Related papers: The Richberg technique for subsolutions

The quasiharmonic approximation (QHA) is the simplest nontrivial approximation for interacting phonons under constant pressure, bringing the effects of anharmonicity into temperature dependent observables. Nonetheless, the QHA is often…

In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution…

In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fr\'echet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local…

We extend the notion of quasibounded harmonic functions to the plurisubharmonic setting. As an application, using the theory of Jensen measures, we show that certain generalized Dirichlet problems with unbounded boundary data admit unique…

We deduce an effective version of Schmidt's subspace theorem on a smooth projective variety X over function fields of characteristic zero for hypersurfaces located in N-subgeneral position with respect to X.

Let $d$ be a positive integer and $\mathbb H$ be an integrally closed subring of a global function field $F$. The purpose of this paper is to provide a general sieve method to compute densities of subsets of $\mathbb H^d$ defined by local…

This chapter sets out preliminaries for the duality theory in later chapters. An underlying idea is that local cohomology functors are higher derived functors of colocalizations (a.k.a.~coreflections). Predominantly well-known facts about…

In this paper, we propose a method of finding all coherent structures supported by a given nonlinear wave equation. It relies on enhancing the recent global bifurcation theory as developed by Dancer, Toland, Buffoni and others, by…

Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq…

Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general…

In this paper, the author derives an $O(h^4)$-superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet boundary…

A new and simple method for quasi-convex optimization is introduced from which its various applications can be derived. Especially, a global optimum under constrains can be approximated for all continuous functions.

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a…

We present an extension of the classical De Giorgi class, and then we show that functions in this new class are locally bounded and locally H\"older continuous. Some applications are given. As a first application, we give a regularity…

The main result of this paper is: {\bf Theorem.} Let $f:\mathbb{R}^k\rightarrow \mathbb{R}$ be a $C^{1}$ function, so that $\nabla f$ is locally Lipschitz continuous. Assume moreover that $f$ is $C^2$ near its generalised saddle points. Fix…

A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \(…

We develop a globalized Proximal Newton method for composite and possibly non-convex minimization problems in Hilbert spaces. Additionally, we impose less restrictive assumptions on the composite objective functional considering…

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…

In the Euclidean setting, the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method…