Fine and quasi connectedness in nonlinear potential theory

Adams, David R.; Lewis, John L.

doi:10.5802/aif.998

Cited by 13 publications

(14 citation statements)

References 16 publications

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“…C. Evans and R. F. Gariepy [137], H. Federer [141], y, G. Maz'ya [308], and W. P. Ziemer [438], and conceming Lebesgue points especially Theorem 4.5.9, statements (16), (21), and (22) in [141], p. 483, and Theorem 5.14.4 in [438]. However, the situation with regard to Lebesgue points is more complicated than in W1,1, and the set of exceptional points can have a-finite (N -1)-dimensional measure.…”

Section: Further Resultsmentioning

confidence: 99%

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Function Spaces and Potential Theory

Adams

Hedberg

1996

Grundlehren Der Mathematischen Wissenschaften

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997

1,156

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Softcover reprint of the hanlcover lst edition 1996 Cover design: MetaDesign plus GmbH, Berlin Photocomposed from the authors' TEX files after editing and reformatting by Kurt Mattes, Heidelberg, using the MathTime fonts and a Springer TEX macro-package Printed on acid-free paper SPIN: ]]304074 41/3111 -5432. 1 Preface Function spaces, especially those spaces that have become known as Sobolev spaces, and their natural extensions, are now a central concept in analysis. In particular, they play a decisive role in the modem theory of partial differential equations (PDE).Potential theory, which grew out of the theory of the electrostatic or gravitational potential, the Laplace equation, the Dirichlet problem, etc., had a fundamental role in the development of functional analysis and the theory of Hilbert space. Later, potential theory was strongly influenced by functional analysis. More recently, ideas from potential theory have enriched the theory of those more general function spaces that appear naturally in the study of nonlinear partial differential equations. This book is motivated by the latter development.The connection between potential theory and the theory of Hilbert spaces can be traced back to C. F. Gauss [181], who proved (with modem rigor supplied almost a century later by O. Frostman [158]) the existence of equilibrium potentials by minimizing a quadratic integral, the energy. This theme is pervasive in the work of such mathematicians as D. Hilbert, Ch.-J. de La Vallee Poussin, M. Riesz, O. Frostman, A. Beurling, and the connection was made particularly clear in the work of H. Cartan [97] in the 1940's. In the thesis of J. Deny [119], and in the subsequent work of J. Deny and J. L. Lions [122] in the early 1950's, this point of view, blended with the L. Schwartz theory of distributions, led to a new understanding of the function spaces of Hilbert type. According to the classical Dirichlet principle, one obtains the solution of Dirichlet's problem for the Laplace equation in a domain a by minimizing the Dirichlet integral, fg IVu(x)1 2 dx, over a certain class of functions taking given values on the boundary aa. The natural explanation is that solutions to the Laplace equation describe an equilibrium state, a state attained when the energy carried by the system is at a minimum.Also, the classical electrostatic capacity (or capacitance, as it is known to physicists) of a compact set K C R 3 differs only by a constant factor from C (K) = inf f g I vu (x) 1 2 dx, where the infimum is taken over all smooth compactly supported functions u, such that u :::: 1 on K.As emphasized by H. Cartan, taking these infima of the Dirichlet integral is equivalent to taking projections in a Hilbert space normed by the square root of the Dirichlet integral. VI PrefaceA slight modification of the nonn defined by the Oirichlet integral leads to the simplest of the function spaces treated in this work, the Sobolev space denoted by H' or W1,2. It is nonned by the square root of lIull~1 = fRN(lu12 + IVuI2)dx. This Hilbert space ...

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Section: Further Resultsmentioning

confidence: 99%

“…L. Lewis [21]. The arcs can even be chosen as unions of line segments parallel to the coordinate axes, and their lengths can be bounded.…”

Section: If E C R N Is (A P)-finely Open and (A P)-finely Connectedmentioning

confidence: 99%

Function Spaces and Potential Theory

Adams

Hedberg

1996

Grundlehren Der Mathematischen Wissenschaften

Self Cite

997

1,156

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“…The fine topology has been extensively studied in the context of non‐linear potential theory. For more details we refer the reader to and to the references therein. We recall here the following result, see [, Theorem 1.4 1.5], which deals with the compatibility of finely open sets with quasi open sets.…”

Section: Quasi Open and Finely Open Setsmentioning

confidence: 99%

A variational characterisation of the second eigenvalue of the p‐Laplacian on quasi open sets

Fusco

Mukherjee

Zhang

2019

Proc. London Math. Soc.

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“…their complement within U is also p-quasiopen), are the sets with zero p-capacity and their complements (within U ). This definition was also used by Adams-Lewis [1] in the nonlinear potential theory. Equivalently, U is pquasiconnected if it cannot be written as a union of two disjoint p-quasiopen sets with positive p-capacity.…”

Section: Introductionmentioning

confidence: 99%

A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

Björn¹,

Björn²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We show that every Sobolev function in W 1,p loc (U ) on a p-quasiopen set U ⊂ R n with a.e.-vanishing p-fine gradient is a.e.-constant if and only if U is p-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality. On unweighted R n , we also obtain the corresponding result for p-finely open sets in terms of p-fine connectedness, using a deep result by Latvala.

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Fine and quasi connectedness in nonlinear potential theory

Cited by 13 publications

References 16 publications

Function Spaces and Potential Theory

Function Spaces and Potential Theory

A variational characterisation of the second eigenvalue of the p‐Laplacian on quasi open sets

A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

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