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Lemma 6.1 (Riesz’s lemma). Let \( E \) be an n.v.s. and let \( M \subset E \) be a closed linear space such that \( M \neq E \) . Then\n\n\[ \forall \varepsilon > 0\;\exists u \in E\text{ such that }\parallel u\parallel = 1\text{ and }\operatorname{dist}\left( {u, M}\right) \geq 1 - \varepsilon . \] | Proof. Let \( v \in E \) with \( v \notin M \) . Since \( M \) is closed, then\n\n\[ d = \operatorname{dist}\left( {v, M}\right) > 0. \]\n\nChoose any \( {m}_{0} \in M \) such that\n\n\[ d \leq \begin{Vmatrix}{v - {m}_{0}}\end{Vmatrix} \leq d/\left( {1 - \varepsilon }\right) . \]\n\nThen\n\n\[ u = \frac{v - {m}_{0}}{\b... | Yes |
The spectrum \( \sigma \left( T\right) \) of a bounded operator \( T \) is compact and \[ \sigma \left( T\right) \subset \left\lbrack {-\parallel T\parallel , + \parallel T\parallel }\right\rbrack . \] | Proof. Let \( \lambda \in \mathbb{R} \) be such that \( \left| \lambda \right| > \parallel T\parallel \) . We will show that \( T - {\lambda I} \) is bijective, which implies that \( \sigma \left( T\right) \subset \left\lbrack {-\parallel T\parallel , + \parallel T\parallel }\right\rbrack \) . Given \( f \in E \), the ... | Yes |
Lemma 6.2. Let \( T \in \mathcal{K}\left( E\right) \) and let \( {\left( {\lambda }_{n}\right) }_{n \geq 1} \) be a sequence of distinct real numbers such that\n\n\[ \n{\lambda }_{n} \rightarrow \lambda \n\]\n\nand\n\n\[ \n{\lambda }_{n} \in \sigma \left( T\right) \smallsetminus \{ 0\} \;\forall n.\n\]\n\nThen \( \lamb... | Proof. We know that \( {\lambda }_{n} \in {EV}\left( T\right) \) ; let \( {e}_{n} \neq 0 \) be such that \( \left( {T - {\lambda }_{n}I}\right) {e}_{n} = 0 \) . Let \( {E}_{n} \) be the space spanned by \( \left\{ {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\} \) . We claim that \( {E}_{n} \subset {E}_{n + 1},{E}_{n} \neq {... | Yes |
Theorem 6.12. Let \( H = {L}^{2}\left( \Omega \right) \) and \( K\left( {x, y}\right) \in {L}^{2}\left( {\Omega \times \Omega }\right) \) . Then the operator\n\n\[ u \mapsto \left( {Ku}\right) \left( x\right) = {\int }_{\Omega }K\left( {x, y}\right) u\left( y\right) {dy} \]\n\n is a Hilbert-Schmidt operator: | Conversely, every Hilbert-Schmidt operator on \( {L}^{2}\left( \Omega \right) \) is of the preceding form for some unique function \( K\left( {x, y}\right) \in {L}^{2}\left( {\Omega \times \Omega }\right) \) . | No |
Proposition 7.2. Let \( A \) be a maximal monotone operator. Then\n\n(a1)\n\n\[{A}_{\lambda }v = A\left( {{J}_{\lambda }v}\right) \;\forall v \in H\\text{ and }\\forall \\lambda > 0,\]\n\n(a2)\n\[{A}_{\lambda }v = {J}_{\lambda }\\left( {Av}\\right) \\;\\forall v \\in D\\left( A\\right) \\text{ and }\\forall \\lambda > ... | Proof.\n\n(a1) can be written as \( v = \\left( {{J}_{\lambda }v}\\right) + {\\lambda A}\\left( {{J}_{\lambda }v}\\right) \), which is just the definition of \( {J}_{\lambda }v \) .\n\n(a2) By \( \\left( {a}_{1}\\right) \) we have\n\n\[{A}_{\lambda }v + A\\left( {v - {J}_{\lambda }v}\\right) = {Av}\]\n\ni.e.,\n\n\[{A}_... | Yes |
Lemma 7.1. Let \( w \in {C}^{1}\left( {\lbrack 0, + \infty }\right) ;H) \) be a function satisfying\n\n\[ \frac{dw}{dt} + {A}_{\lambda }w = 0\text{ on }\lbrack 0, + \infty ). \]\n\nThen the functions \( t \mapsto \left| {w\left( t\right) }\right| \) and \( t \mapsto \left| {\frac{dw}{dt}\left( t\right) }\right| = \left... | Proof. We have\n\n\[ \left( {\frac{dw}{dt}, w}\right) + \left( {{A}_{\lambda }w, w}\right) = 0. \]\n\nBy Proposition 7.2(e) we know that \( \left( {{A}_{\lambda }w, w}\right) \geq 0 \) and thus \( \frac{1}{2}\frac{d}{dt}{\left| w\right| }^{2} \leq 0 \), so that \( \left| {w\left( t\right) }\right| \) is nonincreasing. ... | Yes |
Lemma 7.2. Let \( {u}_{0} \in D\left( A\right) \) . Then \( \forall \varepsilon > 0\exists {\bar{u}}_{0} \in D\left( {A}^{2}\right) \) such that \( \left| {{u}_{0} - {\bar{u}}_{0}}\right| < \varepsilon \) and \( \left| {A{u}_{0} - A{\bar{u}}_{0}}\right| < \varepsilon \) . In other words, \( D\left( {A}^{2}\right) \) is... | Proof of Lemma 7.2. Set \( {\bar{u}}_{0} = {J}_{\lambda }{u}_{0} \) for some appropriate \( \lambda > 0 \) to be fixed later. We have\n\n\[ \n{\bar{u}}_{0} \in D\left( A\right) \;\text{ and }\;{\bar{u}}_{0} + {\lambda A}{\bar{u}}_{0} = {u}_{0}.\n\]\n\nThus \( A{\bar{u}}_{0} \in D\left( A\right) \), i.e., \( {\bar{u}}_{... | Yes |
Proposition 7.6. Let \( A \) be a maximal monotone symmetric operator. Then \( A \) is selfadjoint. | Proof. Let \( {J}_{1} = {\left( I + A\right) }^{-1} \) . We will first prove that \( {J}_{1} \) is self-adjoint. Since \( {J}_{1} \in \) \( \mathcal{L}\left( H\right) \) it suffices to check that\n\n(25)\n\n\[ \left( {{J}_{1}u, v}\right) = \left( {u,{J}_{1}v}\right) \;\forall u, v \in H. \]\n\nSet \( {u}_{1} = {J}_{1}u... | Yes |
Theorem 7.8 (Hille-Yosida). Let \( A \) be \( m \) -accretive. Then given any \( {u}_{0} \in D\left( A\right) \) there exists a unique function\n\n\[ \n u \in {C}^{1}\left( {\lbrack 0, + \infty );E}\right) \cap C\left( {\lbrack 0, + \infty );D\left( A\right) }\right)\n\]\n\nsuch that\n\n(38)\n\n\[ \n\left\{ \begin{arra... | For the proof, see, e.g., P. Lax [1], A. Pazy [1], J. Goldstein [1], E. Davies [1], [2], K. Yosida [1], M. Reed-B. Simon [1], Volume 2, H. Tanabe [1], N. Dunford-J. T. Schwartz [1] Volume 1, M. Schechter [1], A. Friedman [2], R. Dautray-J.- L. Lions [1], Chapter XVII, A. Balakrishnan [1], T. Kato [1], W. Rudin [1]. The... | No |
Proposition 8.1. The space \( {W}^{1, p} \) is a Banach space for \( 1 \leq p \leq \infty \) . It is reflexive \( {}^{3} \) for \( 1 < p < \infty \) and separable for \( 1 \leq p < \infty \) . The space \( {H}^{1} \) is a separable Hilbert space. | Proof.\n\n(a) Let \( \left( {u}_{n}\right) \) be a Cauchy sequence in \( {W}^{1, p} \) ; then \( \left( {u}_{n}\right) \) and \( \left( {u}_{n}^{\prime }\right) \) are Cauchy sequences in \( {L}^{p} \) . It follows that \( {u}_{n} \) converges to some limit \( u \) in \( {L}^{p} \) and \( {u}_{n}^{\prime } \) converges... | Yes |
Lemma 8.1. Let \( f \in {L}_{\text{loc }}^{1}\left( I\right) \) be such that\n\n\[ {\int }_{I}f{\varphi }^{\prime } = 0\;\forall \varphi \in {C}_{c}^{1}\left( I\right) \]\n\nThen there exists a constant \( C \) such that \( f = C \) a.e. on \( I \) . | Proof. Fix a function \( \psi \in {C}_{c}\left( I\right) \) such that \( {\int }_{I}\psi = 1 \) . For any function \( w \in {C}_{c}\left( I\right) \) there exists \( \varphi \in {C}_{c}^{1}\left( I\right) \) such that\n\n\[ {\varphi }^{\prime } = w - \left( {{\int }_{I}w}\right) \psi \]\n\nIndeed, the function \( h = w... | Yes |
Lemma 8.2. Let \( g \in {L}_{\text{loc }}^{1}\left( I\right) \) ; for \( {y}_{0} \) fixed in \( I \), set\n\n\[ v\left( x\right) = {\int }_{{y}_{0}}^{x}g\left( t\right) {dt},\;x \in I. \]\n\nThen \( v \in C\left( I\right) \) and\n\n\[ {\int }_{I}v{\varphi }^{\prime } = - {\int }_{I}{g\varphi }\;\forall \varphi \in {C}_... | Proof. We have\n\n\[ {\int }_{I}v{\varphi }^{\prime } = {\int }_{I}\left\lbrack {{\int }_{{y}_{0}}^{x}g\left( t\right) {dt}}\right\rbrack {\varphi }^{\prime }\left( x\right) {dx} \]\n\n\[ = - {\int }_{a}^{{y}_{0}}{dx}{\int }_{x}^{{y}_{0}}g\left( t\right) {\varphi }^{\prime }\left( x\right) {dt} + {\int }_{{y}_{0}}^{b}{... | Yes |
Proposition 8.3. Let \( u \in {L}^{p} \) with \( 1 < p \leq \infty \) . The following properties are equivalent:\n\n(i) \( u \in {W}^{1, p} \) ,\n\n(ii) there is a constant \( C \) such that\n\n\[ \left| {{\int }_{I}u{\varphi }^{\prime }}\right| \leq C\parallel \varphi {\parallel }_{{L}^{{p}^{\prime }}\left( I\right) }... | Proof.\n\n(i) \( \Rightarrow \) (ii). This is obvious.\n\n(ii) \( \Rightarrow \) (i). The linear functional\n\n\[ \varphi \in {C}_{c}^{1}\left( I\right) \mapsto {\int }_{I}u{\varphi }^{\prime } \]\n\n is defined on a dense subspace of \( {L}^{{p}^{\prime }} \) (since \( {p}^{\prime } < \infty \) ) and it is continuous ... | Yes |
A function \( u \) in \( {L}^{\infty }\left( I\right) \) belongs to \( {W}^{1,\infty }\left( I\right) \) if and only if there exists a constant \( C \) such that\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq C\left| {x - y}\right| \text{ for a.e. }x, y \in I. \] | Proof. If \( u \in {W}^{1,\infty }\left( I\right) \) we may apply Theorem 8.2 to deduce that\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq {\begin{Vmatrix}{u}^{\prime }\end{Vmatrix}}_{{L}^{\infty }}\left| {x - y}\right| \text{ for a.e. }x, y \in I. \]\n\nConversely, let \( \varphi \in {C}_{c}^{1}\left... | Yes |
Proposition 8.5. Let \( u \in {L}^{p}\left( \mathbb{R}\right) \) with \( 1 < p < \infty \) . The following properties are equivalent:\n\n(i) \( u \in {W}^{1, p}\left( \mathbb{R}\right) \) ,\n\n(ii) there exists a constant \( C \) such that for all \( h \in \mathbb{R} \) ,\n\n\[{\begin{Vmatrix}{\tau }_{h}u - u\end{Vmatr... | Proof.\n\n(i) \( \Rightarrow \) (ii). (This implication is also valid when \( p = 1 \) .) By Theorem 8.2 we have, for all \( x \) and \( h \) in \( \mathbb{R} \) ,\n\n\[u\left( {x + h}\right) - u\left( x\right) = {\int }_{x}^{x + h}{u}^{\prime }\left( t\right) {dt} = h{\int }_{0}^{1}{u}^{\prime }\left( {x + {sh}}\right... | Yes |
Theorem 8.6 (extension operator). Let \( 1 \leq p \leq \infty \) . There exists a bounded linear operator \( P : {W}^{1, p}\left( I\right) \rightarrow {W}^{1, p}\left( \mathbb{R}\right) \), called an extension operator, satisfying the following properties:\n\n(i) \( P{u}_{\mid I} = u\forall u \in {W}^{1, p}\left( I\rig... | Proof. Beginning with the case \( I = \left( {0,\infty }\right) \) we show that extension by reflexion\n\n\[ \left( {Pu}\right) \left( x\right) = {u}^{ \star }\left( x\right) = \left\{ \begin{array}{ll} u\left( x\right) & \text{ if }x \geq 0, \\ u\left( {-x}\right) & \text{ if }x < 0, \end{array}\right. \]\n\nworks. Cl... | Yes |
Lemma 8.3. Let \( u \in {W}^{1, p}\left( I\right) \) . Then\n\n\[ \eta \widetilde{u} \in {W}^{1, p}\left( {0,\infty }\right) \;\text{ and }\;{\left( \eta \widetilde{u}\right) }^{\prime } = {\eta }^{\prime }\widetilde{u} + \eta \widetilde{{u}^{\prime }}. \]\n | Proof. Let \( \varphi \in {C}_{c}^{1}\left( \left( {0,\infty }\right) \right) \) ; then\n\n\[ {\int }_{0}^{\infty }\eta \widetilde{u}{\varphi }^{\prime } = {\int }_{0}^{1}{\eta u}{\varphi }^{\prime } = {\int }_{0}^{1}u\left\lbrack {{\left( \eta \varphi \right) }^{\prime } - {\eta }^{\prime }\varphi }\right\rbrack \]\n\... | Yes |
Lemma 8.4. Let \( \rho \in {L}^{1}\left( \mathbb{R}\right) \) and \( v \in {W}^{1, p}\left( \mathbb{R}\right) \) with \( 1 \leq p \leq \infty \) . Then \( \rho \star v \in \) \( {W}^{1, p}\left( \mathbb{R}\right) \) and \( {\left( \rho \star v\right) }^{\prime } = \rho \star {v}^{\prime } \) . | Proof. First, suppose that \( \rho \) has compact support. We already know (Theorem 4.15) that \( \rho \star v \in {L}^{p}\left( \mathbb{R}\right) \) . Let \( \varphi \in {C}_{c}^{1}\left( \mathbb{R}\right) \) ; from Propositions 4.16 and 4.20 we have\n\n\[ \int \left( {\rho \star v}\right) {\varphi }^{\prime } = \int ... | Yes |
Corollary 8.9. Suppose that \( I \) is an unbounded interval and \( u \in {W}^{1, p}\left( I\right) \) with \( 1 \leq p < \infty \) . Then\n\n\[ \mathop{\lim }\limits_{\substack{{x \in I} \\ {\left| x\right| \rightarrow \infty } }}u\left( x\right) = 0 \] | Proof. From Theorem 8.7 there exists a sequence \( \left( {u}_{n}\right) \) in \( {C}_{c}^{1}\left( \mathbb{R}\right) \) such that \( {u}_{n \mid I} \rightarrow u \) in \( {W}^{1, p}\left( I\right) \) . It follows from (5) that \( {\begin{Vmatrix}{u}_{n} - u\end{Vmatrix}}_{{L}^{\infty }\left( I\right) } \rightarrow 0 \... | Yes |
Corollary 8.10 (differentiation of a product). \( {}^{8} \) Let \( u, v \in {W}^{1, p}\left( I\right) \) with \( 1 \leq p \leq \) \( \infty \) . Then\n\n\[ \n{uv} \in {W}^{1, p}\left( I\right)\n\]\n\nand\n\n(10)\n\n\[ \n{\left( uv\right) }^{\prime } = {u}^{\prime }v + u{v}^{\prime }\n\] | Proof. First recall that \( u \in {L}^{\infty } \) (by Theorem 8.8) and thus \( {uv} \in {L}^{p} \) . To show that \( {\left( uv\right) }^{\prime } \in {L}^{p} \) let us begin with the case \( 1 \leq p < \infty \) . Let \( \left( {u}_{n}\right) \) and \( \left( {v}_{n}\right) \) be sequences in \( {C}_{c}^{1}\left( \ma... | Yes |
Corollary 8.11 (differentiation of a composition). Let \( G \in {C}^{1}\left( \mathbb{R}\right) \) be such that \( {}^{9} \( G\left( 0\right) = 0 \), and let \( u \in {W}^{1, p}\left( I\right) \) with \( 1 \leq p \leq \infty \) . Then \[ G \circ u \in {W}^{1, p}\left( I\right) \;\text{ and }\;{\left( G \circ u\right) }... | Proof. Let \( M = \parallel u{\parallel }_{\infty } \) . Since \( G\left( 0\right) = 0 \), there exists a constant \( C \) such that \( \left| {G\left( s\right) }\right| \leq \) \( C\left| s\right| \) for all \( s \in \left\lbrack {-M, + M}\right\rbrack \) . Thus \( \left| {G \circ u}\right| \leq C\left| u\right| \) ; ... | Yes |
Proposition 8.14. Let \( F \in {W}^{-1,{p}^{\prime }}\left( I\right) \) . Then there exist two functions \( {f}_{0},{f}_{1} \in \) \( {L}^{{p}^{\prime }}\left( I\right) \) such that\n\n\[ \langle F, u\rangle = {\int }_{I}{f}_{0}u + {\int }_{I}{f}_{1}{u}^{\prime }\;\forall u \in {W}_{0}^{1, p}\left( I\right) \]\n\nand\n... | Proof. Consider the product space \( E = {L}^{p}\left( I\right) \times {L}^{p}\left( I\right) \) equipped with the norm\n\n\[ \parallel h\parallel = {\begin{Vmatrix}{h}_{0}\end{Vmatrix}}_{p} + {\begin{Vmatrix}{h}_{1}\end{Vmatrix}}_{p}\text{ where }h = \left\lbrack {{h}_{0},{h}_{1}}\right\rbrack . \]\n\nThe map \( T : u... | Yes |
Consider the problem\n\n\[ \left\{ \begin{array}{l} - {u}^{\prime \prime } + u = f\;\text{ on }I = \left( {0,1}\right) , \\ u\left( 0\right) = \alpha, u\left( 1\right) = \beta , \end{array}\right. \]\n\nwith \( \alpha ,\beta \in \mathbb{R} \) given and \( f \) a given function. | - Proposition 8.16. Given \( \alpha ,\beta \in \mathbb{R} \) and \( f \in {L}^{2}\left( I\right) \) there exists a unique function \( u \in {H}^{2}\left( I\right) \) satisfying (16). Furthermore, \( u \) is obtained by\n\n\[ \mathop{\min }\limits_{\substack{{v \in {H}^{1}\left( I\right) } \\ {v\left( 0\right) = \alpha,... | Yes |
Proposition 8.17. Given \( f \in {L}^{2}\left( I\right) \) there exists a unique function \( u \in {H}^{2}\left( I\right) \) satisfying (21). \( {}^{13} \) Furthermore, \( u \) is obtained by \[ \mathop{\min }\limits_{{v \in {H}^{1}\left( I\right) }}\left\{ {\frac{1}{2}{\int }_{I}\left( {{v}^{\prime 2} + {v}^{2}}\right... | Proof. If \( u \) is a classical solution of (21) we have \[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv}\;\forall v \in {H}^{1}\left( I\right) . \] We use \( {H}^{1}\left( I\right) \) as our function space: there is no point in working in \( {H}_{0}^{1} \) as above since \( u\left( 0\right... | Yes |
Proposition 8.18. Given any \( f \in {L}^{2}\left( I\right) \) and \( \alpha ,\beta \in \mathbb{R} \) there exists a unique function \( u \in {H}^{2}\left( I\right) \) satisfying (24). Furthermore, \( u \) is obtained by \[ \mathop{\min }\limits_{{v \in {H}^{1}\left( I\right) }}\left\{ {\frac{1}{2}{\int }_{I}\left( {{v... | Proof. If \( u \) is a classical solution of (24) we have \[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv} - {\alpha v}\left( 0\right) + {\beta v}\left( 1\right) \;\forall v \in {H}^{1}\left( I\right) . \] We use \( {H}^{1}\left( I\right) \) as our function space and we apply the Lax-Milgram... | Yes |
Consider the problem\n\n\[ \left\{ \begin{array}{l} - {u}^{\prime \prime } + u = f\;\text{ on }I = \left( {0,1}\right) , \\ u\left( 0\right) = 0,{u}^{\prime }\left( 1\right) = 0. \end{array}\right. \] | If \( u \) is a classical solution of (25) we have\n\n\[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv}\;\forall v \in {H}^{1}\left( I\right) \text{ with }v\left( 0\right) = 0. \] | No |
Consider the problem\n\n\[ \left\{ \begin{array}{l} - {u}^{\prime \prime } + u = f\;\text{ on }I = \left( {0,1}\right) , \\ u\left( 0\right) = u\left( 1\right) ,{u}^{\prime }\left( 0\right) = {u}^{\prime }\left( 1\right) . \end{array}\right. \] | If \( u \) is a classical solution of (28) we have\n\n\[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv}\;\forall v \in {H}^{1}\left( I\right) \;\text{ with }v\left( 0\right) = v\left( 1\right) . \]\n\nThe appropriate setting for applying Lax-Milgram is the Hilbert space\n\n\[ H = \left\{ {v \... | Yes |
Consider the problem\n\n\[ \left\{ \begin{array}{ll} - {u}^{\prime \prime } + u = f & \text{ on }\mathbb{R}, \\ u\left( x\right) \rightarrow 0 & \text{ as }\left| x\right| \rightarrow \infty , \end{array}\right. \] \n\nwith \( f \) given in \( {L}^{2}\left( \mathbb{R}\right) \) . A classical solution of (30) is a funct... | We have first to prove that any classical solution \( u \) is a weak solution; let us check in the first place that \( u \in {H}^{1}\left( \mathbb{R}\right) \) . Choose a sequence \( \left( {\zeta }_{n}\right) \) of cut-off functions as in the proof of Theorem 8.7. Multiplying (30) by \( {\zeta }_{n}u \) and integratin... | Yes |
Proposition 8.21. Let \( f \in {L}^{2}\left( I\right) \) with \( I = \left( {0,1}\right) \) and let \( u \in {H}^{2}\left( I\right) \) be the solution of the problem\n\n\[ \left\{ \begin{array}{l} - {u}^{\prime \prime } + u = f\;\text{ on }I, \\ {u}^{\prime }\left( 0\right) = {u}^{\prime }\left( 1\right) = 0. \end{arra... | Proof. We have\n\n(36)\n\n\[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv}\;\forall v \in {H}^{1}\left( I\right) . \]\n\nPlug \( v = G\left( {u - K}\right) \) into (36) with \( K = \mathop{\sup }\limits_{I}f \) and the same function \( G \) as above. Then proceed just as in the proof of Theo... | No |
Lemma 9.1. Let \( \rho \in {L}^{1}\left( {\mathbb{R}}^{N}\right) \) and let \( v \in {W}^{1, p}\left( {\mathbb{R}}^{N}\right) \) with \( 1 \leq p \leq \infty \) . Then\n\n\[ \rho \star v \in {W}^{1, p}\left( {\mathbb{R}}^{N}\right) \;\text{ and }\;\frac{\partial }{\partial {x}_{i}}\left( {\rho \star v}\right) = \rho \s... | Proof of Lemma 9.1. Adapt the proof of Lemma 8.4. | No |
Proposition 9.3. Let \( u \in {L}^{p}\left( \Omega \right) \) with \( 1 < p \leq \infty \) . The following properties are equivalent:\n\n(i) \( u \in {W}^{1, p}\left( \Omega \right) \),\n\n(ii) there exists a constant \( C \) such that\n\n\[ \left| {{\int }_{\Omega }u\frac{\partial \varphi }{\partial {x}_{i}}}\right| \... | Proof.\n\n(i) \( \Rightarrow \) (ii). Obvious.\n\n(ii) \( \Rightarrow \) (i). Proceed as in the proof of Proposition 8.3.\n\n(i) \( \Rightarrow \) (iii). Assume first that \( u \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{N}\right) \) . Let \( h \in {\mathbb{R}}^{N} \) and set\n\n\[ v\left( t\right) = u\left( {x + {th}}\r... | Yes |
Proposition 9.4 (differentiation of a product). Let \( u, v \in {W}^{1, p}\left( \Omega \right) \cap {L}^{\infty }\left( \Omega \right) \) with \( 1 \leq p \leq \infty \) . Then \( {uv} \in {W}^{1, p}\left( \Omega \right) \cap {L}^{\infty }\left( \Omega \right) \) and\n\n\[ \frac{\partial }{\partial {x}_{i}}\left( {uv}... | Proof. As in the proof of Corollary 8.10, it suffices to consider the case \( 1 \leq p < \infty \) . By Theorem 9.2 there exist sequences \( \left( {u}_{n}\right) ,\left( {v}_{n}\right) \) in \( {C}_{c}^{\infty }\left( {\mathbb{R}}^{N}\right) \) such that\n\n\[ {u}_{n} \rightarrow u,\;{v}_{n} \rightarrow v\;\text{ in }... | Yes |
Proposition 9.5 (differentiation of a composition). Let \( G \in {C}^{1}\left( \mathbb{R}\right) \) be such that \( G\left( 0\right) = 0 \) and \( \left| {{G}^{\prime }\left( s\right) }\right| \leq M\;\forall s \in \mathbb{R} \) for some constant \( M \) . Let \( u \in {W}^{1, p}\left( \Omega \right) \) with \( 1 \leq ... | Proof. We have \( \left| {G\left( s\right) }\right| \leq M\left| s\right| \;\forall s \in \mathbb{R} \) and thus \( \left| {G \circ u}\right| \leq M\left| u\right| \) ; as a consequence, \( G \circ u \in {L}^{p}\left( \Omega \right) \) and also \( \left( {{G}^{\prime } \circ u}\right) \frac{\partial u}{\partial {x}_{i}... | Yes |
Proposition 9.6 (change of variables formula). Let \( \Omega \) and \( {\Omega }^{\prime } \) be two open sets in \( {\mathbb{R}}^{N} \) and let \( H : {\Omega }^{\prime } \rightarrow \Omega \) be a bijective map, \( x = H\left( y\right) \), such that \( H \in {C}^{1}\left( {\Omega }^{\prime }\right) \) , \( {H}^{-1} \... | \[ \frac{\partial }{\partial {y}_{j}}u\left( {H\left( y\right) }\right) = \mathop{\sum }\limits_{i}\frac{\partial u}{\partial {x}_{i}}\left( {H\left( y\right) }\right) \frac{\partial {H}_{i}}{\partial {y}_{j}}\left( y\right) \;\forall j = 1,2,\ldots, N. \] Proof. When \( 1 \leq p < \infty \), choose a sequence \( \left... | Yes |
Theorem 9.7. Suppose that \( \Omega \) is of class \( {C}^{1} \) with \( \Gamma \) bounded (or else \( \Omega = {\mathbb{R}}_{ + }^{N} \) ). Then there exists a linear extension operator\n\n\[ P : {W}^{1, p}\left( \Omega \right) \rightarrow {W}^{1, p}\left( {\mathbb{R}}^{N}\right) \;\left( {1 \leq p \leq \infty }\right... | We shall begin by proving a simple but fundamental lemma concerning the extension by reflection.\n\nLemma 9.2. Given \( u \in {W}^{1, p}\left( \Omega \right) \) | No |
Lemma 9.3 (partition of unity). Let \( \\Gamma \) be a compact subset of \( {\\mathbb{R}}^{N} \) and let \( {U}_{1},{U}_{2} \) , \( \\ldots ,{U}_{k} \) be an open covering of \( \\Gamma \), i.e., \( \\Gamma \\subset \\mathop{\\bigcup }\\limits_{{i = 1}}^{k}{U}_{i} \) . Then there exist functions \( {\\theta }_{0} \) , ... | Proof. This lemma is classical; similar statements can be found, for example, in S. Agmon [1], R. Adams [1], G. Folland [1], P. Malliavin [1]. | No |
Lemma 9.4. Let \( N \geq 2 \) and let \( {f}_{1},{f}_{2},\ldots ,{f}_{N} \in {L}^{N - 1}\left( {\mathbb{R}}^{N - 1}\right) \) . For \( x \in {\mathbb{R}}^{N} \) and \( 1 \leq i \leq N \) set\n\n\[ \n{\widetilde{x}}_{i} = \left( {{x}_{1},{x}_{2},\ldots ,{x}_{i - 1},{x}_{i + 1},\ldots ,{x}_{N}}\right) \in {\mathbb{R}}^{N... | Proof. The case \( N = 2 \) is trivial (why?). Let us consider the case \( N = 3 \) . We have \n\n\[ \n{\int }_{\mathbb{R}}\left| {f\left( x\right) }\right| d{x}_{3} = \left| {{f}_{3}\left( {{x}_{1},{x}_{2}}\right) }\right| {\int }_{\mathbb{R}}\left| {{f}_{1}\left( {{x}_{2},{x}_{3}}\right) }\right| \left| {{f}_{2}\left... | Yes |
Lemma 9.5. Let \( u \in {W}^{1, p}\left( \Omega \right) \) with \( 1 \leq p < \infty \) and assume that \( \operatorname{supp}u \) is a compact subset of \( \Omega \) . Then \( u \in {W}_{0}^{1, p}\left( \Omega \right) \) . | Proof. Fix an open set \( \omega \) such that \( \operatorname{supp}u \subset \omega \subset \subset \Omega \) and choose \( \alpha \in {C}_{c}^{1}\left( \omega \right) \) such that \( \alpha = 1 \) on supp \( u \) ; thus \( {\alpha u} = u \) . On the other hand (Theorem 9.2), there exists a sequence \( \left( {u}_{n}\... | Yes |
Proposition 9.18. Suppose \( \Omega \) is of class \( {C}^{1} \) . Let \( u \in {L}^{p}\left( \Omega \right) \) with \( 1 < p < \infty \) . The following properties are equivalent:\n\n(i) \( u \in {W}_{0}^{1, p}\left( \Omega \right) \) ,\n\n(ii) there exists a constant \( C \) such that\n\n\[ \left| {{\int }_{\Omega }u... | Proof.\n\n(i) \( \Rightarrow \) (ii). Let \( \left( {u}_{n}\right) \) be a sequence from \( {C}_{c}^{1}\left( \Omega \right) \) such that \( {u}_{n} \rightarrow u \) in \( {W}^{1, p} \) . For \( \varphi \in {C}_{c}^{1}\left( {\mathbb{R}}^{N}\right) \) we have\n\n\[ \left| {{\int }_{\Omega }{u}_{n}\frac{\partial \varphi... | Yes |
Proposition 9.20. Let \( F \in {W}^{-1,{p}^{\prime }}\left( \Omega \right) \) . Then there exist functions \( {f}_{0},{f}_{1},{f}_{2},\ldots ,{f}_{N} \in \) \( {L}^{{p}^{\prime }}\left( \Omega \right) \) such that\n\n\[ \langle F, v\rangle = {\int }_{\Omega }{f}_{0}v + \mathop{\sum }\limits_{{i = 1}}^{N}{\int }_{\Omega... | Proof. Adapt the proof of Proposition 8.14. | No |
Example 1 (homogeneous Dirichlet problem for the Laplacian). Let \( \Omega \subset {\mathbb{R}}^{N} \) be an open bounded set. We are looking for a function \( u : \bar{\Omega } \rightarrow \mathbb{R} \) satisfying\n\n(31)\n\[ \n\begin{cases} - {\Delta u} + u & = f & & \text{ in }\Omega , \\ u & = 0 & & \text{ on }\Gam... | Step A: Every classical solution is a weak solution.\n\nIndeed, \( u \in {H}^{1}\left( \Omega \right) \cap C\left( \bar{\Omega }\right) \) and \( u = 0 \) on \( \Gamma \), so that \( u \in {H}_{0}^{1}\left( \Omega \right) \) by Theorem 9.17 (see also Remark 19). On the other hand, if \( v \in {C}_{c}^{1}\left( \Omega \... | Yes |
Example 2 (inhomogeneous Dirichlet condition). Let \( \Omega \subset {\mathbb{R}}^{N} \) be a bounded open set. We look for a function \( u : \bar{\Omega } \rightarrow \mathbb{R} \) satisfying\n\n(33)\n\n\[ \begin{cases} - {\Delta u} + u & = f & & \text{ in }\Omega , \\ u & = g & & \text{ on }\Gamma , \end{cases} \]\n\... | Proof. We claim that \( u \in K \) is a weak solution of (33) if and only if we have\n\n(35)\n\n\[ {\int }_{\Omega }\nabla u \cdot \left( {\nabla v - \nabla u}\right) + {\int }_{\Omega }u\left( {v - u}\right) \geq {\int }_{\Omega }f\left( {v - u}\right) \;\forall v \in K. \]\n\nIndeed, if \( u \) is a weak solution of ... | Yes |
Let \( \Omega \subset {\mathbb{R}}^{N} \) be an open bounded set. We are given functions \( {a}_{ij}\left( x\right) \in {C}^{1}\left( \bar{\Omega }\right) ,1 \leq i, j \leq N \), satisfying the ellipticity condition\n\n(36)\n\n\[ \mathop{\sum }\limits_{{i, j = 1}}^{N}{a}_{ij}\left( x\right) {\xi }_{i}{\xi }_{j} \geq \a... | A classical solution of (37) is a function \( u \in {C}^{2}\left( \bar{\Omega }\right) \) satisfying (37) in the usual sense. A weak solution of (37) is a function \( u \in {H}_{0}^{1}\left( \Omega \right) \) satisfying\n\n(38)\n\n\[ {\int }_{\Omega }\mathop{\sum }\limits_{{i, j = 1}}^{N}{a}_{ij}\frac{\partial u}{\part... | Yes |
Theorem 9.23. If \( f = 0 \), then the set of solutions \( u \in {H}_{0}^{1} \) of (40) is a finite-dimensional vector space, say of dimension d. Moreover, there exists a subspace \( F \subset {L}^{2}\left( \Omega \right) \) of dimension \( d \) such that \( {}^{24} \n\n\[ \n\left\lbrack {\left( {40}\right) \text{ has ... | Proof. Fix \( \lambda > 0 \), large enough that the bilinear form\n\n\[ \na\left( {u, v}\right) + \lambda {\int }_{\Omega }{uv} \n\]\n\nis coercive on \( {H}_{0}^{1} \) . For every \( f \in {L}^{2} \) there exists a unique \( u \in {H}_{0}^{1} \) satisfying\n\n\[ \na\left( {u,\varphi }\right) + \lambda {\int }_{\Omega ... | Yes |
Let \( \Omega \subset {\mathbb{R}}^{N} \) be a bounded domain of class \( {C}^{1} \). We look for a function \( u : \bar{\Omega } \rightarrow \mathbb{R} \) satisfying\n\n(44)\n\n\[ \begin{cases} - {\Delta u} + u & = f & & \text{ in }\Omega , \\ \frac{\partial u}{\partial n} & = 0 & & \text{ on }\Gamma , \end{cases} \]\... | Step A: Every classical solution is a weak solution.\n\nRecall that by Green's formula we have\n\n(46)\n\n\[ {\int }_{\Omega }\left( {\Delta u}\right) v = {\int }_{\Gamma }\frac{\partial u}{\partial n}{vd\sigma } - {\int }_{\Omega }\nabla u \cdot \nabla v\;\forall u \in {C}^{2}\left( \bar{\Omega }\right) ,\;\forall v \... | No |
For every \( f \in {L}^{2}\left( \Omega \right) \), there exists a unique weak solution \( u \in {H}^{1}\left( \Omega \right) \) of (44). Furthermore, \( u \) is obtained by\n\n\[\n\mathop{\min }\limits_{{v \in {H}^{1}\left( \Omega \right) }}\left\{ {\frac{1}{2}{\int }_{\Omega }\left( {{\left| \nabla v\right| }^{2} + {... | Proof. Apply Lax-Milgram in \( H = {H}^{1}\left( \Omega \right) \) . | No |
Theorem 9.26 (regularity for the Neumann problem). With the same assumptions as in Theorem 9.25 one obtains the same conclusions for the solution of the Neumann problem, i.e., for \( u \in {H}^{1}\left( \Omega \right) \) such that\n\n\[{\int }_{\Omega }\nabla u \cdot \nabla \varphi + {\int }_{\Omega }{u\varphi } = {\in... | We shall prove only Theorem 9.25 ; the proof of Theorem 9.26 is entirely analogous. The main idea of the proof is the following. We consider first the case \( \Omega = {\mathbb{R}}^{N} \) , then the case \( \Omega = {\mathbb{R}}_{ + }^{N} \) . For a general domain \( \Omega \) we proceed in two steps:\n\n1. Interior re... | No |
Lemma 9.6. We have\n\n\[ \n{\\begin{Vmatrix}{D}_{h}v\\end{Vmatrix}}_{{L}^{2}\\left( \\Omega \\right) } \\leq \\parallel \\nabla v{\\parallel }_{{L}^{2}\\left( \\Omega \\right) }\\;\\forall v \\in {H}^{1}\\left( \\Omega \\right) ,\\;\\forall h\\parallel \\Gamma .\n\] | Proof. Start with \( v \\in {C}_{c}^{1}\\left( {\\mathbb{R}}^{N}\\right) \) and follow the proof of Proposition 9.3 (note that \( \\Omega + {th} = \\Omega \) for all \( t \) and all \( h\\parallel \\Gamma ) \) . For a general \( v \\in {H}^{1}\\left( \\Omega \\right) \) argue by density. | No |
Lemma 9.7. Let \( u \in {H}^{2}\left( \Omega \right) \cap {H}_{0}^{1}\left( \Omega \right) \) satisfying (48). Then \( {Du} \in {H}_{0}^{1}\left( \Omega \right) \) and, moreover,\n\n(58)\n\n\[ \int \nabla \left( {Du}\right) \cdot \nabla \varphi + \int \left( {Du}\right) \varphi = \int \left( {Df}\right) \varphi \;\fora... | Proof. The only delicate point consists in proving that \( {Du} \in {H}_{0}^{1}\left( \Omega \right) \), since (58) is derived from (48) by choosing \( {D\varphi } \) instead of \( \varphi \) (with \( \varphi \in {C}_{c}^{\infty }\left( \Omega \right) \) ) and then arguing by density. Let \( h = \left| h\right| {e}_{j}... | Yes |
Lemma 9.8. With the above notation, w belongs to \( {H}_{0}^{1}\left( {Q}_{ + }\right) \) and satisfies\n\n\[ \mathop{\sum }\limits_{{k,\ell = 1}}^{N}{\int }_{{Q}_{ + }}{a}_{k\ell }\frac{\partial w}{\partial {y}_{k}}\frac{\partial \psi }{\partial {y}_{\ell }}{dy} = {\int }_{{Q}_{ + }}\widetilde{g}{\psi dy}\;\forall \ps... | Proof. Let \( \psi \in {H}_{0}^{1}\left( {Q}_{ + }\right) \) and set \( \varphi \left( x\right) = \psi \left( {Jx}\right) \) for \( x \in \Omega \cap {U}_{i} \) . Then \( \varphi \in \) \( {H}_{0}^{1}\left( {\Omega \cap {U}_{i}}\right) \) and\n\n\[ \frac{\partial v}{\partial {x}_{j}} = \mathop{\sum }\limits_{k}\frac{\p... | Yes |
Proposition 9.29. Suppose that the functions \( {a}_{ij} \in {L}^{\infty }\left( \Omega \right) \) satisfy the ellipticity condition (36), and that \( {a}_{i},{a}_{0} \in {L}^{\infty }\left( \Omega \right) \) with \( {a}_{0} \geq 0 \) in \( \Omega \) . Let \( f \in {L}^{2}\left( \Omega \right) \) and \( u \in {H}^{1}\l... | Proof. We prove this result in the case \( {a}_{i} \equiv 0,1 \leq i \leq N \) ; the general case is more delicate (see D. Gilbarg-N. Trudinger [1], Theorem 8.1). To establish (79) is the same as showing that\n\n\( \left( {79}^{\prime }\right) \)\n\n\[ \n\left\lbrack {u \leq 0\text{ on }\Gamma \text{ and }f \leq 0\text... | No |
Proposition 9.30 (maximum principle for the Neumann problem). Let \( f \in \) \( {L}^{2}\left( \Omega \right) \) and \( u \in {H}^{1}\left( \Omega \right) \) be such that\n\n\[ \n{\int }_{\Omega }\nabla u \cdot \nabla \varphi + {\int }_{\Omega }{u\varphi } = {\int }_{\Omega }{f\varphi }\;\forall \varphi \in {H}^{1}\lef... | Proof. Analogous to the proof of Theorem 9.27. | No |
Let \( u \in {L}^{p}\left( \Omega \right) \cap {W}^{2, r}\left( \Omega \right) \) with \( 1 \leq p \leq \infty \) and \( 1 \leq r \leq \infty \). Then \( u \in {W}^{1, q}\left( \Omega \right) \), where \( q \) is the harmonic mean of \( p \) and \( r \), i.e., \( \frac{1}{q} = \frac{1}{2}\left( {\frac{1}{p} + \frac{1}{... | \n\[
\parallel {Du}{\parallel }_{{L}^{q}} \leq C\parallel u{\parallel }_{{W}^{2, r}}^{1/2}\parallel u{\parallel }_{{L}^{p}}^{1/2}.
\] | Yes |
Lemma 9.9. Let \( \Omega = {\mathbb{R}}_{ + }^{N} \) . There exists a constant \( C \) such that\n\n\[{\left( {\int }_{{\mathbb{R}}^{N - 1}}{\left| u\left( {x}^{\prime },0\right) \right| }^{p}d{x}^{\prime }\right) }^{1/p} \leq C\parallel u{\parallel }_{{W}^{1, p}\left( \Omega \right) }\;\forall u \in {C}_{c}^{1}\left( ... | Proof. Let \( G\left( t\right) = {\left| t\right| }^{p - 1}t \) and let \( u \in {C}_{c}^{1}\left( {\mathbb{R}}^{N}\right) \) . We have\n\n\[G\left( {u\left( {{x}^{\prime },0}\right) }\right) = - {\int }_{0}^{+\infty }\frac{\partial }{\partial {x}_{N}}G\left( {u\left( {{x}^{\prime },{x}_{N}}\right) }\right) d{x}_{N}\]\... | Yes |
Theorem 9.36 (Hopf). Let \( u \in C\left( \bar{\Omega }\right) \cap {C}^{2}\left( \Omega \right) \) satisfy\n\n(91)\n\n\[ \n- \mathop{\sum }\limits_{{i, j}}\frac{\partial }{\partial {x}_{j}}\left( {{a}_{ij}\frac{\partial u}{\partial {x}_{i}}}\right) + \mathop{\sum }\limits_{i}{a}_{i}\frac{\partial u}{\partial {x}_{i}} ... | For the proof, see, e.g., L. Bers-F. John-M. Schechter [1], D. Gilbarg-N. Tru-dinger [1], M. Protter-H. Weinberger [1], and P. Pucci-J. Serrin [1]. | No |
If \( {u}_{0} \in {H}_{0}^{1}\left( \Omega \right) \) then the solution \( u \) of (1),(2),(3) satisfies\n\n\[ u \in C\left( {\lbrack 0,\infty );{H}_{0}^{1}\left( \Omega \right) }\right) \cap {L}^{2}\left( {0,\infty ;{H}^{2}\left( \Omega \right) }\right) \]\n\nand\n\n\[ \frac{\partial u}{\partial t} \in {L}^{2}\left( {... | Proof of (a). We work here in the space \( {H}_{1} = {H}_{0}^{1}\left( \Omega \right) \) equipped with the scalar product\n\n\[ {\left( u, v\right) }_{{H}_{1}} = {\int }_{\Omega }\nabla u \cdot \nabla v + {\int }_{\Omega }{uv}. \]\n\nIn \( {H}_{1} \) consider the unbounded operator \( {A}_{1} : D\left( {A}_{1}\right) \... | Yes |
Corollary 10.5. Let \( {u}_{0} \in C\left( \bar{\Omega }\right) \cap {L}^{2}\left( \Omega \right) \) with \( {u}_{0} = 0 \) on \( \Gamma \cdot {}^{6} \) Then the solution \( u \) of (1),(2),(3) belongs to \( C\left( \bar{Q}\right) \) . | Proof of Corollary 10.5. Let \( \left( {u}_{0n}\right) \) be a sequence of functions in \( {C}_{c}^{\infty }\left( \Omega \right) \) such that \( {u}_{0n} \rightarrow {u}_{0} \) in \( {L}^{\infty }\left( \Omega \right) \) and in \( {L}^{2}\left( \Omega \right) \) (the existence of such a sequence is easily established)... | Yes |
Theorem 10.8 (regularity). Assume that the initial data satisfy\n\n\[ \n{u}_{0} \in {H}^{k}\left( \Omega \right) ,{v}_{0} \in {H}^{k}\left( \Omega \right) \;\forall k, \n\]\n\nand the compatibility conditions\n\n\[ \n{\Delta }^{j}{u}_{0} = 0\;\text{ on }\Gamma \;\forall j \geq 0, j\text{ integer,}\n\]\n\n\[ \n{\Delta }... | Proof of Theorem 10.7. As in Section 10.1 we consider \( u\left( {x, t}\right) \) as a vector-valued function defined on \( \lbrack 0,\infty ) \) ; more precisely, for each \( t \geq 0, u\left( t\right) \) denotes the map \( x \mapsto u\left( {x, t}\right) \) . We write (27) in the form of a system of first-order equat... | Yes |
Theorem 10.9 (J.-L. Lions). Given \( f \in {L}^{2}\left( {0, T;{V}^{ \star }}\right) \) and \( {u}_{0} \in H \), there exists a unique function \( u \) satisfying\n\n\[ u \in {L}^{2}\left( {0, T;V}\right) \cap C\left( {\left\lbrack {0, T}\right\rbrack ;H}\right) ,\;\frac{du}{dt} \in {L}^{2}\left( {0, T;{V}^{ \star }}\r... | For a proof see, e.g., J.-L. Lions-E. Magenes [1]. | No |
Theorem 10.10. Assume \( {u}_{0} \in {L}^{2}\left( \Omega \right) \) and \( f \in {C}^{\infty }\left( {\bar{\Omega } \times \left\lbrack {0, T}\right\rbrack }\right) \) . Then the solution \( u \) of (43) belongs to \( {C}^{\infty }\left( {\bar{\Omega } \times \left\lbrack {\varepsilon, T}\right\rbrack }\right) \) for ... | For a proof, see, e.g., J.-L. Lions-E. Magenes [1], A. Friedman [1], [2], and O. Ladyzhenskaya-V. Solonnikov-N. Uraltseva [1]; it is based on estimates very similar to those presented in Chapter 7 and in Section 10.1. | No |
Theorem 10.11 ( \( {L}^{2} \) -regularity). Given \( f \in {L}^{2}\left( {\Omega \times \left( {0, T}\right) }\right) \) and \( {u}_{0} \in {H}_{0}^{1}\left( \Omega \right) \) , there is a unique solution of (44) satisfying\n\n\[ u \in C\left( {\left\lbrack {0, T}\right\rbrack ;{H}_{0}^{1}\left( \Omega \right) }\right)... | The proof is easy; see, e.g., J.-L. Lions-E. Magenes [1]. | No |
Theorem 10.12 ( \( {L}^{p} \) -regularity). Given \( f \in {L}^{p}\left( {\Omega \times \left( {0, T}\right) }\right) \) with \( 1 < p < \infty \) and \( {u}_{0} = 0,{}^{15} \) there exists a unique solution of (44) satisfying | \[ u,\frac{\partial u}{\partial t},\frac{\partial u}{\partial {x}_{i}},\frac{{\partial }^{2}u}{\partial {x}_{i}\partial {x}_{j}} \in {L}^{p}\left( {\Omega \times \left( {0, T}\right) }\right) \;\forall i, j. \] | No |
Theorem 10.14 (J.-L. Lions). Given \( f \in {L}^{2}\left( {0, T;H}\right) ,{u}_{0} \in V \), and \( {v}_{0} \in H \), there exists a unique function \( u \) satisfying\n\n\[ u \in C\left( {\left\lbrack {0, T}\right\rbrack ;V}\right) ,\;\frac{du}{dt} \in C\left( {\left\lbrack {0, T}\right\rbrack ;H}\right) ,\;\frac{{d}^... | For a proof, see, e.g., J.-L. Lions-E. Magenes [1]. | No |
Proposition 11.1. Let \( E \) be a Banach space and let \( X \subset E \) be a finite-dimensional space. Then \( X \) is closed. | Proof. Assume that \( \left( {x}_{n}\right) \) is a sequence in \( X \) such that \( {x}_{n} \rightarrow x \) in \( E \) . Then \( \left( {x}_{n}\right) \) is a Cauchy sequence in \( X \) and thus \( \left( {x}_{n}\right) \) converges to a limit in \( X \) . Hence \( x \in X \) . | Yes |
Proposition 11.2. Assume that \( X \) is finite-dimensional and \( F \) is a Banach space. Then every linear operator \( T : X \rightarrow F \) must be bounded. | Proof. Let \( \left( {e}_{i}\right) \) be a basis in \( X \) and write \( x = \mathop{\sum }\limits_{{i = 1}}^{p}{x}_{i}{e}_{i} \) . Then \( {Tx} = \mathop{\sum }\limits_{{i = 1}}^{p}{x}_{i}T{e}_{i} \) , so that \( \parallel {Tx}\parallel \leq \mathop{\sum }\limits_{{i = 1}}^{p}\left| {x}_{i}\right| \begin{Vmatrix}{T{e... | Yes |
Proposition 11.3. Assume that \( X \) is a Banach space (with \( \dim X \leq \infty \) ) such that \( {X}^{ \star } \) is finite-dimensional. Then \( X \) is finite-dimensional and \( \dim X = \dim {X}^{ \star } \) . | Proof. We need Hahn-Banach, or more precisely Corollary 1.4. Let \( J : X \rightarrow {X}^{\star \star } \) be the canonical injection defined in Section 1.3. Since \( \dim {X}^{ \star } < \infty \), we deduce from the above discussion that \( \dim {X}^{\star \star } < \infty \) . But \( X \) is isomorphic to \( J\left... | Yes |
Proposition 11.5. Let \( E \) be a Banach space and let \( M \) be a closed subspace of \( E \) of finite codimension. Then any subspace \( \widetilde{M} \) of \( E \) containing \( M \) must be closed. | Proof. The space \( M \) admits an algebraic complement in \( \widetilde{M} \), say \( X \) . Clearly \( \dim X < \) \( \infty \), and \( \widetilde{M} = X + M \) . Applying Proposition 11.4, we see that \( \widetilde{M} \) is closed. | Yes |
Proposition 11.6. Let \( E \) be a Banach space and let \( M \) be a closed subspace of \( E \) of finite codimension. Let \( D \) be a dense subspace of \( E \) . Then there exists a complement \( X \) of \( M \) with \( X \subset D \) . | Proof. Let \( d \) be the codimension of \( M \) in \( E \) . If \( d = 0 \), we have \( M = E \) and we may take \( X = \{ 0\} \) . Hence we may assume that \( d \geq 1 \) . Fix any \( {x}_{1} \in D \) with \( {x}_{1} \notin M \) ; this is possible, for otherwise \( D \subset M \) implies \( E = \bar{D} \subset M \neq... | Yes |
Proposition 11.7. Let \( E \) be a Banach space and let \( G, L \subset E \) be closed subspaces. Assume that there exist finite-dimensional spaces \( {X}_{1},{X}_{2} \subset E \) such that\n\n(1)\n\[ G + L + {X}_{1} = E \]\n\n(2)\n\[ G \cap L \subset {X}_{2}\text{.} \]\n\nThen \( G \) (resp. \( L \) ) admits a complem... | Proof. We divide the proof into two steps.\n\nStep 1: The conclusion of Proposition 11.7 holds when \( {X}_{2} = \{ 0\} \) .\n\nLet \( {\widetilde{X}}_{1} \) be a complement of \( \left( {G + L}\right) \cap {X}_{1} \) in \( {X}_{1} \) . We already know by Proposition 11.4 that \( \left( {L + {\widetilde{X}}_{1}}\right)... | Yes |
Proposition 11.8. The quotient space \( E/M \) equipped with the norm \( \parallel {\parallel }_{E/M} \) is a Banach space. | Proof. Let \( \left( {\pi \left( {x}_{k}\right) }\right) \) be a Cauchy sequence in \( E/M \) . We have to show that \( \left( {\pi \left( {x}_{k}\right) }\right) \) converges, and since \( \left( {\pi \left( {x}_{k}\right) }\right) \) is Cauchy, it suffices to prove that a subsequence converges. Passing to a subsequen... | Yes |
Proposition 11.9. Let \( M \) be a closed subspace of \( E \) and let \( {\pi }^{ \star } : {\left( E/M\right) }^{ \star } \rightarrow {E}^{ \star } \) be the adjoint of \( \pi : E \rightarrow E/M \) . Then \( R\left( {\pi }^{ \star }\right) = {M}^{ \bot } \), and more precisely, \( {\pi }^{ \star } \) is bijective fro... | Proof. With \( \xi \in {\left( E/M\right) }^{ \star } \) and \( x \in E \), write\n\n\[ \n\left\langle {{\pi }^{ \star }\left( \xi \right), x}\right\rangle = \langle \xi ,\pi \left( x\right) \rangle \n\]\n\nIf \( x \in M \) we have \( \pi \left( x\right) = 0 \) and thus \( \left\langle {{\pi }^{ \star }\left( \xi \righ... | Yes |
Proposition 11.10. For any Banach space \( E \) and any closed subspace \( M \) of \( E \), the operator \( \widetilde{T} \) is a bijective isometry from \( {E}^{ \star }/{M}^{ \bot } \) onto \( {M}^{ \star } \) . | Proof. We have only to show that \( \widetilde{T} \) is an isometry. Given any \( f \in {E}^{ \star } \), consider the functional \( {f}_{\mid M} \) on \( M \) . By Corollary 1.2 we know that there exists a functional \( \widetilde{f} \in {E}^{ \star } \) such that \( {\widetilde{f}}_{\mid M} = {f}_{\mid M} \) and \( \... | Yes |
Proposition 11.11. Assume that \( E \) is a reflexive Banach space and \( M \) is a closed subspace. Then \( E/M \) is reflexive. | Proof. We know that \( {E}^{ \star } \) is reflexive (see Corollary 3.21) and thus \( {M}^{ \bot } \) is also reflexive (being a closed subspace of \( {E}^{ \star } \) ; see Proposition 3.20). On the other hand, \( {M}^{ \bot } \) is isomorphic to \( {\left( E/M\right) }^{ \star } \) (by Proposition 11.9). Therefore \(... | Yes |
Proposition 11.12. Assume that \( E \) is a uniformly convex Banach space and \( M \) is a closed subspace. Then \( E/M \) is uniformly convex. | Proof. Let \( \pi \left( x\right) ,\pi \left( y\right) \in E/M \) be such that \( \parallel \pi \left( x\right) \parallel \leq 1,\parallel \pi \left( y\right) \parallel \leq 1 \), and \( \parallel \pi \left( x\right) - \) \( \pi \left( y\right) \parallel > \varepsilon \) . Since \( E \) is reflexive, we know (see Corol... | Yes |
Proposition 11.13. Let \( E \) be a Banach space and let \( M \subset E \) be a closed subspace. Then\n\n(a) \( \dim M < \infty \) if and only if \( \operatorname{codim}{M}^{ \bot } < \infty \), and in that case\n\n\[ \dim M = \operatorname{codim}{M}^{ \bot } \]\n\n(b) \( \operatorname{codim}M < \infty \) if and only i... | Proof.\n\n(a) We know by Proposition 11.10 that \( {E}^{ \star }/{M}^{ \bot } \) is always isomorphic to \( {M}^{ \star } \) . Thus \( \dim {M}^{ \star } < \infty \Leftrightarrow \dim \left( {{E}^{ \star }/{M}^{ \bot }}\right) < \infty \) . By Proposition 11.3 we know that \( \dim M < \infty \Leftrightarrow \dim {M}^{ ... | Yes |
Proposition 11.14. Let \( N \subset {E}^{ \star } \) be a closed subspace. Then \( \dim N < \infty \) if and only if \( \operatorname{codim}{N}^{ \bot } < \infty \), and in that case \( \dim N = \operatorname{codim}{N}^{ \bot } \) . It is also true that \( \dim {N}^{ \bot } \leq \operatorname{codim}N \), but it may hap... | Proof. Recall that\n\n\[ \n{N}^{ \bot } = \{ x \in E;\langle f, x\rangle = 0\;\forall f \in N\} .\n\]\n\nClearly \( \bar{N} \subset {N}^{ \bot \bot } \) ; but it may happen that \( \bar{N} \neq {N}^{ \bot \bot } \) (see Remark 6 in Chapter 1). For example, take \( \xi \in {E}^{\star \star } \) with \( \xi \notin E \) a... | Yes |
Proposition 11.15. The space \( {\ell }^{p} \) is reflexive, and even uniformly convex, for \( 1 < p < \infty \) . | Proof. Apply Theorem 4.10 and Exercise 4.12 with \( \Omega = \mathbb{N} \) . | No |
Proposition 11.16. The spaces \( c,{c}_{0} \), and \( {\ell }^{p} \), with \( 1 \leq p < \infty \), are separable. | Proof. Let\n\n\[ D = \left\{ {x = \left( {x}_{k}\right) ;{x}_{k} \in \mathbb{Q}\;\forall k,\text{ and }{x}_{k} = 0\text{ for }k\text{ sufficiently large }}\right\} .\n\]\n\nIt is clear that \( D \) is countable; moreover, \( D \) is dense in \( {\ell }^{p} \) when \( 1 \leq p < \infty \) and in \( {c}_{0} \) . The set ... | Yes |
Proposition 11.17. The space \( {\ell }^{\infty } \) is not separable. | Proof. Assume that \( A \subset {\ell }^{\infty } \) is countable. We will check that \( A \) cannot be dense in \( {\ell }^{\infty } \) . Write \( A = \left( {a}^{k}\right) \), where each \( {a}^{k} \in {\ell }^{\infty } \), so that \( {a}^{k} = \left( {{a}_{1}^{k},{a}_{2}^{k},\ldots }\right) \) . For each integer \( ... | Yes |
Proposition 11.18. Let \( 1 \leq p < \infty \) . Given any \( \phi \in {\left( {\ell }^{p}\right) }^{ \star } \), there exists a unique \( u \in {\ell }^{{p}^{\prime }} \) such that\n\n\[ \langle \phi, x\rangle = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}{x}_{k}\;\forall x \in {\ell }^{p}. \]\n\nMoreover,\n\n\[ \... | Proof. Let \( {e}_{k} = \left( {0,0,\ldots ,1,0,0,\ldots }\right) \) . Set \( {u}_{k} = \phi \left( {e}_{k}\right) \) . We claim that \( u = \left( {u}_{k}\right) \in \) \( {\ell }^{{p}^{\prime }} \) and\n\n(6)\n\n\[ \parallel u{\parallel }_{{p}^{\prime }} \leq \parallel \phi {\parallel }_{{\left( {\ell }^{p}\right) }^... | Yes |
Proposition 11.19. Given any \( \phi \in {\left( {c}_{0}\right) }^{ \star } \), there exists a unique \( u \in {\ell }^{1} \) such that\n\n\[ \langle \phi, x\rangle = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}{x}_{k}\;\forall x \in {c}_{0} \]\n\nMoreover,\n\n\[ \parallel u{\parallel }_{1} = \parallel \phi {\paral... | Proof. This is an easy adaptation of the proof of Proposition 11.18 (with \( p = \infty \) and \( {p}^{\prime } = 1 \) ); the last part of the proof holds since \( D \) is dense in \( {c}_{0} \) (but not in \( {\ell }^{\infty } \) ). | No |
Proposition 11.20. Given \( \phi \in {\left( c\right) }^{ \star } \), there exists a unique pair \( \left( {u,\lambda }\right) \in {\ell }^{1} \times \mathbb{R} \) such that\n\n\[ \langle \phi, x\rangle = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}{x}_{k} + \lambda \mathop{\lim }\limits_{{k \rightarrow \infty }}{x... | Proof. Applying Proposition 11.19 to \( {\phi }_{\mid {c}_{0}} \), we find some \( u \in {\ell }^{1} \) such that\n\n\[ \phi \left( y\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }{u}_{k}{y}_{k}\;\forall y \in {c}_{0} \]\n\nIf \( x \in c \) write \( x = y + {ae} \), where \( e = \left( {1,1,1,\ldots }\right), a = \... | Yes |
Proposition 11.21. The spaces \( {\ell }^{1},{\ell }^{\infty }, c \), and \( {c}_{0} \) are not reflexive. | Proof. From Propositions 11.19 and 11.18 we know that \( {\left( {c}_{0}\right) }^{ \star } \) is \( {\ell }^{1} \) and \( {\left( {\ell }^{1}\right) }^{ \star } \) is \( {\ell }^{\infty } \) . Therefore the identity map from \( {c}_{0} \) into \( {\ell }^{\infty } \) corresponds to the canonical injection \( J : {c}_{... | Yes |
Proposition 11.23. Let \( G \subset E \) be a linear subspace. If \( g : G \rightarrow \mathbb{C} \) is a continuous linear functional, then there exists \( f \in {E}^{ \star } \) that extends \( g \), and such that | Proof. Set \( \psi = \operatorname{Re}g \), so that \( \psi \) is an element of \( {G}_{\mathbb{R}}^{ \star } \) and \( \parallel \psi {\parallel }_{{G}_{\mathbb{R}}^{ \star }} = \parallel g{\parallel }_{{G}^{ \star }} \) . By Corollary 1.2 there exists some \( \varphi \in {E}_{\mathbb{R}}^{ \star } \) that extends \( ... | Yes |
Proposition 11.24. Let \( A, B \subset E \) be two nonempty convex subsets of \( E \) such that \( A \cap B = \varnothing \) . Assume that one of them is open. Then there exists a closed real hyperplane that separates \( A \) and \( B \) . | Proof. Applying Theorem 1.6 to \( {E}_{\mathbb{R}} \) yields a hyperplane \( H = \left\lbrack {\varphi = \alpha }\right\rbrack \) for some \( \varphi \in {E}_{\mathbb{R}}^{ \star } \) that separates \( A \) and \( B \) in the usual sense. Then use Proposition 11.23 to assert that \( \varphi = \operatorname{Re}f \) for ... | No |
Proposition 11.25. Assume that \( \varphi : E \rightarrow ( - \infty , + \infty \rbrack \) is convex, l.s.c., and \( \varphi ≢ \) \( + \infty \) . Then \( {\varphi }^{\star \star } = \varphi \) . | Proof. There are two methods. Either one can apply Theorem 1.11 to \( \widetilde{\varphi } = \varphi \) viewed on \( {E}_{\mathbb{R}} \), in conjunction with Proposition 11.22. Or one can repeat the proof of Theorem 1.11; when Hahn-Banach is used, one can separate the convex sets \( A \) and \( B \) using a real hyperp... | No |
Proposition 11.27. Given any \( \varphi \in {H}^{ \star } \) there exists a unique \( f \in H \) such that\n\n\[ \varphi \left( u\right) = \left( {u, f}\right) \;\forall u \in H. \]\n\nMoreover,\n\n\[ \left| f\right| = \parallel \varphi {\parallel }_{{H}^{ \star }} \] | Proof. Applying Theorem 5.5 to \( \operatorname{Re}\varphi \) in \( {H}_{\mathbb{R}} \), we find some \( f \in H \) such that\n\n\[ \operatorname{Re}\varphi \left( u\right) = \operatorname{Re}\left( {u, f}\right) \;\forall u \in H. \]\n\nApplying this to \( {iu} \) yields \( \operatorname{Im}\varphi \left( u\right) = \... | Yes |
Proposition 11.28. Assume that a satisfies (13), (14), and (15). Let \( K \) be a nonempty closed convex set in \( H \). Then given any \( \varphi \in {H}^{ \star } \) there exists a unique \( u \in K \) such that\n\n(16)\n\n\[ \operatorname{Re}a\left( {u, v - u}\right) \geq \operatorname{Re}\langle \varphi, v - u\rang... | Moreover, if \( a\left( {v, w}\right) = \overline{a\left( {w, v}\right) }\forall v, w \in H \), then \( u \) is characterized by the property\n\n\[ u \in K\;\text{ and }\;\frac{1}{2}a\left( {u, u}\right) - \operatorname{Re}\langle \varphi, u\rangle = \mathop{\min }\limits_{{v \in K}}\left\{ {\frac{1}{2}a\left( {v, v}\r... | Yes |
Proposition 11.29 (Lax-Milgram). Assume that \( T \in \mathcal{L}\left( H\right) \) satisfies\n\n(18)\n\n\[ \left| \left( {{Tu}, u}\right) \right| \geq \alpha {\left| u\right| }^{2}\;\forall u \in H,\text{ for some }\alpha > 0. \]\n\nThen \( T \) is bijective. | Proof. See Remark 8 in Chapter 5. | No |
Proposition 11.30. The spectrum \( \sigma \left( T\right) \) is a nonempty compact set and\n\n\[ \sigma \left( T\right) \subset \{ \lambda \in \mathbb{C};\left| \lambda \right| \leq \parallel T\parallel \} \] | Proof. The main novelty is that \( \sigma \left( T\right) \) is nonempty. The proof relies on the theory of analytic functions on \( \mathbb{C} \) (more precisely Liouville’s theorem) and we will not present it here. The interested reader may consult A. Taylor-D. Lay [1], W. Rudin [2], or A. Knaap [2]. | No |
Proposition 11.31. For every \( T \in \mathcal{L}\left( E\right) \) we have\n\n\[ r\left( T\right) = \max \{ \left| \lambda \right| ;\lambda \in \sigma \left( T\right) \} . \] | For the proof we refer again to A. Taylor–D. Lay [1], W. Rudin [2], or A. Knaap [2]. The argument relies heavily on the fact that \( E \) is a Banach space over \( \mathbb{C} \) through the theory of power series on \( \mathbb{C} \) . When \( E \) is a Banach space over \( \mathbb{R} \) we can say only that \( \max \{ ... | No |
Proposition 11.32. We have\n\n\[ Q\left( {{EV}\left( T\right) }\right) = {EV}\left( {Q\left( T\right) }\right) \]\n\nand\n\n\[ Q\left( {\sigma \left( T\right) }\right) = \sigma \left( {Q\left( T\right) }\right) \] | Proof. We already know that (20) holds (the argument is the same as in Exercise 6.22). Assume by contradiction that the inclusions are strict. Then there exists \( \mu \in {EV}\left( {Q\left( T\right) }\right) \) such that \( \mu \notin Q\left( {{EV}\left( T\right) }\right) \) . Write\n\n\[ Q\left( t\right) - \mu = \al... | Yes |
Proposition 11.33. We have\n\n\\[ \n\\sigma \\left( T\\right) \\subset \\overline{W\\left( T\\right) },\n\\]\n\nand more precisely, if \\( \\lambda \\notin \\overline{W\\left( T\\right) } \\), then \\( \\lambda \\in \\rho \\left( T\\right) \\) with\n\n(23)\n\\[ \n\\begin{Vmatrix}{\\left( T - \\lambda I\\right) }^{-1}\\... | Proof. Assume that \\( \\lambda \\notin \\overline{W\\left( T\\right) } \\) and set \\( \\alpha = \\operatorname{dist}\\left( {\\lambda, W\\left( T\\right) }\\right) \\) . We have\n\n\\[ \n\\left| {\\left( {{Tu}, u}\\right) - \\lambda }\\right| \\geq \\alpha \\;\\forall u \\in H\\text{ with }\\left| u\\right| = 1.\n\\]... | Yes |
Proposition 11.35. Let \( H \) be a Hilbert space over \( \mathbb{C} \) and let \( T \) be a normal operator. Then\n\n\[ \max \{ \left| \lambda \right| ;\lambda \in \sigma \left( T\right) \} = \parallel T\parallel . \] | Proof. Since \( T \) is normal, we have\n\n\[ \begin{Vmatrix}{T}^{p}\end{Vmatrix} = \parallel T{\parallel }^{p}\text{for every integer}p \geq 1\text{.} \]\n\nThis is proved in Problem 43 when \( H \) is a Hilbert space over \( \mathbb{R} \), and the same argument remains valid when \( H \) is a Hilbert space over \( \m... | Yes |
Proposition 11.36. Let \( H \) be a separable Hilbert space over \( \mathbb{C} \) and let \( T \) be a compact normal operator, then there exists a Hilbert basis composed of eigenvectors of \( T \) (but the corresponding eigenvalues need not be real). | Proof. If \( T \) is normal, so is \( \left( {T - {\lambda I}}\right) \) for any \( \lambda \in \mathbb{C} \) . Therefore (as in Problem 43) we have \( N\left( {T - {\lambda I}}\right) = N\left( {\left( T - \lambda I\right) }^{ \star }\right) = N\left( {{T}^{ \star } - \bar{\lambda }I}\right) \) . It follows that \( N\... | Yes |
Proposition 11.37. Let \( T \) be an isometry. Then\n\n\[ \n{EV}\left( T\right) \subset {S}^{1} = \{ \lambda \in \mathbb{C};\left| \lambda \right| = 1\} .\n\]\n\nIf \( T \) is a unitary operator, then\n\n\[ \n\sigma \left( T\right) \subset {S}^{1}\n\]\n\nand if \( T \) is not a unitary operator, then\n\n\[ \n\sigma \le... | The proof is an easy adaptation of the one given in the solution of Problem 44, question 6. | No |
2. Separating \( \{ 0\} \) and \( {C}_{1} \) we find some \( {x}_{1} \in E \) and a constant \( \alpha \) such that \( 0 < \alpha < \) \( \left\langle {f,{x}_{1}}\right\rangle \forall f \in {C}_{1} \) . If needed, replace \( {x}_{1} \) by a multiple of \( {x}_{1} \) . | 3. One has to find a finite subset \( A \subset E \) such that \( A \subset \left( {1/{d}_{1}}\right) {B}_{E} \) and \( {Y}_{A} = \varnothing \) . We first claim that \( \mathop{\bigcap }\limits_{{A \in \mathcal{F}}}{Y}_{A} = \varnothing \), where \( \mathcal{F} \) denotes the family of all finite subsets \( A \) in \(... | No |
Problem 13 | \[ \text{- A -} \] 1. By question 5 of Exercise 1.25 we know that \[ \mathop{\lim }\limits_{\substack{{\lambda \rightarrow 0} \\ {\lambda > 0} }}\frac{1}{2\lambda }\left( {\parallel x + {\lambda y}{\parallel }^{2} - \parallel x{\parallel }^{2}}\right) = \langle {Fx}, y\rangle \] If \( \lambda < 0 \) set \( \mu = - \lam... | No |
Theorem 1.15.1 Free ultrafilters over any infinite set exist. | Proof: The proof is given in the appendix. | Yes |
Theorem 1.16.1 Let \( M \) be any infinite subset of \( \mathbb{N} \), then there is a free ultrafilter \( U \) over \( \mathbb{N} \) such that \( M \in U \) . | Proof: Let \( D = \mathbb{N} - M \), then if \( Q \subseteq \mathbb{N} \) then \( Q = {Q}^{\prime } \cup R \) for unique \( {Q}^{\prime } \subseteq M \) and \( R \subseteq D \) . Let \( {U}^{\prime } \) be a free ultrafilter over \( M \), and let \( U \) be the set of all \( Q \) such that \( {Q}^{\prime } \in {U}^{\pr... | No |
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