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Proposition 6.3. Let \( E, F \), and \( G \) be three Banach spaces. Let \( T \in \mathcal{L}\left( {E, F}\right) \) and \( S \in \mathcal{K}\left( {F, G}\right) \) [resp. \( T \in \mathcal{K}\left( {E, F}\right) \) and \( S \in \mathcal{L}\left( {F, G}\right) \) ]. Then \( S \circ T \in \mathcal{K}\left( {E, G}\right)... | The proof is obvious. 紧算子与连续算子的复合是紧算子 \( \; \) 于是 \( \;\xi \) 结合 | No |
Theorem 6.4 (Schauder). If \( T \in \mathcal{K}\left( {E, F}\right) \), then \( {T}^{ \star } \in \mathcal{K}\left( {{F}^{ \star },{E}^{ \star }}\right) \) . And conversely. | Proof. We have to show that \( {T}^{ \star }\left( {B}_{{F}^{ \star }}\right) \) has compact closure in \( {E}^{ \star } \) . Let \( \left( {v}_{n}\right) \) be a \( {}^{ \dagger } * \( x \rightarrow x \) - \( y \) - \( z \) sequence in \( {B}_{{F}^{ \star }} \) . We claim that \( \left( {{T}^{ \star }\left( {v}_{n}\ri... | Yes |
Lemma 6.1 (Riesz’s lemma). Let \( E \) be an \( \mathrm{n} \) .v.s. and let \( M \subset E \) be a closed linear space such that \( M \neq E \) . Then 尤其在一列空间时使用较多\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 - \vareps... | Proof. Let \( v \in E \) with \( v \notin M \) . Since \( M \) is closed, then \( {ZOCM} \) dist(u, M)<I\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... | No |
Proposition 6.7. The spectrum \( \sigma \left( T\right) \) of a bounded operator \( T \) is compact and \[ \sigma \left( T\right) \subset \left\lbrack {-\parallel T\parallel , + \parallel T\parallel }\right\rbrack .\text{有界闭集} \] | 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\[{\lambda }_{n} \rightarrow \lambda\]\n\nand\n\n\[{\lambda }_{n} \in \sigma \left( T\right) \smallsetminus \{ 0\} \;\forall n.\]\n\nThen \( \lambda = 0 \). | 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 |
Proposition 6.9. Let \( T \in \mathcal{L}\left( H\right) \) be a self-adjoint operator. Set\n\n\[ m = \mathop{\inf }\limits_{\substack{{u \in H} \\ {\left| u\right| = 1} }}\left( {{Tu}, u}\right) \;\text{ and }\;M = \mathop{\sup }\limits_{\substack{{u \in H} \\ {\left| u\right| = 1} }}\left( {{Tu}, u}\right) .\n\]\nThe... | Proof. Let \( \lambda > M \) ; we will prove that \( \lambda \in \rho \left( T\right) \) . We have\n\n\[ \left( {{Tu}, u}\right) \leq M{\left| u\right| }^{2}\;\forall u \in H \]\n\n\[ \left( {{\lambda u} - {Tu}, u}\right) \geq \left( {\lambda - M}\right) {\left| u\right| }^{2} = \frac{\alpha {\left| u\right| }^{2}}{\ex... | No |
Proposition 7.2. Let \( A \) be a maximal monotone operator. Then\n\n(a1)\n\n\[ \n{A}_{\lambda }v = A\left( {{J}_{\lambda }v}\right) \;\forall v \in H\text{ and }\forall \lambda > 0, \n\]\n\n(a2)\n\[ \n{A}_{\lambda }v = {J}_{\lambda }\left( {Av}\right) \;\forall v \in D\left( A\right) \text{ and }\forall \lambda > 0, \... | 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\[ \n{A}_{\lambda }v + A\left( {v - {J}_{\lambda }v}\right) = {Av} \n\]\n\ni.e.,\n\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 \( g \in {L}_{\mathrm{{loc}}}^{1}\left( I\right) \) ; 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 {... | 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{Vmatrix... | 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\[ \eta \widetilde{u} \in {W}^{1, p}\left( {0,\infty }\right) \;\text{ and }\;{\left( \eta \widetilde{u}\right) }^{\prime } = {\eta }^{\prime }\widetilde{u} + \eta \widetilde{{u}^{\prime }}. \] | 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\n\n\[ \mathop{\min }\limits_{{v \in {H}^{1}\left( I\right) }}\left\{ {\frac{1}{2}{\int }_{I}\left( {{v}^{\prime 2} + {v}^{2}}\ri... | Proof. If \( u \) is a classical solution of (21) we have\n\n(22)\n\n\[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv}\;\forall v \in {H}^{1}\left( I\right) . \]\n\nWe use \( {H}^{1}\left( I\right) \) as our function space: there is no point in working in \( {H}_{0}^{1} \) as above since \( u... | 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\n\n\[ \mathop{\min }\limits_{{v \in {H}^{1}\left( I\right) }}\left\{ {\frac{1}{2}{\int }_{I}\left( ... | Proof. If \( u \) is a classical solution of (24) we have\n\n\[ {\int }_{I}{u}^{\prime }{v}^{\prime } + {\int }_{I}{uv} = {\int }_{I}{fv} - {\alpha v}\left( 0\right) + {\beta v}\left( 1\right) \;\forall v \in {H}^{1}\left( I\right) . \]\n\nWe use \( {H}^{1}\left( I\right) \) as our function space and we apply the Lax-M... | 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 functi... | 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 9.4 (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 \[ 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 \( {ex} \) - tension by reflection. | 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 \( {\theta }_{0} \) , \( {\theta... | 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 B: Existence and uniqueness of a weak solution.\n\nThis is the content of the following basic result.\n\n- Theorem 9.21 (Dirichlet, Riemann, Poincaré, Hilbert). Given any \( f \in {L}^{2}\left( \Omega \right) \) , there exists a unique weak solution \( u \in {H}_{0}^{1}\left( \Omega \right) \) of (31). Furthermore... | Yes |
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\[ \begin{cases} - {\Delta u} + u & = f & & \text{ in }\Omega , \\ u & = g & & \text{ on }\Gamma , \end{cases} \]\n\nwhere \( f \) is given on \( \Omega \) and \( g \) is ... | Proof. We claim that \( u \in K \) is a weak solution of (33) if and only if we have\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 (33) it ... | No |
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 |
Example 4 (homogeneous Neumann problem). 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} & =... | 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 \... | Yes |
Proposition 9.24. 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\[ \mathop{\min }\limits_{{v \in {H}^{1}\left( \Omega \right) }}\left\{ {\frac{1}{2}{\int }_{\Omega }\left( {{\left| \nabla v... | Proof. Apply Lax-Milgram in \( H = {H}^{1}\left( \Omega \right) \). | 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. | Yes |
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}\;\fora... | 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}{... | \[\parallel {Du}{\parallel }_{{L}^{q}} \leq C\parallel u{\parallel }_{{W}^{2, r}}^{1/2}\parallel u{\parallel }_{{L}^{p}}^{1/2}.\n\] | 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 |
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\n\[\nG + L + {X}_{1} = E\n\]\n\n(2)\n\[\nG \cap L \subset {X}_{2}.\n\]\n\nThen \( G \) (resp. \( L \) ) admits a compleme... | 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 |
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 from \( {\left( E/M\r... | 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\[ D = \left\{ {x = \left( {x}_{k}\right) ;{x}_{k} \in \mathbb{Q}\;\forall k}\right. \text{, and}\left. {{x}_{k} = 0\text{for}k\text{sufficiently large}}\right\} \text{.} \]\n\nIt is clear that \( D \) is countable; moreover, \( D \) is dense in \( {\ell }^{p} \) when \( 1 \leq p < \infty \) and in \( {c}... | 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 {\para... | 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 11.26. Let \( K \subset H \) be a nonempty closed convex set. Then for every \( f \in H \) there exists a unique element \( u \in K \) such that\n\n\[ \left| {f - u}\right| = \mathop{\min }\limits_{{v \in K}}\left| {f - v}\right| = \operatorname{dist}\left( {f, K}\right) . | 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) = \o... | No |
Proposition 11.28. Assume that a satifies (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}\right... | 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. 点 \( \left( {U - {P}_{{R}_{f\left( T\right) }}\left( U\right), U}\right) = \varnothing \;\forall U \) e R(T) \( \; \Rightarrow \left( {{U}_{0},{\mathcal{T}}_{{U}_{1}}}\right) : 0 \) (T) \( U \) e R(T) \( \Rightarrow U \) e P(T) | No |
Proposition 11.30. The spectrum \( \sigma \left( T\right) \) is a nonempty compact set and ## \(\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 11.35. Let \( H \) be a Hilbert space over \( \mathbb{C} \) and let \( T \) be a normal operator. Then\n\[ \max \{ \left| \lambda \right| ;\lambda \in \sigma \left( T\right) \} = \parallel T\parallel . \] | Proof. Since \( T \) is normal, we have \( \left| {\tau }^{p}\right| \leq {\left| \tau 0\right| }^{p} \) 显然 \( = |{T}^{F + 1}u|\left( {{T}^{F - 1}u}\right) \) \( \leq \parallel {T}^{\mid c + 1}\parallel \left( {\mid {T}^{\mid c - 1}}\right) \parallel u \)\n\[ \begin{Vmatrix}{T}^{p}\end{Vmatrix} = \parallel T{\parallel ... | No |
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) 有 \( \lambda \left( {u, v}\right) \) \( = \left( {{Tu}, v}\right) \) we have \( N\left( {T - {\lambda I}}\right) = N\left( {\left( T - \lambda I\right) }^{ \star }\right) = N\left... | No |
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 |
1. \( {B}_{{E}^{ \star }} \) is compact and metrizable for the topology \( \sigma \left( {{E}^{ \star }, E}\right) \) (see Theorem 3.28). It follows, by a standard result in point-set topology, that there exists a subset in \( {B}_{{E}^{ \star }} \) that is countable and dense for \( \sigma \left( {{E}^{ \star }, E}\ri... | Since \( B \) is compact for \( \sigma \left( {E,{E}^{ \star }}\right) \), it follows (by Corollary 2.4) that \( B \) is bounded. Thus \( B \) is a compact (metric) space for the topology \( \mathcal{T} \) and, moreover, the topologies \( \sigma \left( {E,{E}^{ \star }}\right) \) and \( \mathcal{T} \) coincide on \( B ... | Yes |
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 | \[ - \mathbf{A} - \] 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 = - \l... | No |
3. Choose \( \bar{\lambda } \in A \) and \( \bar{\mu } \in B \) such that\n\n\[ \mathop{\min }\limits_{{\lambda \in A}}\mathop{\max }\limits_{{\mu \in B}}F\left( {\lambda ,\mu }\right) = \mathop{\max }\limits_{{\mu \in B}}F\left( {\bar{\lambda },\mu }\right) \;\text{ and }\;\mathop{\max }\limits_{{\mu \in B}}\mathop{\m... | 2. The sets \( {B}_{u} \) and \( {A}_{v} \) are compact for the weak topology. Applying the convexity of \( K \) in \( u \) and the concavity of \( K \) in \( v \), we obtain\n\n\[ K\left( {\mathop{\sum }\limits_{i}{\lambda }_{i}{u}_{i},{v}_{j}}\right) \leq \mathop{\sum }\limits_{i}{\lambda }_{i}K\left( {{u}_{i},{v}_{j... | No |
Theorem 1.1. Let \( X \) be a Banach space, and assume that \( K \) is an operator on \( X \) (i.e., maps \( X \) into itself) such that\n\n\[ \n\text{a)}K\left( {v + w}\right) = {Kv} + {Kw}\text{,}\n\]\n\n\[ \n\text{b)}K\left( {-v}\right) = - {Kv}\n\]\n\n\[ \n\text{c)}\parallel {Kv}\parallel \leq M\parallel v\parallel... | The uniqueness in Theorem 1.1 is trivial. In fact, suppose there were two solutions \( {f}_{1} \) and \( {f}_{2} \) of (1.22). Set \( f = {f}_{1} - {f}_{2} \) . Then by a) and b) we have\n\n\[ \nf = {Kf}\n\]\n\nFrom this we get\n\n\[ \nf = {K}^{2}f = {K}^{3}f = \cdots = {K}^{n}f\n\]\n\nfor each \( n \) . Thus,\n\n\[ \n... | Yes |
Lemma 1.2. Inequality (1.28) follows from the aforementioned properties. | Proof. Let \( \alpha \) be any scalar. Then\n\n\[ \left( {{\alpha f} + g,{\alpha f} + g}\right) = {\alpha }^{2}\left( {f, f}\right) + {2\alpha }\left( {f, g}\right) + \left( {g, g}\right) \]\n\n\[ = \left( {f, f}\right) \left\lbrack {{\alpha }^{2} + {2\alpha }\frac{\left( f, g\right) }{\left( f, f\right) } + \frac{{\le... | Yes |
Lemma 1.3. If a vector space \( X \) has a scalar product \( \left( {f, g}\right) \), then it is a normed vector space with norm \( \parallel f\parallel = {\left( f, f\right) }^{\frac{1}{2}} \) . | Proof. Again, the only thing that is not immediate is the triangle inequality. This follows from (1.28) since\n\n\[ \parallel f + g{\parallel }^{2} = \parallel f{\parallel }^{2} + \parallel g{\parallel }^{2} + 2\left( {f, g}\right) \]\n\n\[ \leq \parallel f{\parallel }^{2} + \parallel g{\parallel }^{2} + 2\parallel f\p... | No |
We want \( {l}_{\infty } \) to be a Banach space with norm \[ \parallel f\parallel = \mathop{\sup }\limits_{i}\left| {\alpha }_{i}\right| \] | As one will find in most examples, the only properties which are not immediately obvious are the triangle inequality and completeness. In this case the triangle inequality is not far from it, since \[ \mathop{\sup }\limits_{i}\left| {{\alpha }_{i} + {\beta }_{i}}\right| \leq \mathop{\sup }\limits_{i}\left( {\left| {\al... | Yes |
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