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LECTURENOTE for the COURSE PARTIAL DIFFERENTIAL EQUATIONS 2, MATS340, 9 POINTS Contents 1. Introduction 3 2. Sobolev Spaces 3 2

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LECTURENOTE for the COURSE PARTIAL DIFFERENTIAL EQUATIONS 2, MATS340, 9 POINTS Contents 1. Introduction 3 2. Sobolev Spaces 3 2 LECTURENOTE FOR THE COURSE PARTIAL DIFFERENTIAL EQUATIONS 2, MATS340, 9 POINTS MIKKO PARVIAINEN UNIVERSITY OF JYVASKYL¨ A¨ Contents 1. Introduction3 2. Sobolev spaces3 2.1. Notations3 2.2. Reminders (from the Measure and Integration)4 2.3. Weak derivatives6 2.4. Approximations 17 2.5. Global approximation in Sobolev space 22 k;p 2.6. Sobolev spaces with zero boundary values: W0 (Ω) 24 2.7. Properties of W 1;p(Ω), 1 ≤ p < 1 26 2.8. Difference quotient characterization of Sobolev spaces 29 2.9. Sobolev type inequalities 33 2.9.1. Gagliardo-Nirenberg-Sobolev inequality, 1 ≤ p < n 33 2.9.2. Poincare's inequalities 39 2.9.3. Morrey's inequality, p > n 43 2.9.4. Lipschitz functions and W 1;1 45 2.10. Compactness theorem 46 3. Elliptic linear PDEs 47 3.1. Weak solutions 49 3.2. Existence 53 3.3. Variational method 56 3.4. Uniqueness 60 3.5. Regularity 62 3.5.1. Local L2-regularity 63 3.5.2. Global L2-regularity 70 3.5.3. Local Lp-regularity 72 3.5.4. Cα regularity using De Giorgi's method 73 3.6. Comparison and max principles 83 3.6.1. Strong maximum principle 85 Date: March 13, 2017. 1 2 PDE 2 4. Linear parabolic equations 87 4.1. Existence: Galerkin method 93 4.2. Standard time mollification 98 4.3. Steklov averages 99 4.4. Uniqueness 100 4.5. Cα regularity using Moser's method 102 4.6. H¨oldercontinuity 112 4.7. Remarks 114 5. Schauder estimates 115 6. Notes and comments 126 PDE 2 3 1. Introduction This lecture note contains a sketch of the lectures. More illustrations and examples are presented during the lectures. Partial differential equations (PDEs) have a great variety of applica- tions to mechanics, electrostatics, quantum mechanics and many other fields of physics as well as to finance. In addition, PDEs have a rich mathematical theory. In the ICM at 1900, a German mathematician David Hilbert published part of a nowadays legendary list of 23 mathematical problems that have been very influential for 20th century mathematics. We are interested in particular with the problems: (1) 20th problem: Has not every regular variational problem a solu- tion provided certain assumptions regarding the given boundary conditions, and provided that, if needed, the notion of solutions shall be suitably extended? (2) 19 th problem: Are the solutions of regular problems in the calculus of variations always necessarily analytic? Comments: • Variational problems and PDEs have a tight connection. We will return to this later. • As Hilbert suggested, in most of the cases we will have to relax the definition of the solution to PDEs to obtain existence of a solution. Still we would like to preserve the uniqueness and to some extend regularity and stability. These are the question swe will deal with in this course. 2. Sobolev spaces 2.1. Notations. DOM = Lebesgue's dominated convergence theorem; n Ω ⊂ R open set, bounded unless otherwise stated q 2 2 n jxj = x1 + ::: + xn for x 2 R ; m(E) = jEj = a Lebesgue measure of a set E Z 1 Z : : : dy = : : : dy B(0;") jB(0;")j B(0;") f :Ω ! R a function spt f = fx 2 Ω: f(x) 6= 0g = the support of f C(Ω) = ff : f continuous in Ωg 4 PDE 2 C0(Ω) = ff 2 C(Ω) : spt f is compact subset of Ωg Ck(Ω) = ff 2 C(Ω) : f is k times continuously differentiableg k k C0 (Ω) = C (Ω) \ C0(Ω) 1 1 k C (Ω) = \k=1C (Ω) = smooth functions 1 1 C0 (Ω) = C (Ω) \ C0(Ω) = compactly supported smooth functions Remark 2.1. Recall that u 2 Ck(Ω) () Dαu 2 C(Ω) n for multi-index α = (α1; : : : ; αn) 2 N and jαj := α1 + ::: + αn ≤ k, where @α1 @αn Dαu := ::: : α1 αn @x1 @xn Example 2.2 (Warning). It is not always the case that spt f ⊂ Ω. Example 2.3. (1) ( x2; x ≥ 0 f : R ! R; f(x) = −x2; x < 0 f 2 C1(Ω) n C2(Ω) (2) ( 1=(jxj2−1) n e ; jxj < 1 ' : R ! R;'(x) = 0; jxj ≥ 1: 1 ' 2 C0 (Ω); spt ' ⊂ B(0; 1) Exercise. 2.2. Reminders (from the Measure and Integration). Let E be Lebesgue measurable, 1 ≤ p ≤ 1, and f : E ! [−∞; 1] a Lebesgue measurable function. Then we define ( R p 1=p E jfj dx ; p < 1 jjfjjLp(E) = ess supE jfj; p = 1: where ess sup jfj := inffM : jfj ≤ M a.e. in Eg: E PDE 2 5 Then we define Lp(E) to be a a linear space of all Lebesgue measur- able functions f : E ! [−∞; 1] for which jjfjjLp(E) < 1: If we identify functions that coincide a.e., then this will be a Banach space with the norm defined above. We also recall p p Lloc(E) := ff : E ! [−∞; 1]: f 2 L (F ) for each F b Eg; where b means that F is a compact subset of E. Remark 2.4. There is usually no inclusions between Lp spaces: p q q p L * L L * L : This can be seen by recalling that xα 2 L1((0; 1)) () α > −1 xα 2 L1((1; 1)) () α < −1: Thus if we let 1 ≤ p < q ≤ 1 and choose β > 0 such that 1 1 − > −β > − q p we have x−β 2 Lp((0; 1)); but x−β 2= Lq((0; 1)) x−β 2= Lp((1; 1)); but x−β 2 Lq((1; 1)): Nonetheless, H¨older'sinequality is often a useful tool: jjfgjjL1 ≤ jjfjjLp jjgjjLq ; that is Z Z Z jfgj dx ≤ jfjp dx1=p jgjq dx1=q; where 1 ≤ p; q ≤ 1 are H¨older-conjugates that is 1 1 + = 1: q p This implies, in particular, for 1 ≤ p0 < q0 ≤ 1 and for a set jEj < 1 that f 2 Lq0 (E) ) f 2 Lp0 (E) 6 PDE 2 because (1 − p0=q0 = (q0 − p0)=q0) Z 0 Z 0 0 0 Z 0 0 0 jfjp dx ≤ 1q0=(q0−p0) dx(q −p )=q jfjq dxp =q E E E (q0−p0)=q0 p0 ≤ jEj jjfjj 0 : Lq (E) Also the following inequalities are worth recalling. Young's inequality: for each " > 0, 1 < p; q < 1; 1=p + 1=q = 1 and a; b ≥ 0 it holds ab ≤ "ap + Cbq; where C = C("; p; q) (meaning that C depends on the quantities in the parenthesis). Minkowski's inequality: for 1 ≤ p ≤ 1 and f; g 2 Lp(E) it holds that jjf + gjjLp(E) ≤ jjfjjLp(E) + jjgjjLp(E) : 1 1 2.3. Weak derivatives. Let u 2 C (Ω) and ' 2 C0 (Ω). Then by integrating by parts Z @' Z @u u dx = − ' dx; for i = 1; : : : ; n: Ω @xi Ω @xi Observe that ' vanishes at the boundary and thus there is no boundary term above. More generally for multi-index α, jαj ≤ k, and u 2 Ck(Ω), we have Z Z uDα' dx = (−1)jαj Dαu' dx: Ω Ω Remark 2.5. Observe that the left hand side does not require u to be continuously differentiable. This will be our starting point for defining weak derivatives for functions that are not continuous differentiable. 1 Definition 2.6. Let u; v 2 Lloc(Ω) and α a multi-index. Then v is αth weak partial derivative of u if Z Z uDα' dx = (−1)jαj v' dx; Ω Ω 1 for every test function ' 2 C0 (Ω). We denote Dαu := v: We denote weak partial derivatives with the familiar notation @u : @xi We also use @u @u Du = ( ;:::; ) @x1 @xn PDE 2 7 for the weak gradient. Example 2.7. ( x; 0 < x ≤ 1 u : (0; 2) ! R; u(x) = 1; 1 < x < 2: We claim that the weak derivative is ( 1; 0 < x ≤ 1 u0 = v = 0; 1 < x < 2: 1 This is in Lloc so by definition, the task is to show that Z Z v' dx = −1 u'0 dx: (0;2) (0;2) To see this, we calculate Z Z Z u'0 dx = u'0 dx + u'0 dx (0;2) (0;1) (1;2) = u(1)'(1) − u(0)'(0) + u(2)'(2) −u(1)'(1) | {z } | {z } 0 0 Z Z − u0 ' dx − u0 ' dx |{z} |{z} (0;1) 1 (1;2) 0 Z = − ' dx (0;1) Z = − v' dx: (0;2) Note that above u2 = C1((0; 2)) and u0 2= C((0; 2)). Also observe that weak derivatives are only defined a.e. and thus it is irrelevant what is the point value for example at 1. We found one weak derivative but could there be several? Answer: No, weak derivatives are unique up to a set of measure zero. Theorem 2.8. A weak αth derivate of u is uniquely defined up to a set of measure zero. 1 Proof. Suppose that v; v 2 Lloc(Ω) satisfy Z Z uDα' dx = (−1)jαj v' dx Ω Ω Z = (−1)jαj v' dx Ω 8 PDE 2 1 for all ' 2 C0 (Ω). It follows that Z (v − v)' dx = 0 Ω 1 for every ' 2 C0 (Ω). This implies that v = v a.e. by the following reason: 0 1 0 1 0 Let Ω b Ω and observe that C0 (Ω ) is dense in L (Ω ). Indeed, then there exists 1 0 'i 2 C0 (Ω ); j'ij ≤ 2 such that 0 'i ! sign(v − v) a.e. in Ω ; (more about approximations later) where 8 1 x > 0 <> sign(x) = 0 x = 0 :>−1 x < 0: Then Z 0 = lim (v − v)'i dx i Ω0 DOM, below Z = lim(v − v)'i dx Ω0 i Z = (v − v) sign(v − v) dx Ω0 Z = jv − vj dx; Ω0 1 0 where the use of DOM is based on j(v − v)'ij ≤ 2(jvj + jvj) 2 L (Ω ).
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