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Topics in Harmonic Analysis and Partial Differential Equations: Extension Theorems and Geometric Maximum Principles

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Topics in Harmonic Analysis and Partial Differential Equations: Extension Theorems and Geometric Maximum Principles TOPICS IN HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS: EXTENSION THEOREMS AND GEOMETRIC MAXIMUM PRINCIPLES A Thesis presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment of the Requirements for the Degree Master of Arts by RYAN ALVARADO Dr. Marius Mitrea, Thesis Supervisor MAY 2011 The undersigned, appointed by the Dean of the Graduate School, have examined the thesis entitled TOPICS IN HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS: EXTENSION THEOREMS AND GEOMETRIC MAXIMUM PRINCIPLES presented by Ryan Alvarado, a candidate for the degree of Master of Arts and hereby certify that in their opinion it is worthy of acceptance. Professor Marius Mitrea Professor Steve Hofmann Professor Dorina Kosztin ACKNOWLEDGEMENTS I would first like to thank my friends and family members for their continuous support over the years. Specifically, I would like to thank my mother and father for their guidance and encouragement; without them I certainly would not be where I am today. I must also give credit to Mayumi Sakata- Derendinger for cultivating my mathematical sense during my undergraduate career. On a final note, with up most sincerity, I would like to thank my advisor Marius Mitrea for his work ethic and advice throughout this entire process. ii Contents ACKNOWLEDGEMENTS ii LIST OF ILLUSTRATIONS iv ABSTRACT v CHAPTERS 1 1 Introduction 1 2 Structure of Quasi-Metric Spaces 22 3 Extensions of H¨olderFunctions on General Quasi-Metric Spaces 27 4 Separation Properties of H¨older Functions 34 5 Whitney-like Partitions of Unity via H¨olderFunctions 37 6 Whitney Decomposition in Geometrically Doubling Quasi-Metric Spaces 44 7 Extension of H¨olderFunctions in Geometrically Doubling Quasi-Metric Spaces 52 8 The Geometry of Pseudo-Balls 61 9 Sets of Locally Finite Perimeter 69 10 Measuring the Smoothness of Euclidean Domains in Analytical Terms 73 11 Characterization of Lipschitz Domains in Terms of Cones 87 12 Characterizing Lyapunov Domains in Terms of Pseudo-Balls 101 13 A Historical Perspective on the Hopf-Oleinik Boundary Point Principle 114 14 Boundary Point Principle for Semi-elliptic Operators with Singular Drift 123 15 Sharpness of the Boundary Point Principle Formulated in Theorem 14.3 146 16 The Strong Maximum Principle for Non-uniformly Elliptic Operators with Singular Drift 158 17 Applications to Boundary Value Problems 163 BIBLIOGRAPHY 170 VITA 175 iii LIST OF ILLUSTRATIONS Illustrations Page 1. Hour-glass shapes . 10 2. Threading of a surface between the two rounded components of an hour-glass shape . 11 3. One-component circular cones . 61 4. Pseudo-ball shape . 62 5. Overlapping pseudo-balls . 65 6. Link between the cone property and the direction of the unit normal . 71 7. Threading of a surface between two pseudo-balls . 104 iv TOPICS IN HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS: EXTENSION THEOREMS AND GEOMETRIC MAXIMAL PRINCIPLES Ryan Alvarado Dr. Marius Mitrea, Thesis Supervisor ABSTRACT The present thesis consists of two main parts. In the first part, we prove that a function defined on a closed subset of a geometrically doubling quasi-metric space which satisfies a H¨older-type condition may be extended to the entire space with preservation of regularity. The proof proceeds along the lines of the original work of Whitney in 1934 and yields a linear extension operator. A similar extension result is also proved in the absence of the geometrically doubling hypothesis, albeit the resulting extension procedure is nonlinear in this case. The results presented in this part are based upon work done in [8] in collaboration M. Mitrea. In the second part of the thesis we prove that an open, proper, nonempty subset of Rn is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The latter is a property of a purely geometrical nature, which amounts to the ability of threading the boundary, at any location, in between the two rounded components (referred to as pseudo-balls) of a certain fixed region, whose shape resembles that of an ordinary hour-glass, suitably re-positioned. The limiting cases of the result (pertaining to the curvature of the hour-glass at the contact point) are as follows: Lipschitz domains may be characterized by a uniform double cone condition, whereas domains of class C 1;1 may be characterized by a uniform two-sided ball condition. Additionally, we discuss a sharp generalization of the Hopf-Oleinik Boundary Point Principle for domains satisfying a one- sided, interior pseudo-ball condition, for semi-elliptic operators with singular drift. This, in turn, is used to obtain a sharp version of Hopf's Strong Maximum Principle for second-order, non-divergence v form differential operators with singular drift. This part of my thesis originates from a recent paper in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziad´e[9]. vi Chapter 1 Introduction This thesis consists of two distinct yet related parts. A common feature that both parts share is the emphasis on the geometrical aspects of the problem. In the first part, we address under what minimal geometrical conditions are we gauranteed an operator which extends functions with preservation of smoothness. In the second part of this thesis, we characterize various classes of domains in terms which are purely geometric. In turn, this allows us to investigate how the geometry of a given domain affects the nature of solutions to various partial differential equations. The work in Part I of my thesis is based on an article in partnership with M. Mitrea [8]. While the results in Part I are based on joint work with Mitrea, the second part originates from a recent paper in collaboration with D. Brigham, V. Maz'ya, M. Mitrea, and E. Ziad´e[9]. Over the years, results pertaining to extending classes of functions satisfying certain regularity properties (e.g. continuity, Lipschitzianty) from a subset of a space to the entire space while retaining the regularity properties have played a basic role in analysis. As is well known, a continuous function defined on a closed subset of a metric space may be extended with preservation of continuity to the entire space (see, e.g., [19, Exercise 4.1.F]). Indeed if E is a closed subset of a metric space (X; d) and f : E ! R is a continuous function then a concrete formula for a continuous extension of f to 1 X is given by Hausdorff's formula ( n o inf f(y) + d(x;y) − 1 : y 2 E if x 2 X n E; F (x) := distd(x;E) (1.1) f(x) if x 2 E where for x 2 X we define distd(x; E) := inf fd(x; y): y 2 Eg. Although for each fixed set E, the recipe presented in (1.1) yields a concrete formula for the extension function, the operator f 7! F is nonlinear. However this is an issue that was resolved , under suitable background assumptions on the ambient, in the proceeding years (cf. [91], on which we shall comment more on later). Regarding the extension of functions with higher regularity properties we wish to mention the pioneering work of E.J. McShane [68], H. Whitney [91] and M.D. Kirszbraun [58]. The main result in [68] is to the effect that if (X; d) is a metric space and if E ⊆ X is an arbitrary nonempty set then for any Lipschitz function g 2 Lip (E; d) there exists f 2 Lip (X; d) such that fjE = g and kgkLip (E;d) = kfkLip (X;d). Indeed, in [68], E.J. McShane took f to be either ∗ f (x) := sup g(y) − kgkLip (E;d) d(x; y): y 2 E ; 8 x 2 X (1.2) or similarly, f∗(x) := inf g(y) + kgkLip (E;d) d(x; y): y 2 E ; 8 x 2 X: (1.3) In fact, the extension functions constructed in (1.2) and (1.3) are maximal and minimal (respectively) in the following sense. If f 2 Lip (X; d) such that fjE = g and kgkLip (E;d) = kfkLip (X;d) then ∗ ∗ f ≤ f ≤ f∗ on X. However as before with (1.1), the extension operators g 7! f and g 7! f∗ are nonlinear. One of the important aspects of this result is that it is applicable to any nonempty subset of a general metrizable space. Concerning the non-linearity character of McShane's extension , in 1934, H. Whitney succeeded in constructing a linear extension operator in the Euclidean setting, which also preserves higher degrees of smoothness. Somewhat more specifically, in [91], Whitney gave necessary and sufficient conditions 2 α n on an array of functions ff gjα|≤m defined on a closed subset E of R ensuring the existence of m m α α a functions F 2 C (R ) with the property that (@ F )jE = f whenever jαj ≤ m. In addition, α Whitney's extension operator ff gjα|≤m 7! F is universal (in the sense that it simultaneously preserves all orders of smoothness), as well as linear. A timely exposition of this result may be found in [87]. The proof presented there makes use of three basic ingredients, namely: 1. the existence of a Whitney decomposition of an open set (into Whitney balls of bounded overlap), 2. the existence of a smooth partition of unity subordinate (in an appropriate, quantitative man- ner) to such a decomposition, and 3. differential calculus in open subsets of Rn along with other specific structural properties of the Euclidean space. Extension theorems are useful for a tantalizing array of purposes. On the theoretical side, such results constitute a powerful tool in the areas of harmonic analysis and partial differential equations, (cf, e.g., the discussion in [87], [62], [38]) while on the practical side they have found to be useful in applied math (cf.
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