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It is well-known that solutions to the Hamilton-Jacobi equation fail to be everywhere differentiable. Nevertheless, suppose a solution $$u$$ turns out to be differentiable at a given point $$(t,x)$$ in the interior of its domain. May then one deduce that $$u$$ must be continuously differentiable in a neighborhood of $$(t,x)$$? Although this question has a negative answer in general, our main result shows that it is indeed the case when the proximal subdifferential of $$u(t,\cdot )$$ at $$x$$ is nonempty. Our approach uses the representation of $$u$$ as the value function of a Bolza problem in the calculus of variations, as well as necessary conditions for such a problem. [ABSTRACT FROM AUTHOR]

Let $$M$$ be a Riemannian manifold and let $$\varOmega $$ be a bounded open subset of $$M$$. It is well known that significant information about the geometry of $$\varOmega $$ is encoded into the properties of the distance, $$d_{\partial \varOmega }$$, from the boundary of $$\varOmega $$. Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $$x_0$$ is a singular point of $$d_{\partial \varOmega }$$ then the generalized characteristic starting at $$x_0$$ stays singular for all times. As an application, we deduce that the singular set of $$d_{\partial \varOmega }$$ has the same homotopy type as $$\varOmega $$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and regularity of solutions for Bolza optimal control problems in infinite dimension governed by a class of semilinear evolution equations. Our results apply to systems exhibiting hereditary properties, as heat propagation in real conductors and isothermal viscoelasticity, described by equations with memory terms which account for the past history of the variables in play. [ABSTRACT FROM AUTHOR]

The system of partial differential equations arises in the analysis of mathematical models for sandpile growth and in the context of the Monge–Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form with f≥ 0, and h≥ 0 possibly non-convex, is also included. [ABSTRACT FROM AUTHOR]

Let two second order evolution equations be coupled via the zero order terms, and suppose that the first one is stabilized by a distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behaviour of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay polynomially at infinity, and that this decay rate is, in some sense, optimal. The stabilization result for abstract evolution equations is also applied to study the asymptotic behaviour of various systems of partial differential equations. [ABSTRACT FROM AUTHOR]

Studies properties of the value function and of optimal solutions of semilinear Mayer problem in infinite dimensions. Description of applications concerning systems governed by a state equation of parabolic type; Focus on the issues of the joint Lipschitz continuity and semiconcavity of the value function treated in order to investigate the differentiability of the value function along optimal trajectories.

Given a semi-convex function u: ω⊂ R → R and an integer k≡[0,,n], we show that the set ∑ defined by is countably ℋ i.e., it is contained (up to a ℋ-negligible set) in a countable union of C hypersurfaces of dimensions ( n− k). Moreover, we show that for any open set ω′⊂⊂ω. [ABSTRACT FROM AUTHOR]