=====Lecture Plan=====

  * I will update the plan as I lecture.

^ Week  ^ Topic  ^ Chapter  ^ Pages  ^ Exercises  ^ Remarks  ^
^ 2  | **Introducution**  \\ **The basics:**  \\ Metric and Banach spaces, duals,  \(\ell^p\),  \\ Hahn-Banach and consequences  | 1  \\ 2  | \\ \\ p. 5-8,  \\ Theorem A.1 (appendix),  \\ Brezis chp 1.1: Corollay 1.2-1.4  \\ \\ | E1: Let \(a,b\geq 0, \frac1p+\frac1q=1\). Prove Young's ineq: \[ab\leq \dfrac {a^p}p +\dfrac{b^q}q.\] Hint: \(ab=e^{\frac1p\ln a^p+\frac1q\ln b^q}\) + convexity of exp.  \\ \\ E2: Let \(\varepsilon>0\). Prove \(ab\leq \varepsilon\frac {a^p}p +\frac{b^q}{q\varepsilon^{q-1}}\)  \\ E3: Prove H\"older in \(\ell^p\): \(|\sum_n x_ny_n|\leq \|x\|_p\|y\|_q\). Hint: Use Young and/or Google.  \\ \\ E4: Prove \(\|x\|_\infty\leq\|x\|_p\leq\|x\|_q\leq\|x\|_1\) for \(1<p<q<\infty\).  \\ \\ E5: Holden Ex 4 p. 33 (see also hints in Appendix): Prove that \(\ell^p\) and \(\ell^q\) are dual.   \\ | **Introduction meeting.**  \\ 1 lecture.  \\ \\ \\ \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week2.pdf|Solutions]] to the exercises (by Fredrik). |
^ 3  | Conclusion to Hahn-Banach,  \\ separable and reflexive spaces, \\ \\ Compactness in metric spaces,  \\ weak convergence, relation to strong convergence, \\ Banach-Steinhaus without proof  \\ \\  |  | p. 7-10, 15-16  | E1: Show that \((c_0)'=\ell^1\) (Holden Ex 4 p 33) \\ \\  E2: Show \(\ell^\infty\) is not separable.  \\ Hint: For any countable \(\{x_i\}\subset \ell^\infty\), find \(y\in\ell^\infty\) s.t. \(\|x_i-y\|_\infty\geq 1\) for all \(i\).  \\ \\ E3: Show that \(D\) is dense in \(\ell^p\) for \(p\in[1,\infty)\) when \[D=\{x=\{x_k\}_k : x_k\in \mathbb{Q}, \text{finite number of } x_k\neq 0\}\] Hint: The set can be seen as a countable union of countable sets (why?) and is hence countable (you do not need to provethe latter claim).  \\ \\ E4: Let \(x_n=\{x_{n,k}\}_n\in \ell^1\) be defined by \(x_{n,k}=1\) for \(n=k\) and \(0\) otherwise.  \\ \\ (a) Show that \(x_n\) does not converge weakly in \(\ell^1\).   \\ \\ (b) Show that \(x_n\) converge weakly * in \(\ell^\infty\). Explain what this convergence is and why it does not imply weak convergence in this case.  \\ \\ E5: Prove that the weak limit is unique (Ex 2 in Holden p 33)  \\ \\ E6: Ex 5 in Holden p 33    | **Conclusion** of cl. subsp. \(X_0\) of reflexive sp. is reflexive (Prop. 20):  \\ \\ For any \(x_0''\in X_0''\), we showed that for there is \(x\in X_0\) s.t. \[<x_0'',x'>=<x',x>\ \forall\ x'\in RX'.\]  //To conclude we note that// \(RX'=X_0'\).  \\ \\ \\ **Holden p 16 - Proof of Prop 2.18:** The inductively defined balls must be chosen such that they contain infinitely many elements of the sequence that lies in the intersection of all the previously defined balls.  \\ \\ \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week3.pdf|Solutions]] to the exercises (by Sebastian).  \\ \\ |
^ 4  | Weak compactnes, Eberlein-Smuljan's thm,  \\ weak * convergence, Helley's thm,  \\ \\ Alaoglu's thm, strong compactness for functions: Arzela-Ascoli's thm  \\ \\   | \\   | p 11-17\\ \\ | E1: State and prove Proposition 2.12 p. 14 in the Holden note.  \\ \\ E2: State and prove Proposition 2.14 p. 15 in the Holden note.  \\ \\ E3: Let \(\{x_n\}_n\subset\ell^1\) be defined by \(x_{n,k}=1\) when \(n=k\) and \(0\) otherwise. Prove that \(x_n\overset{*}{\rightharpoonup} 0\) in \(\ell^1\).  \\ \\ Note that by last weeks problems, it does not converge weakly in \(\ell^1\)!  \\ \\ //Hence we have an example showing that weak * convergence is weaker than weak convergence.//   \\ \\ E4: Let \(x:[0,T]\to I\!R^n\) be the solution of the ODE \(\dot x=f(x),\ x(0)=x_0\). We will work with the integral form \[x(t)=x_0+\int_0^tf(x(s))ds,\quad t\in[0,T].\] The corresponding forward Euler discretisation is \[y(t)=y(n\Delta t) + (t-n\Delta t)f(y(n\Delta t)),\ t\in[n\Delta t,(n+1)\Delta t],\] \(\Delta t=\frac TN\), and \(y(0)=x_0\). Note that $y$ is a continuous function coinciding with the Euler approximation at the points \(n\Delta t\). Assume that \(f\) is Lipschitz, \[|f(x)-f(y)|\leq L_f|x-y|, \quad x,y\in I\!R^n.\] Prove the convergence of this method by the following steps:  \\ \\ (a) Show by a direct argument that \[|y(t)|\leq |x_0|e^{L_fT}+|f(0)|\int_0^Te^{L_ft}dt=:M,\ t\in[0,T].\]  (b) Show by a direct argument that \[|y(t)-y(s)|\leq |t-s|\max_{|r|\leq M}|f(s)|,\ t\in[0,T].\] c) Let \(\Delta t=\frac TN, N=1,2,3,\dots\) and \(y=y_{\Delta t}=y_N\).  \\ \\ Use the Arzela-Ascoli theorem to find a subsequence of \(\{y_{N_k}\}_{N_k}\subset\{y_N\}_{N}\) and continuous function \(\bar y\) such that \(y_{N_k}\to \bar y\) uniformly on \([0,T]\).  \\ \\ (d) Verify that the uniform limit \(\tilde y\) of //any// subsequence \(\{y_N\}_N\) from c) is a solution of the ODE in integral form.  \\ (Hence also the subsequence found in c)).  \\ \\  (e) Since the ODE has a unique solution (\(f\) is Lipschitz), conclude that the whole sequence converges.  \\ \\ Hint: Use the argument for the corollary/2nd part of the Eberlein-Smuljan theorem.  \\ \\ | **Solutions** to the exercises (by David and Sølve):  \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week4_1.pdf| E1-E3]] and [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week4_2.pdf| E4]].  \\ \\ |
^ 5  | **Distribution theory**  \\ Definitions, properties,  \\ operations, regular/singular  \\ \\ operations(cont.), derivatives of regular distr.,  \\ the fundamental theorem, convolution  \\ \\  | 3  | \\ \\ | E1: Holden Ex 1 p 52.  \\ \\ E2: Prove that \(T\in D'\) continuous iff \(T\in D'\) continuous at \(0\).  \\ \\ E3: Prove that \(D'\) is a vector space.  \\ I.e. prove that it is closed under addition and scalar multiplication.  \\ \\ E4: Prove that for a regular distribution, \(T_f=0\) iff \(f=0\) a.e. (\(f\in L^1_{loc}\)). \\ This is Ex 3 p 52 in Holden.  \\ \\ E5: Prove that \(T_3=\sum_{n=1}^{\infty}\delta_{\frac1n}\) belongs to \(D'(0,2)\).  \\ (Note that it does //not// belong to \(D'(IR)\)).  \\ \\ E6: Prove that \(\partial^\alpha T \in D'\) for any \(T\in D'\). \\ Hint: Verify that is it well-defined, linear and continuous.  \\ \\ E7: Prove that \(T(\phi)=\sum_{n=1}^\infty \phi^{(n)}(n)\) defines a distribution on \(I\!R\).  \\ \\ E8: Holden Ex 5 p 52, first derivative only.  \\ \\  | **Remark:** We define convolution both as a function as in Holden and as a distribution (see e.g. Wikipedia). The two definitions are related, one is the "density function" of the other.  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week5.pdf|Solutions]] to the exercises (by Abdullah).  |
^ 6  | Convolutions (cont.), convergence,  \\ approximations  \\ \\ Primitive in 1D, equations in \(D'\),  \\ fundamental solutions  | | | E1: Prove that \(C_ST=C_TS\) for all \(S,T\in D'\) with compact support.  \\ \\ E2: Prove that \(f,f_n\in L^1_{loc}\) and \(\int_{|x|<R}|f(x)-f_n(x)|dx\to 0\) for all \(R>0\), implies that \(f_n\to f\) in \(D'\).  \\ \\ E3: Prove that \(\eta(\frac xn)(\psi_n*T) \to T\) in \(D'\) when \(\eta,\psi\in C_c^\infty\), \(\eta=1\) for \(|x|<1\), \(\psi\geq0\), \(\int \psi=1\), \(\psi_n(x)=n^d\psi(nx)\).  \\ Hint: Use that fact that the result holds if \(\eta\) is replaced by \(1\).  \\ \\ E4: Define \(D'\) as the \(D'\) limits of \(C_c^\infty\) functions (\(T\in D'\) if there is \(\{\psi_n\}\subset C_c^\infty\) such that \(\psi_n\to T\) in \(D'\)). By a theorem in class, such a limit is a continuous and linear functional on \(C_c^\infty\). Define the derivative of \(T\) in the following way: \[\partial_i T(\phi)=\lim_n \int\partial_i\psi_n\phi\,dx.\] Show that then \(\partial_tT(\phi)=-T(\partial \phi)\) by passing to the limit.  \\ Conclude that this definition of derivative does not depend on the approximating sequence \(\{\psi_n\}_n\).  \\ \\  E5: Show that \(\frac12 e^{-|x|}, x\in I\!R\) belongs to \(L^1_{loc}\) and is a fundamental solution of \(L=1-\partial^2\), i.e. \[\int u (1-\partial^2)\phi dx = \phi(0) \quad \text{for all}\quad \phi\in C_c^\infty.\] Hint: Use similar ideas as in the Holden note: truncation of domain and integration by parts.  \\ \\ E6: Solve the equation \[T''-2T'=\delta''\quad\text{in}\quad D'.\] Hint: Integrate once, then use integrating factor. The answer should be \(T=\delta+2e^{2t}H+K_1+e^{2t}K_2\) where \(K_1,K_2\) are arbitrary constants.  \\ \\ | **OBS:** Mistake in definition of convolution of distributions. Correct definition should be: \[C_ST(\phi)=T(\phi*S_\sigma)\] for \(T,S\in D'\) and \(S\) has compact support.  \\ \\ **OBS2:** Correction in problem E6.  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week6.pdf|Solutions]] to the exercises (by Fredrik Hoeg).  |
^ 7  | \\ **Lebesgue Spaces**  \\ Strong and weak \(L^p\), properties, inequalities. \\ \\ Convolutions, compactness,  \\ Arzela-Ascoli and Kolmogorov-Riesz.   | \\ 4  |   \\ \\ | E1: Prove Holder's inequality \(\|fg\|_1\leq\|f\|_p\|g\|_q\).  \\ Hint: Exponential + Young, proof as in  \(\ell^p\).  \\ \\ E2: (Ex (4) in Holden) Prove Minkowski's inequality \(\|f+g\|_p\leq \|f\|_p+\|g\|_p\).  \\ Hint: \(|f+g|^p\leq |f+g|^{p-1}(|f|+|g|)\) + Holder.  \\ \\ E3: (Ex (3) in Holden) Prove the generalized Holder inequality.  \\ Hint: We did the case of two functions in class, use this result and induction.  \\ \\ E4: Prove \(\|f*g\|_1\leq \|f\|_1\|g\|_1\). Hint: Tonelli.  \\ \\ E5: Prove Young's 2nd inequality \(\|f(g*h)\|_1\leq\|f\|_p\|g\|_q\|h\|_r\) using Holder's inequality and Young's 1st inequality for convolutions.  \\ \\  | [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week7.pdf|Solutions]] to the exercises (by Sondre).  |
^ 8  | Approximation in \(L^p\), Kolmogorov (cont.).  \\ \\ Kolmogorov (cont.), modes of convergence     | | p 86-89  \\ \\ p 59-63  | E1: Prove Lemma 26 from the lectures: Define the cut-off function \(\phi_j\) and show that \[\|f\phi_j\|_p\leq \|f\|_p,\quad\|f-f\phi_j\|_p\to 0,\quad p\in[0,\infty). \] Explain also the \(L^\infty\)-case.  \\ \\ E2: Kolmogorov-Riesz 1: \(L^p\)-equiboundedness and -equicontinuity of \(\mathcal F\subset L^p\) implies total boundedness of the restriction of \(\mathcal F\) to any bounded set \(\Omega\).  Formulate the precise result and prove it using Kolmogorov-Riesz 2 (the result in whole space).  \\ \\ Hint: See hints in class - multiply \(\mathcal F\) by a cut-off function which is 1 on \(\Omega\).  \\ \\ E3: (Mass escaping to infinity) Let \(f_k(x)=\chi_{[k,k+1]}(x)\). \\ \\ (a) Show that \(f_k\to0\) point wise, but that \(\{f_k\}_k\) does not converge in \(L^1(I\!R)\) nor in measure.  \\ \\ (b) Show that \(f_k\to0\) in \(L^1_{loc}\) and locally in measure (only consider points in compact subsets of \(I\!R\) - give a definition!).  \\ \\ c) Show that \(\{f_k\}_k\) is equibounded and -continuous in \(L^1\). Is it tight?  \\ Prove that there is a convergent subsequence and explain what type of convergence we get.  \\ \\ E4: Show that \(\mathcal F:=\{f(x)=\chi_{[a,b]}(x) : -1<a<b<1\}\) is totally bounded in \(L^p\) for \(p\in[1,\infty)\).  \\ \\ E5: (Challange) Let \(p\in[1,\infty)\) and \(\mathcal F\subset L^\infty(\Omega)\) where \(\Omega\subset I\!R^d\) is bounded and open. Assume (C1) \(\sup_f\|f\|_\infty <\infty\) and (C2) \(\sup_f\|f-\tau_yf\|_{L^p((\Omega-y)\cap\Omega)}\to 0\) as \(y\to0\). Then \(\mathcal F\) totally bounded in \(L^p(\Omega)\).  \\ \\ Hint: Extend functions to \(I\!R^d\) by \(0\) in \(\Omega^c\). Show that extension satisfies condition in Kolmogorov-Riesz theorem (inspiration can be found in E4...).  \\ \\   | **Proof of Kolmogorov:** We take the classical proof and not the one in the Holden notes, see e.g. Theorem A.5 in Holden-Risebro: Front Tracking for Hyperbolic Conservation Laws.  \\ \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week8.pdf|Solutions]] to the exercises E1-E4 and [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week8_E5.pdf|solution]] to E5 (all by Sebastian).  |
^ 9  | Modes of convergence (cont.),  \\ convergence and compactness in \(L^p,\ p\in(1,\infty)\).  \\ \\ Limiting cases: Convergence and compactness in\(L^1\),  \\ Dunford-Pettis with proof, uniform integrability, de la Vallee-Poussin.   | | p 60-65  \\ \\ p 65-70  \\ \\  | E1: Verify the claims in Remark 4.8 p 63 in Holden.  \\ \\ E2: Give the details to the proof of Thm 4.9 p 64 in Holden.  \\ \\ Hint: 1. Show \(\int f_n\phi \to \int f\phi\) for all simple \(\phi\). 2. Use the fact that simple functions are dense in \(L^q\) for \(q\in[1,\infty)\) to conclude.  \\ \\ E3: (Holden Ex 9 p 93) Show that \(f_n=n\chi_{(0,\frac1n)}\) is not uniformly integrable on \(X=(0,1)\) in two ways:  \\ \\ (a) Using directly the definition, and  \\ \\ (b) Showing that the sequence converge to \(\delta_0\) in distributions - and hence has no convergence subsequence in \(L^1(0,1)\) - and then conclude by Dunford-Pettis.  \\ \\ | **My lecture notes** on Dunford-Pettis and equiintegrability [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote02.03.2017_DunfordPettis_and_Equiintegrability.pdf|PDF]] \\ \\ **Proof of Dunford-Pettis:** The proof given in the lecture was correct up till part 5) - the conclusion. \\ \\ A correct conclusion is given in the lecture note above.  \\ \\ Note that the proof in the notes of Holden lack the conclusion.  \\ \\  [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week9.pdf|Solutions]] to the exercises (by Styrbjørn).  \\ \\  |
^ 10  | Convergence and compactness in\(L^1\) (cont.), \\ Convergence and compactness in \(L^\infty\)  \\ Examples.  \\ \\ Radon measures, the space \(\mathcal M\),\\ Weak * compactness in \(\mathcal M\) and the subspace \(L^1\)    | | p 69-75  \\ \\ p 75-78  | E1: (see also Holden Ex 10 p 93) Let \(f_n(x)=g(x+n)\) for some \(0\not\equiv g\in L^p(I\!R)\), \(p\in(1,\infty)\). Show that \(f_n\to 0\) weakly in \(L^p(I\!R)\).  \\ Hint: Show convergence of means on intervals.  \\ \\ E2: (Holden Ex 11 p 93) Show the \(p=1\) case in Example 4.25 (i) p 72 in Holden.  \\ \\  E3: Show that \(f_n\to f\) in \(L^1\) implies \(f_n\to f\) in \(M\).  \\ Explain how to interpret the latter convergence.  \\ \\ E4: Prove that Thm 74 in my notes implies Thm 75.  \\ Hint: This is easy, approximate \(C_0\) functions by \(C_c\) functions.  \\ \\  | This material is partly taken from Folland: //Real Analysis// chp 7, partly from Holden.  \\ \\ **Note:** Riesz representation theorem, the space \(M\) needs be defined as the space of //finite// Radon measures.  \\ \\ **My lecture notes** on Radon measure and compactness: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote09.03.2017_RadonMeasuresAndCompactness.pdf|PDF]].  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week10.pdf|Solutions]] to the exercises (by David).  \\ \\ |
^ 11  | **Sobolev Spaces** \\ Definitions, smooth approximations  \\ \\ Smooth approximations (cont.) \\ straightening the boundary, extensions  | 5  | 95-97  \\ \\ 97-97, 179 (App B.3)  | E1: Prove that the following two statements are equivalent:  \\ \\ (a) \(f_n\to f\) in \(W^{1,p}(K^o), \ \forall K\subset\Omega\) compact.  \\ \\ (b) \(f_n\phi\to f\phi\) in \(W^{1,p}(\Omega), \ \forall \phi\in C_c^\infty(\Omega)\).  \\ \\ E2: Prove that \(W^{m,p}(\Omega)\) is a Banach space.  \\ \\ Hint: Show 1) normed space and 2) completeness.  \\ Use the completeness of \(L^p\).  \\ \\ E3: Let \(\phi\in C_b^m(\Omega)\) (all derivatives of order less than \(m\) are bounded, continuous) and \(f\in W^{m,p}(\Omega)\), show that \[\|\phi f\|_{m,p}\leq C\|\phi\|_{m,\infty}\|f\|_{m,p}.\] Hint: Product rule, induction.  \\ \\ E4: Let \(\Omega_1,\Omega_2\subset I\!R^d\) be bounded open and \(\Phi:\overline\Omega_1\to \overline\Omega_2\) and \(\Phi^{-1}:\overline\Omega_2\to \overline\Omega_1\) be invertible \(C^m\) transformations. \\ \\ If \(f\in W^{m,p}(\Omega_1)\), show that then \(g(x)=f(\Phi^{-1}(x)), \ x\in \Omega_2\) belongs to \(W^{m,p}(\Omega_2)\) and that there is \(c>1\) such that \[\frac1c\|f\|_{\Omega_1,m,p}\leq \|g\|_{\Omega_2,m,p}\leq c\|f\|_{\Omega_1,m,p}.\] Hint: Chain rule + change of variables in multiple integral formula (involving e.g. \(\text{det}(D\Phi^{-1})\)) + only do proof for \(m=1\) where you may assume (by invertibility and continuity) that there is \(\lambda>1\) such that \[\frac1\lambda\leq |\text{det} D\Phi|+|\text{det} (D\Phi^{-1})|\leq \lambda.\] E5: Prove Lemma 13 from the lectures (see scan in the right collums) when \(m=1\).  \\ \\ Hint: Use E4 and Obs 10 from the lectures.  \\ \\   | **OBS:** //Global approx up to boundary// - the proof in Holden using straightening is not optimal and only gives \(C^1\) approximate functions. In the lectures I used the proof from //Evans: PDEs// chapter 5 that avoids straightening and give \(C^\infty\) approximations.  \\ \\ **My lecture notes** on smooth approximation up the boundary and straightening the boundary: [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote16.03.2017_SobolevSmApproStraightening.pdf|PDF]].  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week11.pdf|Solutions]] to the exercises (by Abdullah).   |
^ 12  | Extensions, restrictions  \\ \\ Restrictions (cont),  \\ Sobolev inequalities, Gagliardo-Nirenberg-Sobolev  | | 98-102 \\ \\ 102-104  | E1: (Evans P3 p 306) Let \(Q=(-1,1)^2\) (open square) and let \(f\) be the tent function (pyramid) supported on \(Q\) defined by \[f(x)=\begin{cases}1-x_1 & \text{for } x_1>0, \ x_1>|x_2|, \\ 1+x_1 & \text{for } x_1<0, \ -x_1>|x_2|,\\ 1-x_2 & \text{for } x_2>0, \ x_2>|x_1|, \\ 1+x_2 & \text{for } x_2<0, \ -x_2>|x_1|. \end{cases}\] Show that \(f\in W^{1,p}(Q)\) for all \(p\in[1,\infty]\).  \\ \\ Hint: Find explicitly the weak derivative. \\ \\ E2: (Evans P7 p 306) Assume \(1\leq p<\infty\), \(\Omega\subset I\!R\) open, bounded, and there exists a \(C^1\) vector field \(\gamma\) along \(\partial \Omega\) such that \(\gamma\cdot n\geq1\) where \(n\) is the outward unit normal.  \\ \\ Apply the divergence theorem to \(\int_{\partial\Omega}|f|^p\,\gamma\cdot n\, dS\) to derive a new proof of the trace inequality \[\int_{\partial \Omega}|f|^p dS\leq C\int_\Omega (|Df|^p+|f|^p)dx.\]  E3: (Evans P8 p 307) Show that there can be no trace operator for general functions in \(L^p\), i.e. a bounded linear operator \(T:L^p(\Omega)\to L^{p}(\partial\Omega)\) such that \(Tf=f|_{\partial\Omega}\) for every \(f\in L^p(\Omega)\cap C(\overline\Omega)\).  \\ \\ Hint: Let \(\Omega=B(0,1)\), the unit ball, and define \(f_n(r)=\min(n,-\log(1-r))^{\frac1p}\) in polar coordinates. Show \(0\leq f\in C(\overline\Omega)\) and that \[\frac{\|f_n\|^p_{L^p(\partial\Omega)}}{\|f_n\|^p_{L^p(\Omega)}}\to\infty,\] and hence \(T\) can not be bounded.  \\ \\ E4: (Evans P4 p 306) Let \(d=1, 1\leq p<\infty\), and \(f\in W^{1,p}(0,1)\).  \\ Show that \[|f(x)-f(y)|\leq (\int_0^1|f'|_pdx)^{\frac1p}|x-y|^{1-\frac1 p}.\] Hint: Show \(f(x)=\int_0^xf'(x)dx+C=:g(x)\ a.e.\) for a constant \(C\) [They have the same weak/distributional derivatives...]. Check that \(g\) is absolutely continuous. Use the fundamental theorem of calculus and H\"older's inequality.  \\ \\   | [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week12.pdf|Solutions]] to the exercises (by Sondre).  \\ \\ |
^ 13  | Sobolev inequalities: Gagliardo, Poincare.  \\ \\ H\"older spaces, Morrey's inequality.  \\ \\    | | 104-107, 111-112.  \\ \\ 110-111, 118.  | E1: Let \(\Omega\subset I\!R^d\) be open, \(k\in I\!N\), and \(\gamma\in(0,1]\). Prove that \(C^{0,\gamma}(\overline\Omega)\) is a Banach space.  \\ \\ E2: Let \(|f|_{1,p}=(\int|\nabla f|^p)^{1/p}\). Prove that there is \(C>0\) such that for all \(f\in W^{1,p}_0\), \[|f|_{1,p}\leq\|f\|_{1,p}\leq C|f|_{1,p}.\] What is \(C\)? Hint: Poincare.  \\ \\ E3: Interpolation inequalities (Evans P10 p 307). Let \(\Omega\subset I\!R^d\) be open, bounded.  \\ \\ (a) Integrate by parts to prove \[\|\nabla f\|_{p,\Omega}\leq C\|f\|_{p,\Omega}^{1/2}\|\nabla^2f\|_{p,\Omega}^{1/2}\] for \(2\leq p<\infty\) and all \(f\in C_c^\infty(\Omega)\).  \\ \\ Hint: \(\int_\Omega |\nabla f|^p = \sum_{i=1}^d\int_{\Omega}f_{x_i}f_{x_i}|\nabla f|^{p-2}.\)  \\ \\ (b) Assume also \(p=2\) and \(\partial\Omega\) is \(C^2\).  \\ Prove by approximation that the interpolation inequality of part (a) holds also for \(f\in W^{2,2}\cap W^{1,2}_0(\Omega)\)  \\ \\ Hint: You may use that there exist approximations \(f_m\in C^2(\overline\Omega)\) converging to \(f\) in \(W^{2,2}\). Use also a sequence \(g_m\in C_c^\infty\) converging to \(f\) in \(W^{1,2}\) [why can you find such a sequence?]. Redo proof the of (a).  \\ \\ c) Prove \[\|\nabla f\|_{2p,\Omega}\leq C\|f\|^{1/2}_{\infty,\Omega}\|\nabla^2f\|^{1/2}_{p,\Omega}\] for \(1\leq p<\infty\) and all \(f\in C_c^\infty(\Omega)\).    | [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week13.pdf|Solutions]] to the exercises (by Fredrik Høeg).  \\ \\ |
^ 14  | General Sobolev inequalities,  \\ embedding, compactness in \(W^{1,p}\)  \\ \\ Compactness (cont.),  \\ Sobolev chain rule, finite differences.  \\ \\ | | 116-118, 18-19, 112-114  \\ \\ 114-116, 126, 128  \\ \\ | E1: Let \(X,Y\) be normed spaces, \(\Phi:X\to Y\) be a continuous embedding (embedding=linear, bounded, \(1-1\)), and \(\Phi(X)\subset Y\) dense. Prove that the adjoint \(\Phi':Y'\to X'\) defined by \[(\Phi x,y')_{Y,Y'}=(x, \Phi'y')_{X,X'},\quad y'\in Y',\] is a continuous embedding.  \\ \\ E2: Let \(\Omega\subset I\!R^d\) is an open bounded set, and \(0<\gamma_1<\gamma_2<1\).  \\ \\ (a) Prove the following interpolation inequality for Hölder spaces: \[\|f\|_{C^{0,\gamma_1}(\overline\Omega)}\leq 2^{1-\frac{\gamma_1}{\gamma_2}}\|f\|_{C_b(\overline\Omega)}^{1-\frac{\gamma_1}{\gamma_2}}\|f\|_{C^{0,\gamma_2}(\overline\Omega)}^{\frac{\gamma_1}{\gamma_2}} \]  (b) Prove that \(C^{0,\gamma_2}(\overline{\Omega})\) is compactly embedded in \(C^{0,\gamma_1}(\overline{\Omega})\).  \\ \\ Hint: Use Arzela-Ascoli's theorem and part (a).    \\ \\ E3: (Holden Ex 1, p 119) Let \(\Omega\subset I\!R^d\) be a bounded open set. Show that \(W^{k,2}(\Omega)\) is compactly embedded in \(W^{k-1,2}(\Omega)\) for \(k\in I\!N\).  \\ \\ Hint: Use Rellich-Kondrachov, do it for all values of \(d\).  \\ \\ E4: Let \(f\in L^2(I\!R^d)\) and let \(u\in W^{1,2}(I\!R^d)\) be the weak solution of \[u-\Delta u=f\quad\text{in}\quad I\!R^d.\] Use finite differences to show that \(u\in W^{2,2}(I\!R^d)\).  \\ \\ Hint: Follow the steps layed out in Example 49 in my lecture notes (see leftmost collumn). Here you also find the definition of a weak solution of this equation. OBS: A key step is to use that \[\|u\|_{W^{1,2}}^2=|\int fu| \leq \|u\|_{W^{1,2}}\sup_{0\neq \phi \in W^{1,2}}\frac{|\int f\phi|}{\|\phi\|_{W^{1,2}}}:=\|u\|_{W^{1,2}}\|f\|_{(W^{1,2})'}.\] Note that \(f\) defines an element in \((W^{1,2})'\) through \(F(\phi)=\int f\phi\) - i.e. a regular distribution.  \\ \\  | **Rellich-Kondrachov:** In the lectures I give a stronger version of this result than presented in the notes of Holden. I follow the book of Evans, and show compact embedding into Hoelder spaces \(C^{0\,\gamma}\).  \\ \\ **The proof** of the first part of Rellich-Kondarchov's compactness theorem follows from extension, Kolmogorov-Riesz compactness theorem, and interpolation in \(L^p\). This way avoids the long regularization + Arzela-Ascoli argument used in the Holden notes (and in PDE book by Evans). In fact the the regularization + Arzela-Ascoli argument is exactly the argument we used in class to proof Kolmogorov-Riesz in the first place (but the proof in the notes of Holden is slightly different).  \\ \\ **My lecture notes** from this week:  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote03.04.2017_GenSobolevIneq_and_RellichKondrachovCompThm.pdf|General Sobolev inequalities and Strong comactness in \(W^{1,p}\)]]  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote06.04.2017_Rellich_ChainRule_and_DifferenceQuotients.pdf|Strong comactness, chain rule, and difference quotients]]  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/solutions/ma8105-ex-week14.pdf|Solutions]] to the exercises (by Fredrik Hildrum).  \\ \\ |
^ 15  ^ EASTER HOLIDAYS  ^^^^ 
^ 16  | Application: Convergence of finite difference approximation for the Porous Medium Equation (PME).  \\ \\ A. About PME (eq'n, background, self-similar solutions, derivation, well-posedness and a priori estimates).  \\ \\ B. The explicit monotone finite difference approximation.  \\  \\ C. A priori estimates for the approximation.  \\ \\  | Lecture note,  \\ see right collumn   |  \\ \\ | | **My lecture note** from this week:  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote19.04.2017_Application:ConvergenceOfFDMforPorousMediumEqn_I.pdf|Convergence of finite difference approximation for the Porous Medium Equation I]]  \\ \\  |
^ 17  | Application: Convergence of finite difference approximation for the Porous Medium Equation (PME).  \\ \\ D. Interpolation in time and compactness.  \\ \\ E. Convergence of the method.  \\ \\ Plan the oral exam.  \\ Dates and organization. | Lecture note,  \\ see right collumn   | | | **My lecture note** from this week:  \\ \\ [[http://www.math.ntnu.no/emner/MA8105/2017v/notes/LecNote25.04.2017_Application:ConvergenceOfFDMforPorousMediumEqn_II.pdf|Convergence of finite difference approximation for the Porous Medium Equation II]]  \\ \\ |