scholarly journals A Differential Perspective on Gradient Flows on $$\textsf {CAT} (\kappa )$$-Spaces and Applications

2021 ◽  
Author(s):  
Nicola Gigli ◽  
Francesco Nobili
Keyword(s):  
Differential Inclusion ◽  
Lower Semicontinuous ◽  
Energy Functional ◽  
Gradient Flows ◽  
Minimal Norm ◽  
Formula One ◽  
Basic Properties

AbstractWe review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on $$\textsf {CAT} (\kappa )$$ CAT ( κ ) -spaces and prove that they can be characterized by the same differential inclusion $$y_t'\in -\partial ^-\textsf {E} (y_t)$$ y t ′ ∈ - ∂ - E ( y t ) one uses in the smooth setting and more precisely that $$y_t'$$ y t ′ selects the element of minimal norm in $$-\partial ^-\textsf {E} (y_t)$$ - ∂ - E ( y t ) . This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar–Schoen energy functional on the space of $$L^2$$ L 2 and (0) valued maps: we define the Laplacian of such $$L^2$$ L 2 map as the element of minimal norm in $$-\partial ^-\textsf {E} (u)$$ - ∂ - E ( u ) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is $$L^2$$ L 2 -dense. Basic properties of this Laplacian are then studied.

Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

2019 ◽  
Vol 19 (3) ◽  
pp. 625-637 ◽  
Author(s):  
Bin Ge ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Variational Principle ◽  
Differential Inclusion ◽  
Energy Functional ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Locally Lipschitz Functions ◽  
Fountain Theorem ◽  
Inclusion Problems ◽  
Locally Lipschitz ◽  
Symmetric Mountain Pass Lemma

Abstract In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.

scholarly journals Trajectory approximately controllability and optimal control for noninstantaneous impulsive inclusions without compactness

2021 ◽  
pp. 1-31
Author(s):  
Shengda Liu ◽  
JinRong Wang ◽  
Donal O'Regan
Keyword(s):  
Optimal Control ◽  
Differential Inclusion ◽  
Regularity Condition ◽  
Measure Of Noncompactness ◽  
Lower Semicontinuous ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Semilinear Differential Inclusion ◽  
The Stability ◽  
Noninstantaneous Impulses

In this paper, a noninstantaneous impulsive differential inclusion model is established based on the heating phenomenon of the rod. The controllability problem for this system governed by a semilinear differential inclusion with noninstantaneous impulses is studied in a Banach space and in this differential inclusion system we assume that the semigroup generated by the linear part of the inclusion is not compact. We suppose that the set-valued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness and some sufficient conditions for approximately controllability for both upper and almost lower semicontinuous types of nonlinearity are presented. Also we discuss existence and the stability of optimal control. As an application, the controllability for a differential inclusion system governed by a heat equation is considered.

scholarly journals A variational formulation of the BDF2 method for metric gradient flows

2019 ◽  
Vol 53 (1) ◽  
pp. 145-172 ◽  
Author(s):  
Daniel Matthes ◽  
Simon Plazotta
Keyword(s):  
Variational Formulation ◽  
Metric Spaces ◽  
Compact Riemannian Manifold ◽  
Energy Functional ◽  
Convergence Order ◽  
Well Posedness ◽  
Gradient Flows ◽  
Wasserstein Metric ◽  
Discrete Approximations ◽  
Variational Form

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity – but no smoothness – of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the L2-Wasserstein metric.

scholarly journals Regular and singular perturbations of upper semicontinuous differential inclusion

1997 ◽  
Vol 20 (4) ◽  
pp. 699-706 ◽  
Author(s):  
Tzanko Donchev ◽  
Vasil Angelov
Keyword(s):  
Differential Inclusion ◽  
Singular Perturbations ◽  
Differential Inclusions ◽  
Existence Result ◽  
Lower Semicontinuous ◽  
Solution Set ◽  
Singularly Perturbed ◽  
Solution Sets ◽  
Upper Semicontinuous ◽  
Continuity Properties

In the paper we study the continuity properties of the solution set of upper semicontinuous differential inclusions in both regularly and singularly perturbed case. Using a kind of dissipative type of conditions introduced in [1] we obtain lower semicontinuous dependence of the solution sets. Moreover new existence result for lower semicontinuous differential inclusions is proved.

scholarly journals A Bidirectional Flow Joint Sobolev Gradient for Image Interpolation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yi Zhan ◽  
Sheng Jie Li ◽  
Meng Li
Keyword(s):  
Texture Features ◽  
Energy Functional ◽  
Inner Product ◽  
Real Image ◽  
Gradient Flows ◽  
Sobolev Gradient ◽  
Forward Diffusion ◽  
Bidirectional Flow ◽  
Staircase Artifacts ◽  
Sobolev Inner Product

An energy functional with bidirectional flow is presented to sharpen image by reducing its edge width, which performs a forward diffusion in brighter lateral on edge ramp and backward diffusion that proceeds in darker lateral. We first consider the diffusion equations asL2gradient flows on integral functionals and then modify the inner product fromL2to a Sobolev inner product. The experimental results demonstrate that our model efficiently reconstructs the real image, leading to a natural interpolation with reduced blurring, staircase artifacts and preserving better the texture features of image.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

A Combination of the Work Formalism for Exchange with an Optimized Correlation Energy Functional for Atoms

1995 ◽  
Vol 5 (9) ◽  
pp. 1277-1287 ◽  
Author(s):  
N. A. Cordero ◽  
K. D. Sen ◽  
J. A. Alonso ◽  
L. C. Balbás
Keyword(s):  
Correlation Energy ◽  
Energy Functional

On the Essence and Tipology of International Public Goods

2005 ◽  
pp. 131-141
Author(s):  
V. Mortikov
Keyword(s):  
Economic Policy ◽  
Social Construction ◽  
Public Goods ◽  
Public Good ◽  
Global Public Goods ◽  
Club Goods ◽  
Basic Properties ◽  
International Public Goods ◽  
Regional Public Goods ◽  
Joint Products

The basic properties of international public goods are analyzed in the paper. Special attention is paid to the typology of international public goods: pure and impure, excludable and nonexcludable, club goods, regional public goods, joint products. The author argues that social construction of international public good depends on many factors, for example, government economic policy. Aggregation technologies in the supply of global public goods are examined.

Sign in / Sign up

Export Citation Format

Share Document