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An Introductory View of the Weak Solution of the P-Laplacian Equation

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An Introductory View of the Weak Solution of the P-Laplacian Equation An Introductory View of the Weak Solution of the p-Laplacian Equation Zhengyuan(Albert). Dong June 2017 A thesis submitted for the degree of Honours of the Australian National University Declaration The work in this thesis is my own except where otherwise stated. Zhengyuan(Albert).Dong Acknowledgements The author wishes to thank Prof.John Urbas for supervising this thesis and pro- viding numerous helps and guides throughout the honours year; the Mathematical Science Institute at the Australian National University for providing an office and all related amenities. v Abstract In this paper we will explore the various property of the weak solutions of the p-Laplacian equation: div(jDujp−2Du) = 0 for 1 < p < 2, including existence, uniqueness theory, differentiability and regu- larity results. vii Contents Acknowledgements v Abstract vii Notation and terminology xi 1 Weak Solutions and Preliminary Results 1 2 Regularity Theory 9 2.1 The case p > n ............................ 11 2.2 The case p = n ............................ 12 2.3 The case 1 < p < n .......................... 14 3 Differentiability 19 4 Regularity of the Derivatives 29 4.1 An apriori estimate on the oscillations of jDuj ........... 29 4.2 An A Priori H¨olderEstimate for Du ................ 41 4.3 Proof of Theorem 4.1 . 43 Bibliography 48 ix Notation and terminology In this paper I will mainly follow the notation convention used in [2]. Following are some main comment worth pointing out. • I will stick to the symbol Du and avoid using ru for the gradient of the function u, and the letter U for an open subset of Rn. • The letter C denotes various constants, it may not be the same constant when appearing in different lines of computation. In cases when I need to keep track of the constant, I will use C1;C2;::: to represent different constants. • For function spaces that specifies functions with compact support, I will stick to the subscript 0 notation instead of subscript c as preferred by some k k author. For example, C0 (U) denotes functions in C (U) with compact support. • For functions that are not pointwisely defined, we write \sup" and \inf" notation to mean the essential supreme and infimum, which are defined as: sup f ≡ ess sup f ≡ inffµ 2 Rj measff > µg = 0g inf f ≡ ess inf f ≡ − sup(−f): Since this definition coincides with the normal supremum and infimum in the classical sense, using the same notation causes no trouble. Other common notations are listed below. Notation 4pu the p-Laplacian operator, defined as p−2 4pu ≡ div(jDuj Du) = 0. xi xii NOTATION AND TERMINOLOGY th [u]C0,α(U) the α -Holder seminorm of u on U, defined as ju(x)−u(y)j [u]C0,α(U) ≡ supx;y2U jx−yjα . Usually I will write [u]Cα(U) for [u]C0,α(U). th kukC0,α(U) α -Holder norm, defined as kukC0,α(U) ≡ supx2U ju(x)j + [u]C0,α(U) = kukC(U) + [u]C0,α(U). (u)U Average of u on U, defined as 1 (u)U = U u(x)dx = jUj U u(x)dx. ffl ´ k;p kukW k;p(U) W norm of u : U ! R, defined as 8 1=p P α p < jα≤kj U jD uj dx ; (1 ≤ p < 1) kukW k;p(U) = : P ´ α : jα|≤k ess supU jD uj; (p = 1) !(n) Volume of n-dimensional unit ball. (this term will normally be absorbed into the constant term so we do not need the formula) Terminology compact embedding Let A and B be Banach spaces, we say A is compactly embedded in B, written A ⊂⊂ B provided 1. kakB ≤ CkakA (a 2 A) for some constant C and 2. each bounded sequence in A is precompact in B. Chapter 1 Weak Solutions and Preliminary Results The p-Laplacian equation div(jDujp−2Du) = 0 (1.1) reduces to the well-known Laplacian equation when p = 2, suggesting which it is generalised from, as the Laplacian equation is the Euler-Lagrange equation∗ for the Diricihlet integral 1 D(u) = jDuj2dx: (1.2) 2 ˆU Changing the square to a pth power we have the integral 1 I[u] = L(Du(x); u(x); x)dx = jDujpdx; 1 < p < 1: (1.3) ˆU p ˆU We now demonstrate that (1.1) is a the Euler-Lagrange equation for (1.3). 1 Let η 2 C0 (U) and suppose u is a minimizer of (1.3). Let i(τ) := I[u + τη](τ 2 R): (1.4) ∗See [2, Section 8.1 and 8.2] for a systematic introduction on Euler-Lagrange equation. 1 2 CHAPTER 1. WEAK SOLUTIONS AND PRELIMINARY RESULTS Computing the first variation (i.e. the derivative) explicitly, we get 1 i(τ) = jDu + τDηjpdx p ˆU !p=2 1 X = (D u + τD η)2 dx p ˆ i i U i p−2 ! 2 ! 1 X X ) i0(τ) = (D u + τD η)2 2(D u + τD η)D η dx ˆ 2 i i i i i Ω i i ! p−2 X = jDu + τDηj Du · Dη + τ jD ηj2 dx: ˆ i U i Since i(·) has a minimum at τ = 0, we have i0(0) = 0 = jDujp−2 Du · Dηdx: (1.5) ˆU If we assume, for now, that u is a smooth solution, then using integration by parts, we have jDujp−2 Du · Dηdx = η div(jDujp−2Du)dx ˆU ˆU = 0; (1.6) 1 there is no boundary term as η 2 C0 . Since (1.6) has to hold for all test functions η, we must have p−2 4pu ≡ div(jDuj Du) = 0 (1.7) showing that the minimizer of (1.3) is a solution of (1.1). Note that the computation above gives us a motivation for defining a weak solution as requiring only u 2 C1 or even u 2 C1 is too narrow for the treatment of such problem and clearly the less smoothness we assume of u to start with, more theory can be developed, hence a notion of weak derivative is suitable in this case: 1 Definition 1.1. Suppose u; v 2 Lloc(U). and α ia a multiindex. We say that v is the αth-weak partial derivative of u, written Dαu = v; provided uDαφ dx = (−1)jαj vφ dx ˆU ˆU 1 for all test functions φ 2 C0 (U). 3 To study function with this property systematically, we define the function space called the Sobolev space: Definition 1.2. The Sobolev space, denoted by W k;p(U) consists of all locally summable functions u : U ! R such that for each multiindex α wth jαj ≤ k, Dαu exists in the weak sense and belongs to Lp(U), equipped with the norm 8 1=p P α p < jα|≤k U jD uj dx (1 ≤ p < 1) kukW k;p(U) := P ´ α : jα|≤k ess supU jD uj (p = 1): Remark 1.3. • From the definition, it is clear that W k;p is a subspace of Lp and hence functions in Sobolev space are defined up to sets of measure zero. Moreover, the Sobolev space is also Banach and, in particular, is Hilbert when p = 2. • The weak derivative bears many properties the normal derivative has such as linearity (with respect to addition and constant multiplication); u 2 W k;p ) Dαu 2 W k−|αj;p for jαj ≤ k; product rule and chain rule. For a detailed exploration of the properties of Sobolev space, see [2, chapter 5]. It is clear that for the right side of (1.5) to make sense, the integrand needs to 1 p−1 be in L , in other words, we need at least Du 2 Lloc (U), which means the natural 1;p−1 space to seek a weak solution is Wloc (U). Unfortunately, little is known about the weak solution in this space, thus for the interest of this paper, we will instead 1;p study weak solutions in the space Wloc (U). We now have a precise meaning of the weak solution: Definition 1.4. Let U be a domain in Rn. We say that u 2 W 1;p is a weak solution of (1.1) in U, if jDujp−2 Du · Dη dx = 0 (1.8) ˆU 1 for each η 2 C0 (U). If, in addition, u is continuous, then we say that u is a p-harmonic function. We conclude the above computation in the next theorem: Theorem 1.5. The following conditions are equivalent for u 2 W 1;p(U): 4 CHAPTER 1. WEAK SOLUTIONS AND PRELIMINARY RESULTS (i). u is minimizing : jDujpdx ≤ jDvjpdx; when v − u 2 W 1;p(U) ˆ ˆ 0 (ii). the first variation vanishes: jDujp−2Du · Dη dx = 0; when η 2 W 1;p(U) ˆ 0 If, in addition, 4pu is continuous, the the conditions are equivalent to 4pu in U. Proof. \(i) ) (ii)" is already shown in the above computation. \(ii) ) (i)" Recall that for a convex function f : R ! R, f is convex if and only if for any a; b 2 R f(b) ≥ f(a) + f 0(a)(b − a): In the case where f : Rn ! R, the inequality becomes: f(b) ≥ f(a) + Df(a) · (b − a): Since j · jp is convex for p ≥ 1, then for f : x 7! jxjp, we have jbjp ≥ jajp + pjajp−2a · (b − a) (1.9) It follows that jDvjpdx ≥ jDujpdx + p jDujp−2Du · D(v − u): ˆU ˆU ˆU By letting η = v − u, (ii) implies jDvjpdx ≥ jDujpdx; ˆU ˆU which is (i) as needed.
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