viscosity sense
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Author(s):  
Raffaele Grande

AbstractThe evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with applications in IT and neurogeometry [see Citti et al. (SIAM J Imag Sci 9(1):212–237, 2016)]. Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we obtain a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr et al. (Commun Pure Appl Math 9(2):307–326, 2010).


2021 ◽  
pp. 2150053
Author(s):  
Khaled Bahlali ◽  
Brahim Boufoussi ◽  
Soufiane Mouchtabih

We consider a system of semilinear partial differential equations (PDEs) with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized PDEs, which converges to our initial problem. Since the coefficients we consider may be discontinuous, we use the notion of solution in the [Formula: see text]-viscosity sense. The method we use is based on backward stochastic differential equations and their [Formula: see text]-tightness. This work is motivated by the fact that many PDEs in physics have discontinuous coefficients. As a consequence, it follows that if the uniqueness holds, then the solution can be constructed by a penalization.


Author(s):  
Giulia Cavagnari ◽  
Antonio Marigonda ◽  
Marc Quincampoix

AbstractThis study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on $${\mathbb {R}}^d$$ R d representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space.


Author(s):  
Olivier Bokanowski ◽  
Athena Picarelli ◽  
Christoph Reisinger

AbstractWe study a second order Backward Differentiation Formula (BDF) scheme for the numerical approximation of linear parabolic equations and nonlinear Hamilton–Jacobi–Bellman (HJB) equations. The lack of monotonicity of the BDF scheme prevents the use of well-known convergence results for solutions in the viscosity sense. We first consider one-dimensional uniformly parabolic equations and prove stability with respect to perturbations, in the $$L^2$$ L 2 norm for linear and semi-linear equations, and in the $$H^1$$ H 1 norm for fully nonlinear equations of HJB and Isaacs type. These results are then extended to two-dimensional semi-linear equations and linear equations with possible degeneracy. From these stability results we deduce error estimates in $$L^2$$ L 2 norm for classical solutions to uniformly parabolic semi-linear HJB equations, with an order that depends on their Hölder regularity, while full second order is recovered in the smooth case. Numerical tests for the Eikonal equation and a controlled diffusion equation illustrate the practical accuracy of the scheme in different norms.


2021 ◽  
pp. 1-33
Author(s):  
S. Buccheri ◽  
J.V. da Silva ◽  
L.H. de Miranda

In this work, given p ∈ ( 1 , ∞ ), we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector ( u p , v p ), for the following local/nonlocal PDE system (0.1) − Δ p u + ( − Δ ) p r u = 2 α α + β λ | u | α − 2 | v | β u in  Ω − Δ p v + ( − Δ ) p s v = 2 β α + β λ | u | α | v | β − 2 v in  Ω u = 0 on  R N ∖ Ω v = 0 on  R N ∖ Ω , where Ω ⊂ R N is a bounded open domain, 0 < r , s < 1 and α ( p ) + β ( p ) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ ∞ , and the uniformly convergence of the pair ( u p , v p ) to the ∞-eigenvector ( u ∞ , v ∞ ). Finally, the triple ( u ∞ , v ∞ , λ ∞ ) verifies, in the viscosity sense, a limiting PDE system.


Author(s):  
Félix del Teso ◽  
Erik Lindgren

AbstractWe prove a new asymptotic mean value formula for the p-Laplace operator, $$\begin{aligned} \Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u), \quad 1<p<\infty \end{aligned}$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , 1 < p < ∞ valid in the viscosity sense. In the plane, and for a certain range of p, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.


Author(s):  
Nils Dabrock ◽  
Martina Hofmanová ◽  
Matthias Röger

Abstract We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. For the first time, we are able to construct martingale solutions which satisfy the equation pointwise and not only in a generalized (distributional or viscosity) sense. Moreover, we study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $$L^{\infty }_{\omega ,x,t}$$ L ω , x , t ∞ estimate for the gradient and an $$L^{2}_{\omega ,x,t}$$ L ω , x , t 2 bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.


2020 ◽  
Vol 26 ◽  
pp. 66 ◽  
Author(s):  
Julien Bernis ◽  
Piernicola Bettiol

We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.


2020 ◽  
Vol 26 ◽  
pp. 36 ◽  
Author(s):  
Alexander Quaas ◽  
Ariel Salort ◽  
Aliang Xia

We study existence of principal eigenvalues of a fully nonlinear integro-differential elliptic equations with a drift term via the Krein–Rutman theorem and regularity estimates up to boundary of viscosity solutions. We also show simplicity of eigenfunctions in the viscosity sense by using a nonlocal version of the ABP estimate and a “sweeping lemma”.


Author(s):  
Tat Dat Tô

Abstract We study the Kähler–Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler–Ricci flow has long-time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler–Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge–Ampère flows in big classes that we develop, extending and refining the approach of Eyssidieux–Guedj–Zeriahi.


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