Necessary Conditions for Nonsmooth Optimization Problems with Operator Constraints in Metric Spaces

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of Banach spaces, which admit uniformly Gateaux differentiable renorms (in particular, in any separable spaces)

Existence of Biharmonic Curves and Symmetric Biharmonic Maps

The existence of curves and symmetric maps with minimal total tension is proved. Such curves and maps satisfy a class of fourth order differential equations. 1 Definitions of biharmonic maps and curves Let N be a Riemannian manifold embedded into the Euclidean space R m, m ≥ 2, and Ω a smooth bounded domain in R n, n ≥ 1. Given maps ϕ: ∂Ω → N and ψ: ∂Ω → TϕN (i.e., ψ (x) is tangent to N at ϕ (x) for x ∈ ∂Ω), we look for an “optimal ” map u: Ω → N such that u = ϕ, ∂u = ψ on ∂Ω, (1) ∂n where n is the exterior normal direction of ∂Ω. In other words, we look for a “best ” way to extend the boundary value ϕ with the prescribed normal derivative ψ. Typical examples of Ω and N are the unit ball and the unit sphere, respectively. In this case, ψ: ∂Ω → TϕN means ϕ (x) · ψ (x) = 0 for all |x | = 1. With the given Dirichlet data ϕ, the most natural extension is perhaps the harmonic map. Recall that a map u: Ω → N is harmonic if and only if its tension field T (u) vanishes. In terms of the second fundamental form A of N ⊂ R m, T (u) can be expressed as T (u) ≡ ∆u − A (u) (∇u, ∇u) , (2) where u is considered as a vector valued function from Ω to Rm, ∆u is the ordinary Laplacian of u, ∇u is the gradient of u, and A (u) (∇u, ∇u) is understood as the trace of A. However, with the normal derivative being prescribed, it is easy to see that a harmonic extension does not generally exist. In fact, it was shown in [ 6] that for almost all ϕ: ∂Ω → N, there is a unique energy minimizing harmonic extension u: Ω → N; therefore, ∂u ∂n has been determined by ϕ. In this paper, we seek an extension u of ϕ with ∂u ∂n = ψ that is as close to a harmonic map as possible. Specifically, we consider the total tension of u T (u) = |T (u) | 2 dx, (3

Removability of singular sets of harmonic maps

Abstract. We prove that a harmonic map with small energy and monotonicity property is smooth if its singular set is rectifiable and has a finite uniform density; moreover, the monotonicity property holds if the singular set has a lower dimension or its gradient has higher integrability. This work generalizes the results in [CL][DF][LG12], which were proved under the assumptions that the singular sets are isolated points or smooth submanifolds. § 1. Introduction. Suppose that m, n ≥ 2 are integers and 1 < p < ∞. Let Ω ⊂ R m be a bounded smooth domain and N ⊂ R n be a smooth compact submanifold. Denote by W 1,p (Ω, N) the set of all functions u ∈ L p (Ω, R n) with image in N and finite (p-)energy

General Algebraic and Differential Riccati Equations from Stochastic LQR Problems with Infinite Horizon

This is a continuation of the paper [12]. We consider general matrix Riccati equations, including those from stochastic linear regulator problems with infinite horizon. For differential Riccati equations, we prove a monotonicity of solutions, which leads to a necessary and sufficient condition for the existence of solutions to algebraic Riccati equations. For solutions to the algebraic Riccati equations, we obtain results on their comparison, uniqueness, stabilizability and approximation. 1

Harmonic maps with fixed singular sets

Here, for a smooth domain Ω in R m and a compact smooth Riemannian manifold N we study a space H consisting of all harmonic maps u: Ω → N that have a singular set being a fixed compact subset Z of Ω having finite m − 3 dimensional Minkowski content. This holds if, for example, Z is m − 3 rectifiable [F, 3.2.14]. We define a suitable topology on H using Hölder norms on derivatives weighted by powers of th

Multiple Solutions and Regularity of H-systems

The main result of this paper proves the existence of multiple solutions to a class of generalized constant mean curvature equations, called H-systems. Also contained is a regularity for conformal nharmonic maps