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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]

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]

We find bounds for the principal eigenvalue and eigenfunction associated to the existence of self-similar solutions to a fully nonlinear parabolic problem. In contrast with the local case, the upper and lower bounds for the eigenfunction are of the same polynomial order | x | - (N + 2 s) . In order to accomplish this we prove a Harnack inequality for general nonlocal elliptic equations with zero order terms. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show the existence of global-in-time solutions to the problem. Our condition to ensure the global existence is specific to the fully nonlinear parabolic system. [ABSTRACT FROM AUTHOR]

We study monotone P1 finite element methods on unstructured meshes for fully non-linear, degenerately parabolic Isaacs equations with isotropic diffusions arising from stochastic game theory and optimal control and show uniform convergence to the viscosity solution. Elliptic projections are used to manage singular behaviour at the boundary and to treat a violation of the consistency conditions from the framework by Barles and Souganidis by the numerical operators. Boundary conditions may be imposed in the viscosity or in the strong sense, or in a combination thereof. The presented monotone numerical method has well-posed finite dimensional systems, which can be solved efficiently with Howard’s method. [ABSTRACT FROM AUTHOR]

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space R N in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when t → + ∞ , i.e., they behave like λ * t + ϕ (x) where λ * is the maximal ergodic constant and ϕ is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution problem ut(x,t)−λj(D2u(x,t))=0,inΩ×(0,+∞),u(x,t)=g(x,t),on∂Ω×(0,+∞),u(x,0)=u0(x),inΩ,where Ω is a bounded domain in RN (which verifies a suitable geometric condition on its boundary) and λj(D2u) stands for the jth eigenvalue of the Hessian matrix D2u. We assume that u0 and g are continuous functions with the compatibility condition u0(x)=g(x,0), x∈∂Ω. We show that the (unique) solution to this problem exists in the viscosity sense and can be approximated by the value function of a two‐player zero‐sum game as the parameter measuring the size of the step that we move in each round of the game goes to zero. In addition, when the boundary datum is independent of time, g(x,t)=g(x), we show that viscosity solutions to this evolution problem stabilize and converge exponentially fast to the unique stationary solution as t→∞. For j=1, the limit profile is just the convex envelope inside Ω of the boundary datum g, while for j=N, it is the concave envelope. We obtain this result with two different techniques: with partial differential equations (PDE) tools and with game‐theoretical arguments. Moreover, in some special cases (for affine boundary data), we can show that solutions coincide with the stationary solution in finite time (which depends only on Ω and not on the initial condition u0). [ABSTRACT FROM AUTHOR]

In a recent paper (Carlier et al. in ESAIM Control Optim Calc Var 18(3):611–620, 2012), an interpolation flow between an evolution by convexity and the geometric flow of motion by principal negative curvature was informally proposed. It is also expected that the geometric flow will eventually convexify the sub-level sets of the initial function u 0 , yielding the quasiconvex envelope of u 0 . In this note, we establish existence and uniqueness of the interpolation flow under appropriate conditions and provide a rigorous proof for its limit behaviour. In addition, we show by example that, contrary to intuition, the proposed geometric flow does not always convexify the sub-level sets of u 0 . [ABSTRACT FROM AUTHOR]

We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solution preserves a strong version of star-shapedness. We also show that the solution converges to a ball as time goes to infinity. For asymptotic behavior of the solution we use the gradient flow structure of the problem, whereas a modified notion of viscosity solutions is introduced to study the geometric properties of the flow by moving planes method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation ∂th(t,x)=νΔh(t,x)+λ|∇h(t,x)|2+Dη(t,x),(t,x)∈R+×Rdin d≥3 dimensions in the perturbative regime, i.e. for λ>0 small enough and a smooth, bounded, integrable initial condition h0=h(t=0,·). The forcing term η in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85-121, 1994). Standard large deviation estimates for η make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards-Wilkinson model (λ=0) with renormalized coefficients νeff=ν+O(λ2),Deff=D+O(λ2). [ABSTRACT FROM AUTHOR]

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 as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary. [ABSTRACT FROM AUTHOR]

General envelope methods are introduced which may be used to embed equations withu-dependence into equations without solution dependence. Furthermore, these methods present a rigorous way to consider so-called nodal solutions. That is, ifw(t,x,z) is the viscosity solution of some pde, the nodal solution of an associated pde is a functionu(t,x) so thatw(t,x,u(t,x)) = 0. Examples are given to first- and second-order pdes arising in optimal control, differential games, minimal time problems, scalar conservation laws, geometric-type equations, and forward backward stochastic control. [ABSTRACT FROM PUBLISHER]

The mathematical modeling of tumor growth leads to singular “stiff pressure law” limits for porous medium equations with a source term. Such asymptotic problems give rise to free boundaries, which, in the absence of active motion, are generalized Hele-Shaw flows. In this note we use viscosity solutions methods to study limits for porous medium-type equations with active motion. We prove the uniform convergence of the density under fairly general assumptions on the initial data, thus improving existing results. We also obtain some additional information/regularity about the propagating interfaces, which, in view of the discontinuities, can nucleate and, thus, change topological type. The main tool is the construction of local, smooth, radial solutions which serve as barriers for the existence and uniqueness results as well as to quantify the speed of propagation of the free boundary propagation. [ABSTRACT FROM AUTHOR]

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 $$C^{2,\alpha }$$ 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 dimension and the ellipticity. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of a “steady” spiral moving with forced mean curvature motion. This spiral has a stationary shape and rotates with constant angular velocity. Under appropriate conditions on the initial data, we also show the long time convergence (up to some subsequence in time) of the solution of the Cauchy problem to the steady state. This result is based on a new Liouville result which is of independent interest. [ABSTRACT FROM PUBLISHER]

We obtain the existence and uniqueness results of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate and singular parabolic inhomogeneous equation of the form , where Δ denotes the so-called normalized infinity Laplacian given by $\Delta _\infty ^N u = \frac{1} {{|Du|^2 }}\left\langle {D^2 uDu,Du} \right\rangle $. [ABSTRACT FROM AUTHOR]

We study the regularity of solutions of parabolic fully nonlinear nonlocal equations. We prove C regularity in space and time and, under different assumptions on the kernels, C in space for translation invariant equations. The proofs rely on a weak parabolic ABP and the classic ideas of Tso (Commun. Partial Diff. Equ. 10(5):543-553, ) and Wang (Commun. Pure Appl. Math. 45(1), 27-76, ). Our results remain uniform as σ → 2 allowing us to recover most of the regularity results found in Tso (Commun. Partial Diff. Equ. 10(5):543-553, ). [ABSTRACT FROM AUTHOR]