C 1 , α -regularity for a class of singular/degenerate fully nonlinear elliptic equations. We also derive the existence of a viscosity solution to the Dirichlet problem with the associated operator. [ABSTRACT FROM AUTHOR]

In this paper we prove an asymptotic C 1 , γ -estimate for value functions of stochastic processes related to uniformly elliptic dynamic programming principles. As an application, this allows us to pass to the limit with a discrete gradient and then to obtain a C 1 , γ -result for the corresponding limit PDE. [ABSTRACT FROM AUTHOR]

We prove a local Hölder estimate for any exponent 0 < δ < 1 2 for solutions of the dynamic programming principle u ε (x) = ∑ j = 1 n α j inf dim (S) = j sup v ∈ S | v | = 1 u ε (x + ε v) + u ε (x − ε v) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE ∑ i = 1 n α i λ i (D 2 u) = 0 , where λ1(D2u) ≤⋯ ≤ λn(D2u) are the eigenvalues of the Hessian. [ABSTRACT FROM AUTHOR]

We prove sharp uniform estimates for strong supersolutions of a large class of fully nonlinear degenerate elliptic complex equations. Our findings rely on ideas of Kuo and Trudinger who dealt with degenerate linear equations in the real setting. We also exploit the pluripotential theory for the complex Monge-Ampère operator as well as suitably tailored theory of Lp-viscosity subsolutions. [ABSTRACT FROM AUTHOR]

We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation - div (| ∇ u | p - 2 ∇ u + a (x) | ∇ u | q - 2 ∇ u) = 0 and the normalized double phase parabolic equation u t = | ∇ u | 2 - p div (| ∇ u | p - 2 ∇ u + a (x , t) | ∇ u | q - 2 ∇ u) , 1 < p ≤ q < ∞. This is the first mean value result for such kind of nonuniformly elliptic and parabolic equations. In addition, the results obtained can also be applied to the p(x)-Laplace equations and the variable coefficient p-Laplace type equations. [ABSTRACT FROM AUTHOR]

We consider a discrete version of reaction-diffusion equations. A typical example is the fully overdamped Frenkel–Kontorova model, where the velocity is proportional to the force. We also introduce an additional exterior force denoted by σ . For general discrete and fully nonlinear dynamics, we study traveling waves of velocity c = c (σ) depending on the parameter σ . Under certain assumptions, we show properties of the velocity diagram c (σ) for σ ∈ [ σ - , σ + ] . We show that the velocity c is nondecreasing in σ ∈ (σ - , σ +) in the bistable regime, with vertical branches c ≥ c + for σ = σ + and c ≤ c - for σ = σ - in the monostable regime. [ABSTRACT FROM AUTHOR]

In this paper, we consider the full velocity difference model for traffic flow and we study the existence and uniqueness of traveling wave solutions. First, using the monotony of the car’s interdistance, we derive necessary conditions for the existence of such solutions. Then, in the framework of viscosity solutions, we construct a traveling wave solution by considering an approximate non-local Hamilton–Jacobi equation on a bounded domain. This traveling wave solution can be interpreted as a phase transition between a congested state and a free-flow one. [ABSTRACT FROM AUTHOR]

We consider a jump-diffusion mean field control problem with regime switching in the state dynamics. The corresponding value function is characterized as the unique viscosity solution of a HJB master equation on the space of probability measures. Using this characterization, we prove that the value function, which is not regular, is the limit of a finite agent centralized optimal control problem as the number of agents go to infinity, with an explicit convergence rate. Assuming in addition that the value function is smooth, we establish a quantitative propagation of chaos result for the optimal trajectory of agent states. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation Δ p u = 0 with Dirichlet boundary conditions to the solution of the infinity-Laplace equation Δ ∞ u = 0 as p → ∞ . The rate scales like p − 1 / 4 for general solutions of the Dirichlet problem and like p − 1 / 2 for solutions with positive gradient. An explicit example shows that it cannot be better than p − 1 . The proof of this result solely relies on the comparison principle with the fundamental solutions of the p-Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with Hölder cones and Hölder regularity is available. [ABSTRACT FROM AUTHOR]

We investigate the global structures of the non-selfsimilar solutions for n-dimensional (n-D) non-homogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a (n − 1)-dimensional sphere. We first obtain the expressions of n-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves, we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the n-D shock waves. The asymptotic behaviors with geometric structures are also proved. [ABSTRACT FROM AUTHOR]

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop. [ABSTRACT FROM AUTHOR]

The classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes. [ABSTRACT FROM AUTHOR]

This paper develops a new approach to small time local attainability of smooth manifolds of any dimension, possibly with boundary and to prove Hölder continuity of the minimum time function. We give explicit pointwise conditions of any order by using higher order hamiltonians which combine derivatives of the controlled vector field and the functions that locally define the target. For the controllability of a point our sufficient conditions extend some classically known results for symmetric or control affine systems, using the Lie algebra instead, but for targets of higher dimension our approach and results are new. We find our sufficient higher order conditions explicit and easy to compute for targets with curvature and general control systems. Some cases of nonsmooth targets are also included. [ABSTRACT FROM AUTHOR]

We investigate which elliptic PDEs have the property that every viscosity supersolution belongs to a Sobolev space W loc 1 , q (Ω) , Ω ⊆ R n . The asymptotic cone of the operator's sublevel set seems to be essential. It turns out that much can be said if we know how the cone is related to the sublevel set of the dominative p-Laplacian, with p = n - 1 n q + 1 . We show that, in a certain sense, this is the minimal operator associated to the exponent q. [ABSTRACT FROM AUTHOR]

In this work, we show that for the martingale problem for a class of degenerate diffusions with bounded continuous drift and diffusion coefficients, the small noise limit of non-degenerate approximations leads to a unique Feller limit. The proof uses the theory of viscosity solutions applied to the associated backward Kolmogorov equations. Under appropriate conditions on drift and diffusion coefficients, we will establish a comparison principle and a one-one correspondence between Feller solutions to the martingale problem and continuous viscosity solutions of the associated Kolmogorov equation. This work can be considered as an extension to the work in Borkar and Kumar in (J Theor Probab 23(3): 729–747, 2010). [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity. [ABSTRACT FROM AUTHOR]

We study homogenization for a class of stationary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed. [ABSTRACT FROM AUTHOR]

This current work is presented to deal with the model of double diffusive convection in porous material with variable viscosity, such that the equations for convective fluid motion in a Brinkman type are analysed when the viscosity varies with temperature quadratically. Hence, we carefully find a priori bounds when the coe cients depend only on the geometry of the problem, initial data, and boundary data, where this shows the continuous dependence of the solution on changes in the viscosity. A convergence result is also showen when the variable viscosity is allowed to tend to a constant viscosity.

This article is concerned with the Dirichlet eigenvalue problem associated with the ∞\infty -Laplacian in metric spaces. We establish a direct partial differential equation approach to find the principal eigenvalue and eigenfunctions in a proper geodesic space without assuming any measure structure. We provide an appropriate notion of solutions to the ∞\infty -eigenvalue problem and show the existence of solutions by adapting Perron’s method. Our method is different from the standard limit process via the variational eigenvalue formulation for pp-Laplacian in the Euclidean space.

In this article we prove existence, uniqueness and regularity for the singular equation { | ∇ u | α (F (D 2 u) + h (x) ⋅ ∇ u) + c (x) | u | α u + p (x) u − γ = 0 in Ω u > 0 in Ω , u = 0 on ∂ Ω when p is some continuous and positive function, c and h are continuous, α > − 1 and F is Fully nonlinear elliptic. Some conditions on the first eigenvalue for the operator − | ∇ u | α (F (D 2 u) + h (x) ⋅ ∇ u) − c (x) | u | α u are required. The results generalize the well known results of Lazer and McKenna. [ABSTRACT FROM AUTHOR]

We study the viscosity solutions to the first eigenvalue equation. We consider Ω a bounded B-regular domain in C n and we prove that the Dirichlet problem Λ 1 (D C 2 u) = f in Ω and u = φ on ∂ Ω admits a unique viscosity solution. We also deal with viscosity theory for operators which are comparable to the first eigenvalue operator. [ABSTRACT FROM AUTHOR]