Weakly Differentiable Functions, Softcover reprint of the original 1st ed. 1989
Sobolev Spaces and Functions of Bounded Variation
Graduate Texts in Mathematics Series, Vol. 120

The term "weakly differentiable functions" in the title refers to those inte­ n grable functions defined on an open subset of R whose partial derivatives in the sense of distributions are either LP functions or (signed) measures with finite total variation. The former class of functions comprises what is now known as Sobolev spaces, though its origin, traceable to the early 1900s, predates the contributions by Sobolev. Both classes of functions, Sobolev spaces and the space of functions of bounded variation (BV func­ tions), have undergone considerable development during the past 20 years. From this development a rather complete theory has emerged and thus has provided the main impetus for the writing of this book. Since these classes of functions play a significant role in many fields, such as approximation theory, calculus of variations, partial differential equations, and non-linear potential theory, it is hoped that this monograph will be of assistance to a wide range of graduate students and researchers in these and perhaps other related areas. Some of the material in Chapters 1-4 has been presented in a graduate course at Indiana University during the 1987-88 academic year, and I am indebted to the students and colleagues in attendance for their helpful comments and suggestions.

1 Preliminaries.- 1.1 Notation.- Inner product of vectors.- Support of a function.- Boundary of a set.- Distance from a point to a set.- Characteristic function of a set.- Multi-indices.- Partial derivative operators.- Function spaces—continuous, Hölder continuous, Hölder continuous derivatives.- 1.2 Measures on Rn.- Lebesgue measurable sets.- Lebesgue measurability of Borel sets.- Suslin sets.- 1.3 Covering Theorems.- Hausdorff maximal principle.- General covering theorem.- Vitali covering theorem.- Covering lemma, with n-balls whose radii vary in Lipschitzian way.- Besicovitch covering lemma.- Besicovitch differentiation theorem.- 1.4 Hausdorff Measure.- Equivalence of Hausdorff and Lebesgue measures.- Hausdorff dimension.- 1.5 Lp-Spaces.- Integration of a function via its distribution function.- Young’s inequality.- Hölder’s and Jensen’s inequality.- 1.6 Regularization.- Lp-spaces and regularization.- 1.7 Distributions.- Functions and measures, as distributions.- Positive distributions.- Distributions determined by their local behavior.- Convolution of distributions.- Differentiation of distributions.- 1.8 Lorentz Spaces.- Non-increasing rearrangement of a function.- Elementary properties of rearranged functions.- Lorentz spaces.- O’Neil’s inequality, for rearranged functions.- Equivalence of Lp-norm and (p, p)-norm.- Hardy’s inequality.- Inclusion relations of Lorentz spaces.- Exercises.- Historical Notes.- 2 Sobolev Spaces and Their Basic Properties.- 2.1 Weak Derivatives.- Sobolev spaces.- Absolute continuity on lines.- Lp-norm of difference quotients.- Truncation of Sobolev functions.- Composition of Sobolev functions.- 2.2 Change of Variables for Sobolev Functions.- Rademacher’s theorem.- Bi-Lipschitzian change of variables.- 2.3 Approximation of Sobolev Functions by Smooth Functions.- Partition of unity.- Smooth functions are dense in Wk,p.- 2.4 Sobolev Inequalities.- Sobolev’s inequality.- 2.5 The Rellich-Kondrachov Compactness Theorem.- Extension domains.- 2.6 Bessel Potentials and Capacity.- Riesz and Bessel kernels.- Bessel potentials.- Bessel capacity.- Basic properties of Bessel capacity.- Capacitability of Suslin sets.- Minimax theorem and alternate formulation of Bessel capacity.- Metric properties of Bessel capacity.- 2.7 The Best Constant in the Sobolev Inequality.- Co-area formula.- Sobolev’s inequality and isoperimetric inequality.- 2.8 Alternate Proofs of the Fundamental Inequalities.- Hardy-Littlewood-Wiener maximal theorem.- Sobolev’s inequality for Riesz potentials.- 2.9 Limiting Cases of the Sobolev Inequality.- The case kp=n by infinite series.- The best constant in the case kp = n.- An L?-bound in the limiting case.- 2.10 Lorentz Spaces, A Slight Improvement.- Young’s inequality in the context of Lorentz spaces.- Sobolev’s inequality in Lorentz spaces.- The limiting case.- Exercises.- Historical Notes.- 3 Pointwise Behavior of Sobolev Functions.- 3.1 Limits of Integral Averages of Sobolev Functions.- Limiting values of integral averages except for capacity null set.- 3.2 Densities of Measures.- 3.3 Lebesgue Points for Sobolev Functions.- Existence of Lebesgue points except for capacity null set.- Approximate continuity.- Fine continuity everywhere except for capacity null set.- 3.4 LP-Derivatives for Sobolev Functions.- Existence of Taylor expansions Lp.- 3.5 Properties of Lp-Derivatives.- The Spaces TktkTk,ptk,p.- The implication of a function being in Tk,pat all points of a closed set.- 3.6 An Lp-Version of the Whitney Extension Theorem.- Existence of a C? function comparable to the.- distance function to a closed set.- The Whitney extension theorem for functions in Tk,p and tk,p.- 3.7 An Observation on Differentiation.- 3.8 Rademacher’s Theorem in the Lp-Context.- A function in Tk,peverywhere implies it is in tk,palmost everywhere.- 3.9 The Implications of Pointwise Differentiability.- Comparison of Lp-derivatives and distributional derivatives.- If u ? tk,p(x)for everyxand if the.- LP-derivatives are in Lpthen u ? Wk,p.- 3.10 A Lusin-Type Approximation for Sobolev Functions.- Integral averages of Sobolev functions are uniformly close to their limits on the complement of sets of small capacity.- Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity.- 3.11 The Main Approximation.- Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity and are close in norm.- Exercises.- Historical Notes.- 4 Poincaré Inequalities—A Unified Approach.- 4.1 Inequalities in a General Setting.- An abstract version of the Poincaré inequality.- 4.2 Applications to Sobolev Spaces.- An interpolation inequality.- 4.3 The Dual of WM,p(?).- The representation of (W0M,p(?) )*.- 4.4 Some Measures in (W0M,p(?))*.- Poincaré inequalities derived from the abstract version by identifying Lebesgue and Hausdorff measure with elements in (WM,p(?))*.- The trace of Sobolev functions on the boundary of Lipschitz domains.- Poincaré inequalities involving the trace of a Sobolev function.- 4.5 Poincaré Inequalities.- Inequalities involving the capacity of the set on which a function vanishes.- 4.6 Another Version of Poincaré’s Inequality.- An inequality involving dependence on the set on which the function vanishes, not merely on its capacity.- 4.7 More Measures in (WM,p(?))*.- Sobolev’s inequality for Riesz potentials involving measures other than Lebesgue measure.- Characterization of measures in (WM,p(?))*.- 4.8 Other Inequalities Involving Measures in (WM,p)*.- Inequalities involving the restriction of Hausdorff measure to lower dimensional manifolds.- 4.9 The Case p= 1.- Inequalities involving the L1-norm of the gradient.- Exercises.- Historical Notes.- 5 Functions of Bounded Variation.- 5.1 Definitions.- Definition of BV functions.- The total variation measure ? Du?.- 5.2 Elementary Properties of BV Functions.- Lower semicontinuity of the total variation measure.- A condition ensuring continuity of the total variation measure.- 5.3 Regularization of BV Functions.- Regularization does not increase the BV norm.- Approximation of BV functions by smooth functions Compactness in L1of the unit ball in BV.- 5.4 Sets of Finite Perimeter.- Definition of sets of finite perimeter.- The perimeter of domains with smooth boundaries.- Isoperimetric and relative isoperimetric inequality for sets of finite perimeter.- 5.5 The Generalized Exterior Normal.- A preliminary version of the Gauss-Green theorem.- Density results at points of the reduced boundary.- 5.6 Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal.- Blow-up at a point of the reduced boundary.- The measure-theoretic normal.- The reduced boundary is contained in the measure-theoretic boundary.- A lower bound for the density of ?DXE?.- Hausdorff measure restricted to the reduced boundary is bounded above by ?DXE?.- 5.7 Rectifiability of the Reduced Boundary.- Countably (n — 1)-rectifiable sets.- Countable (n — 1)-rectifiability of the measure-theoretic boundary.- 5.8 The Gauss-Green Theorem.- The equivalence of the restriction of Hausdorff measure to the measure-theoretic boundary and ?DXE?.- The Gauss-Green theorem for sets of finite perimeter.- 5.9 Pointwise Behavior of BV Functions.- Upper and lower approximate limits.- The Boxing inequality.- The set of approximate jump discontinuities.- 5.10 The Trace of a BV Function.- The bounded extension of BV functions.- Trace of a BV function defined in terms of the upper and lower approximate limits of the extended function.- The integrability of the trace over the.- measure-theoretic boundary.- 5.11 Sobolev-Type Inequalities for BV Functions.- Inequalities involving elements in (BV(?))*.- 5.12 Inequalities Involving Capacity.- Characterization of measure in (BV(?))*.- Poincaré inequality for BV functions.- 5.13 Generalizations to the Case p> 1.- 5.14 Trace Defined in Terms of Integral Averages.- Exercises.- Historical Notes.- List of Symbols.