A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms

Let be a bounded domain such that . We obtain existence of sign-changing solutions for the Dirichlet problem on Ω,u=0 on ∂Ω for suitable positive numbers μ and λ.     

Randomized approximation of Sobolev embeddings, III

We continue the study of randomized approximation of embeddings between Sobolev spaces on the basis of function values. The source space is a Sobolev space with nonnegative smoothness order; the target space has negative smoothness order. The optimal order of approximation (in some cases only up to logarithmic factors) is determined. Extensions to Besov and Bessel potential spaces are given and a problem recently posed by Novak and Woźniakowski is partially solved. The results are applied to the complexity analysis of weak solution of elliptic PDE.     

覆盖下近似算子的拓扑性质

在不限制U为有限论域的情况下,研究了覆盖下近似算子XL和CL的拓扑性质。证明了覆盖下近似算子XL是内部算子,而且由XL生成的拓扑TXL为包含由覆盖C本身作为子基生成的拓扑TC的最小Alexandrov拓扑。特别地,当U为有限论域时,TXL=TC.然而,覆盖下近似算子CL不是内部算子。当覆盖C为某拓扑的基时,CL是内部算子,且此时由CL生成的拓扑TCL与TC是同一个拓扑。若进一步要求U为有限论域,则TCL=TXL=TC,进而CL=XL.     

Randomized approximation of Sobolev embeddings, II

We study the approximation of Sobolev embeddings by linear randomized algorithms based on function values. Both the source and the target space are Sobolev spaces of non-negative smoothness order, defined on a bounded Lipschitz domain. The optimal order of convergence is determined. We also study the deterministic setting. Using interpolation, we extend the results to other classes of function spaces. In this context a problem posed by Novak and Woźniakowski is solved. Finally, we present an application to the complexity of general elliptic PDE.     

Sobolev spaces on Lie groups: Embedding theorems and algebra properties

Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure ${\mu }_{\chi }$ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces $L_{\alpha }^p({\mu }_{\chi })$ adapted to X and ${\mu }_{\chi }$ ($1<p<\infty$, $\alpha \ge 0$) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group.     

Approximation numbers of Sobolev embeddings—Sharp constants and tractability

We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d)$H^s\left({\mathbb{T}}^d\right)$ of fractional smoothness s>0$s>0$ for various equivalent norms including the classical one. The error is always measured in _L2(T^d)$L_2\left({\mathbb{T}}^d\right)$. Particular emphasis is given to the dependence of all constants on the dimension d$d$. We capture the exact decay rate in n$n$ and the exact decay order of the constants with respect to d$d$, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.     

Existence of solutions for the p-Laplacian with critical Sobolev exponent and convection

In this article, we study the existence of solutions for the p-Laplacian involving critical Sobolev exponent and convection based on the theory of the Leray–Schauder degree for non-compact mappings.     

Limit holonomy and extension properties of Sobolev and Yang-Mills bundles

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally L^m/2-integrable curvature and trivial limit holonomy extends through singularity of codimension 2. Additionally, if the connection satisfies the Yang-Mills-Higgs equation, the extension also satisfies the equation.     

A Property of Sobolev Spaces and Existence in Optimal Design

   Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W ₀ ^l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary dimension and with general cost functionals.     

A Property of Sobolev Spaces and Existence in Optimal Design

Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W ₀ ^l,p (Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary value problems of arbitrary order, in arbitrary dimension and with general cost functionals.     

Existence of positive solutions for a semilinear elliptic problem with critical Sobolev and Hardy terms

Let , let and let be a bounded domain with a smooth boundary . Our purpose in this paper is to consider the existence of solutions of the problem:

where

     

The Mumford relations and the moduli of rank three stable bundles

The cohomology ring of the moduli space M(n,d) of semistable bundles of coprime rank n and degree d over a Riemann surface M of genus g 2 has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when n>2. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these dual Mumford relations form a complete set for the rational cohomology ring of M(3,d) and show (Sect. 5) that the Pontryagin ring vanishes in degree 12g-8 and above.     

Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Let Ω be a smooth bounded domain in , with N?5, a>0, α?0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem

     

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel‐Lizorkin spaces on manifolds, improving [11].