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Advanced Partial Differential Equations Prof. Dr. Thomas

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Advanced Partial Differential Equations Prof. Dr. Thomas Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1. Existence of solutions 2. Uniqueness of solutions 3. Continuous dependence on the data. Also “properties of solutions” were important (regularity, estimates, mean value formulae, infinite/finite propagation speed and “techniques”). Recall that on existence, two types of results exist (1) Write down explicit solution formula (i.e. derive it – somehow – then prove that it does produce a solution). We did this “several times”, but in very specific cases and n 2 n very limited! For example for the Laplace equation −∆u = f in R , for f 2 Cc (R ). n We studied harmonic polynomials, i.e. ∆u = 0 in R with u a polynomial. Then we studied the problem in the half-space ( n ∆u = 0 in R+ n u = g on @R+ n−1 1 n−1 for g 2 C(R ) \ L (R ). The same for a Ball and g 2 C(@B(0; r)). Or the heat equation ( n ut − ∆u = 0 in R × (0; 1) n u = g on R × ft = 0g n 1 n for g 2 C(R ) \ L (R ). Or ( n ut − ∆u = f in R × (0; 1) n u = 0 on R × ft = 0g and combinations. Also the wave equation was studied ( utt − uxx = 0 in R × (0; 1) u = g; ut = h on R × ft = 0g and on R+ × (0; 1) 8 >utt − uxx = 0 in + × (0; 1) <> R u = g; ut = h on R+ × ft = 0g > :>u = 0 on fx = 0g × (0; 1) and for dimensions n > 2 odd/even. I.e. 3 (a) Equations are very specific (not general elliptic/parabolic/hyperbolic) n (b) The geometry is very specific (all of space (R ), half-space, ball). (2) One exception: Perron’s Method for ( ∆u = 0 in U u = g on @U for g 2 C(@U) guaranteed existence for every g 2 C(@U) provided some (complicated) condition on @U is satisfied (certain simple geometric conditions (on @U) can assure that this holds). More general geometry – but still a very specific equation. Recall that in fact we are interested in (homogeneous/inhomogeneous) Lu = 0, Lu = f (elliptic) n in U ⊆ R or ut − Lu = 0, ut − Lu = f (parabolic) in U × [0;T ] or utt − Lu = 0, utt − Lu = f (hyperbolic) in U × [0;T ] with n n X X Lu = aij(x)uxixj + bi(x)uxi + c(x)u(x) i;j=1 i=1 and (maybe) conditions on aij; bj; c. Hence, in PDE 2, we shall concentrate on methods to existence of solutions – mostly (only...) illustrated on elliptic linear 2nd order equations Lu = f. Possibly we also discuss parabolic/hyperbolic equations. Learning techniques, methods, and ideas that can often be applied also for other equations – different order, and more or less nonlinear. Two main ideas (1) Extend the concept of a “solution” (from “classical solutions”, as in PDE 1) (2) Use some Functional Analysis. Today (and next times), we will start by saying something (more or less vague) about these two ideas: “why & how”. Depending on where/with whom you did PDE 1, you might know some of this. There is no hope to solve “every” PDE by finding a solution. Therefore, we need something more systematic/general, which enters Functional Analysis. Functional Analysis (linear!) is the study of (1) linear (infinite dimensional) spaces, with a topology (i.e. notion of convergence/approxi- mation): Banach spaces, Hilbert spaces, Fréchet-spaces, locally convex topological vector spaces. (a) In general (i.e. all Banach spaces) (b) Specific examples (i.e. Lp(Ω); `p). (2) Linear maps (mostly continuous linear maps) between such spaces: T : X −! Y . The idea of using Functional Analysis is to reformulate the question “Does a solution to F (:::) = 0 (PDE) exist?” as “Does the map F : X −! Y vanish somewhere?” m n m n Note that the study of f : R −! R, f : R −! R (1) (nonlinear) and A: R −! R (2) (linear) in (1) is Analysis I-III and (2) is Linear Algebra. This is also motivated by questions about solutions to equations (so is “all” of abstract algebra). Hence, the “goal” is to reformulate questions in PDE via FA. Nevertheless, this is not an FA-course! We will need 4 certain tools which should be known from FA 1. If need be, we can do some in the Tutorials. We need to fix appropriate vector spaces of functions (“function spaces”). A natural choice is Ck(U) or Ck(U) with supremum-norm (sup-norm). kfk = P kDαfk . Ck(U) jαj6k L1(U) We get a Banach space (i.e. complete). However, the relevant operators (relevant for PDE) T : X −! Y turn out not to be “well-behaved” in this setup (maybe more precise later). One possible replacement is Hölder-spaces. n Definition 0.1. Let U ⊆ R open, 0 < γ 6 1. γ (i) If u: U −! R and there is a constant C > 0 such that ju(x) − u(y)j 6 Cjx − yj for every x; y 2 U then u is Hölder-continuous with exponent γ. (ii) If u: U −! R is bounded and continuous, we let kukC(U) := supx2U ju(x)j and if u is also Hölder-continuous with exponent γ, we let ju(x) − u(y)j [u]C0,γ (U) := sup γ x;y2U jx − yj x6=y (the γ’th Hölder semi-norm) and kukC0,γ (U) := kukC(U) + [u]C0,γ (U) (the γ’th Hölder- norm of u). k,γ (iii) The Hölder-space C (U), k 2 N0; γ 2 (0; 1] is defined as k,γ k C (U) := fu: U −! R j u 2 C (U); kukCk,γ (U) < 1g where the norm k · kCk,γ (U) is X α X α kukCk,γ (U) := kD ukC(U) + [D u]C0,γ (U jαj6k jαj=k k,γ Theorem 0.2. (C (U); k · kCk,γ (U)) is a Banach space. Furthermore, for many problems in PDE (especially elliptic), these spaces are “good” for studying PDE’s (i.e. the operators T : X −! Y one needs to study have “good” properties). For example one can prove “Elliptic Schauder-estimates”. However, this is often very, very hard work – this is not easy to explain, but basically, the reason is the sup-norm is hard to deal/work with – norms given by integrals are much, much “better”. So the idea is to take Ck(U) – or Ck,γ(U) – and use an integral norm – say, Lp-norm (p 2 [1; 1)): P α P α kD uk p instead of kD uk . The problem is: With these norms, we jαj6k L (Ω) jαj6k C(U) do not get a complete space (i.e. Ck(U), or Ck,γ(U) with these norms are not Banach spaces). The solution could be to abstractly “complete” the space (always possible!) but this is not a good solution if we have no concrete representation (of added objects). It turns out that one does have one. Q is not complete – R is – and “added” objects are still numbers. Riemann integrable functions on [0; 1] – R[0; 1] – with L1-norm – are not complete. But L1[0; 1] is complete – the “added” objects are still “functions”. This leads to the definition of “weak derivatives” and “Sobolev spaces” (next chapter). Note that this leads to proofs of existence of solutions to PDE’s which might be one of the “new added functions” – called “weak solutions” – and not a “true” classical solution. Secondly (often relatively independent) is afterwards to study regularity: Prove that this weak solution is in fact a true Ck-function (and that solves equation as a classical solution). I.e. Sobolev spaces are a tool to prove existence of some kind of solutions (“generalised solutions”), but the real interest is in classical solutions (always) regularity studies.
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