Localization in fine potential theory and uniform approximation by subharmonic functions

It is shown how the cone $\text{l}$(U) of superharmonic functions ?0 on an open set U in $\text{R}$ⁿ, n ? 3, can be recovered from the cone $\text{l}$ of superharmonic functions ?0 on the whole of $\text{R}$ⁿ by a process involving the operator of localization associated with U. Actually we treat the more general case where U is open in the Cartan-Brelot fine topology on $\text{R}$ⁿ. As an application we obtain a new proof of a theorem of J. Bliedtner and W. Hansen on uniform approximation by continuous subharmonic functions in open sets containing a given compact set K in $\text{R}$ⁿ.     

Infima of superharmonic functions

Let Ω be a Greenian domain in ℝ ^d , d≥2, or—more generally—let Ω be a connected -Brelot space satisfying axiom D, and let u be a numerical function on Ω, , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and , whenever V is a relatively compact open set in Ω and x∈V.     

Nonlinear superharmonic functions and supersolutions

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations ${-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}$ and their generalizations ${-\nabla \cdot A(x,\nabla u) = \mu}$ ,where ${A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}$ , the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpeläinen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation ${-\nabla \cdot A(x,\nabla u) = f(x)}$ .     

Local strong uniqueness for nonlinear approximation

The local constant of strong uniqueness for nonlinear approximation with respect to a norm is the local constant of strong uniqueness for approximation in the associated problem of linear approximation by the tangent space.     

Compactness principles in nonlinear operator approximation theory

This paper is concerned with the approximate solution of nonlinear operator equations in abstract settings and with applications to integral and differential equations. A given operator with certain continuity and compactness or inverse compactness properties is a suitable limit of a sequence of operators with analogous properties which hold uniformly or asymptotically. Both fixed point equations and inhomogeneous equations are treated. Solutions of approximate problems converge to solutions of the given problem. This is an appropriate type of set convergence when solutions are not unique.     

Nonlinear parabolic capacity and polar sets of superparabolic functions

We extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the $p$ -Laplacian type. We study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential. As an application, we show that polar sets of superparabolic functions are of zero capacity. The main technical tools used include estimates for equations with measure data and obstacle problems.     

Local approximation of functions with several variables

This paper presents the problem of local approximation of scalar functions with several variables, including points of non-differentiability. The procedure considers that the mapping displays rates of change of power type with respect to linear changes in the coordinate domain, and the exponents are not necessarily integer. The approach provides a formula describing the local variability of scalar fields which contains and generalizes Taylor’s formula of first order. The functions giving the contact are Müntz polynomials. The knowledge of their coefficients and exponents enables the finding of local extremes including cases of non-smoothness. Sufficient conditions for the existence of global maxima and minima of concave-convex functions are obtained as well.     

Uniform-metric nonlinear approximation of functions from Besov-Lorentz spaces

A number of results on a nonlinear approximation in the uniform metric of functions in Besov-Lorentz spaces by means of their approximation by ϕ-polynomials and, in particular, by rational functions and splines, is obtained. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 61–81. Translated by O. A. Ivanov.     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.     

The Riesz decomposition of finely superharmonic functions

This paper answers a question of Fuglede about minimal positive harmonic functions associated with irregular boundary points. As a consequence, an old and central problem of fine potential theory, concerning the Riesz decomposition, is resolved. Namely, it is shown that, on certain fine domains, there exist positive finely superharmonic functions which do not admit any positive finely harmonic minorant and yet are not fine potentials.     

A connection between a general class of superparabolic functions and supersolutions

We show for a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super- and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.     

Generalized dirichlet problem in nonlinear potential theory

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA (x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting. Part of this research was performed in 1988–1989 while a visitor at Indiana University, Bloomington, Indiana.     

Stress potential in the nonlinear theory of viscoelasticity

By introducing a functional derivative and the line integral of a functional the author supplies an energy proof of the nonlinear theory of viscoelasticity. The stress potential for small strains, analogous to the potential proposed by Coleman for large strains [7], is considered for comparison.Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 4, pp. 633–642, July–August, 1970.     

Classification of Riemannian manifolds in nonlinear potential theory

The classification theory of Riemann surfaces is generalized to Riemanniann-manifolds in the conformally invariant case. This leads to the study of the existence ofA-harmonic functions of typen with various properties and to an extension of the definition of the classical notions with inclusionsO _G O _HP O _HB O _HD . In the classical case the properness of the inclusions were proved rather late, in the 50's by Ahlfors and Tôki. Our main objective is to show that such inclusions are proper also in the generalized case.This research was supported in part by grants from the Academy of Finland and the U.S. National Science Foundation (NSF DMS 9003438).