Recent zbMATH articles in MSC 35F21 https://www.zbmath.org/atom/cc/35F21 2021-04-16T16:22:00+00:00 Werkzeug Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints. https://www.zbmath.org/1456.35236 2021-04-16T16:22:00+00:00 "Safdari, Mohammad" https://www.zbmath.org/authors/?q=ai:safdari.mohammad Summary: We prove the optimal $$W^{2, \infty}$$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $$C^1$$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $$W^{2, \infty}$$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work. All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function. https://www.zbmath.org/1456.35082 2021-04-16T16:22:00+00:00 "Fujita, Yasuhiro" https://www.zbmath.org/authors/?q=ai:fujita.yasuhiro "Hamamuki, Nao" https://www.zbmath.org/authors/?q=ai:hamamuki.nao "Yamaguchi, Norikazu" https://www.zbmath.org/authors/?q=ai:yamaguchi.norikazu Summary: We determine all the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function, which is everywhere continuous and nowhere differentiable. This result clarifies how singularities of the solution propagate along generalized characteristics. Moreover it turns out that the Takagi function still keeps the validity of the recent results in [\textit{P. Albano} et al., J. Differ. Equations 268, No. 4, 1412--1426 (2020; Zbl 1437.35153)], in which locally Lipschitz continuous initial data are handled. Hamilton-Jacobi equations for neutral-type systems: inequalities for directional derivatives of minimax solutions. https://www.zbmath.org/1456.49009 2021-04-16T16:22:00+00:00 "Lukoyanov, Nikolai Yu." https://www.zbmath.org/authors/?q=ai:lukoyanov.nikolai-yu "Plaksin, Anton R." https://www.zbmath.org/authors/?q=ai:plaksin.anton-romanovich The authors consider the Hamilton-Jacobi (HJ) equations for differential systems of neutral type. They continue a series of study of the dynamical system from the point of view of directional derivatives in optimization context. They give an infinitesimal criterion of the minimax (generalized) solutions for the HJ equations arising in optimal control problems and differential games, using a suitable definition and techniques of directional derivatives. Reviewer: A. Omrane (Cayenne)