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Corollary 34.2.10 Let \( f,{f}^{\prime } \in {L}^{1}\left( {a, b;X}\right) \) so that\n\n\[ f\left( t\right) = f\left( 0\right) + {\int }_{0}^{t}{f}^{\prime }\left( s\right) {ds} \]\n\n(34.2.8)\n\nwhere in this formula, \( t \rightarrow f\left( t\right) \) is the continuous representative of \( f \) . Then there exists... | Proof: From the integral equation 34.2.8,\n\n\[ f\left( t\right) = f\left( s\right) + {\int }_{s}^{t}{f}^{\prime }\left( r\right) {dr} \]\n\n\[ \parallel f\left( t\right) {\parallel }_{X} \leq \parallel f\left( s\right) {\parallel }_{X} + \left| {{\int }_{s}^{t}{\begin{Vmatrix}{f}^{\prime }\left( r\right) \end{Vmatrix}... | Yes |
Corollary 34.2.12 Suppose \( f,{f}^{\prime } \in {L}^{1}\left( {a, b;X}\right) \) and suppose \( \phi \in {C}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) . Then the following integration by parts formula holds. | \[ {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} = f\left( b\right) \phi \left( b\right) - f\left( a\right) \phi \left( a\right) - {\int }_{a}^{b}{f}^{\prime }\left( t\right) \phi \left( t\right) {dt}. \] Proof: From Theorem 34.2.9 \[ {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\rig... | Yes |
Lemma 34.2.13 Let \( \bar{f} \) be given in Definition 34.2.2 and suppose \( f,{f}^{\prime } \in {L}^{1}\left( {a, b;X}\right) \) . Then \( \bar{f},{\bar{f}}^{\prime } \in {L}^{1}\left( {{2a} - b,{2b} - a;X}\right) \) also and\n\n\[{\bar{f}}^{\prime }\left( t\right) \equiv \left\{ \begin{array}{l} {f}^{\prime }\left( t... | Proof: It is clear from the definition of \( \bar{f} \) that \( \bar{f} \in {L}^{1}\left( {{2a} - b,{2b} - a;X}\right) \) and that in fact\n\n\[ \parallel \bar{f}{\parallel }_{{L}^{1}\left( {{2a} - b,{2b} - a;X}\right) } \leq 3\parallel f{\parallel }_{{L}^{1}\left( {a, b;X}\right) }.\]\n\n(34.2.10)\n\n\n\nLet \( \phi \... | Yes |
Theorem 34.2.15 Let \( V \) and \( H \) be a Banach space and Hilbert space as described in Definition 34.2.14. Suppose \( f \in {L}^{p}\left( {0, T;V}\right) \) and \( {f}^{\prime } \in {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \) . Then \( f \) is a.e. equal to a continuous function mapping \( \left\lbrac... | Proof: Let \( \Psi \in {C}_{c}^{\infty }\left( {-T,{2T}}\right) \) satisfy \( \Psi \left( t\right) = 1 \) if \( t \in \left\lbrack {-T/2,{3T}/2}\right\rbrack \) and \( \Psi \left( t\right) \geq 0 \) . For \( t \in \mathbb{R} \), define \[ \widehat{f}\left( t\right) \equiv \left\{ \begin{array}{l} \bar{f}\left( t\right)... | Yes |
Lemma 34.3.1 Let \( \Phi : \left\lbrack {0, T}\right\rbrack \rightarrow E \), be Lebesgue measurable and suppose\n\n\[ \Phi \in K \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;E}\right), p \geq 1 \]\n\nThen there exists a sequence of nested partitions, \( {\mathcal{P}}_{k} \subseteq {\mathcal{P}}_{k + 1} \), ... | Proof: For \( t \in \mathbb{R} \) let \( {\gamma }_{n}\left( t\right) \equiv k/{2}^{n},{\delta }_{n}\left( t\right) \equiv \left( {k + 1}\right) /{2}^{n} \), where\n\n\[ t \in \left( {k/{2}^{n},\left( {k + 1}\right) /{2}^{n}}\right\rbrack \]\n\nand \( {2}^{-n} < T/4 \). Also suppose \( \Phi \) is defined to equal 0 on ... | Yes |
Theorem 34.3.2 Let \( V \subseteq H = {H}^{\prime } \subseteq {V}^{\prime } \) be a Gelfand triple and suppose \( Y \in \) \( {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \equiv {K}^{\prime } \) and\n\n\[ X\left( t\right) = {X}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\text{ in }{V}^{\prime } \]\n\nwhere \( {... | Proof: By Lemma 34.3.1, there exists a sequence of uniform partitions \( {\left\{ {t}_{k}^{n}\right\} }_{k = 0}^{{m}_{n}} = \) \( {\mathcal{P}}_{n},{\mathcal{P}}_{n} \subseteq {\mathcal{P}}_{n + 1} \), of \( \left\lbrack {0, T}\right\rbrack \) such that the step functions\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{{{m}_{n}... | Yes |
Lemma 34.3.3 Let \( s < t \) . Then for \( X, Y \) satisfying 34.3.16\n\n\[{\left| X\left( t\right) \right| }^{2} = {\left| X\left( s\right) \right| }^{2} + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - {\left| X\left( t\right) - X\left( s\right) \right| }^{2}\] | Proof: It follows from the following computations\n\n\[X\left( t\right) - X\left( s\right) = {\int }_{s}^{t}Y\left( u\right) {du}\]\n\n\[- {\left| X\left( t\right) - X\left( s\right) \right| }^{2} = - {\left| X\left( t\right) \right| }^{2} + 2\left( {X\left( t\right), X\left( s\right) }\right) - {\left| X\left( s\right... | Yes |
Lemma 34.3.4 In the above situation,\n\n\\[ \n\\mathop{\\sup }\\limits_{{t \\in \\left\\lbrack {0, T}\\right\\rbrack }}{\\left| X\\left( t\\right) \\right| }_{H} \\leq C\\left( {\\parallel Y{\\parallel }_{{K}^{\\prime }},\\parallel X{\\parallel }_{K}}\\right) \n\\]\n\nAlso, \\( t \\rightarrow X\\left( t\\right) \\) is ... | Proof: From the above formula applied to the \\( {k}^{\\text{th }} \\) partition of \\( \\left\\lbrack {0, T}\\right\\rbrack \\) described above,\n\n\\[ \n{\\left| X\\left( {t}_{m}\\right) \\right| }^{2} - {\\left| {X}_{0}\\right| }^{2} = \\mathop{\\sum }\\limits_{{j = 0}}^{{m - 1}}{\\left| X\\left( {t}_{j + 1}\\right)... | Yes |
Lemma 34.4.1 Let \( V \) be a separable Banach space. Then there exists \( {\left\{ {g}_{k}\right\} }_{k = 1}^{\infty } \) which are linearly independent and whose span is dense in \( V \) . | Proof: Let \( \left\{ {f}_{k}\right\} \) be a countable dense subset. Thus their span is dense. Delete \( {f}_{{k}_{1}} \) such that \( {k}_{1} \) is the first index such that \( {f}_{k} \) is in the span of the other vectors. That is, it is the first which is a finite linear combination of the others. If no such vecto... | Yes |
Lemma 34.4.2 Suppose \( V, W \) are separable Banach spaces such that \( V \) is dense in \( W \) and \( B \in \mathcal{L}\left( {W,{W}^{\prime }}\right) \) satisfies\n\n\[ \langle {Bx}, x\rangle \geq 0,\langle {Bx}, y\rangle = \langle {By}, x\rangle, B \neq 0. \]\n\nThen there exists a countable set \( \left\{ {e}_{i}... | Proof: Let \( {\left\{ {g}_{k}\right\} }_{k = 1}^{\infty } \) be linearly independent vectors of \( V \) whose span is dense in \( V \) . This is possible because \( V \) is separable. Thus, their span is also dense in \( W \) . Let \( {n}_{1} \) be the first index such that \( \left\langle {B{g}_{{n}_{1}},{g}_{{n}_{1}... | Yes |
Theorem 34.4.3 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and let \( Y \in \) \( {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \) and\n\n\[ \n{Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\text{ in }{V}^{\prime },{u}_{0} \in W,{Bu}\left( t\ri... | Proof: By Lemma 34.3.1, there exists a sequence of partitions \( {\left\{ {t}_{k}^{n}\right\} }_{k = 0}^{{m}_{n}} = \) \( {\mathcal{P}}_{n},{\mathcal{P}}_{n} \subseteq {\mathcal{P}}_{n + 1} \), of \( \left\lbrack {0, T}\right\rbrack \) such that the lengths of the sub intervals converge uniformly to 0 as \( n \rightarr... | Yes |
Lemma 34.4.4 Let \( s < t \) . Then for \( u, Y \) satisfying 34.4.22\n\n\[ \langle {Bu}\left( t\right), u\left( t\right) \rangle = \langle {Bu}\left( s\right), u\left( s\right) \rangle \]\n\n\[ + 2{\int }_{s}^{t}\langle Y\left( r\right), u\left( t\right) \rangle {dr} - \langle {Bu}\left( t\right) - {Bu}\left( s\right)... | Proof: It follows from the following computations\n\n\[ {Bu}\left( t\right) - {Bu}\left( s\right) = {\int }_{s}^{t}Y\left( r\right) {dr} \]\n\nand so\n\n\[ 2{\int }_{s}^{t}\langle Y\left( r\right), u\left( t\right) \rangle {dr} - \langle {Bu}\left( t\right) - {Bu}\left( s\right), u\left( t\right) - u\left( s\right) \ra... | Yes |
Corollary 34.4.6 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and \( B \in \) \( \mathcal{L}\left( {W,{W}^{\prime }}\right) \) is nonnegative and self adjoint. Also suppose \( t \rightarrow B\left( {u\left( t\right) }\right) \) has a weak derivative \( {\left( Bu\right) }^{\... | \[ {Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}{\left( Bu\right) }^{\prime }\left( s\right) {ds}\text{ in }{V}^{\prime } \] (34.4.26) Then \( t \rightarrow {Bu}\left( t\right) \) is in \( C\left( {{N}^{C},{W}^{\prime }}\right) \) and also for such \( t \) , \[ \frac{1}{2}\langle {Bu}\left( t\right), u\left( t\right... | Yes |
Proposition 34.4.8 Let\n\n\[ \nX = \left\{ {u \in {L}^{p}\left( {0, T;V}\right) \equiv \mathcal{V} : {Lu} \equiv {\left( Bu\right) }^{\prime } \in {L}^{{p}^{\prime }}\left( {0, T,{V}^{\prime }}\right) }\right\} \n\] \n\nwhere \( V \) is a reflexive Banach space. Let a norm on \( X \) be given by\n\n\[ \n\parallel u{\pa... | Proof: It only remains to verify the last assertion. Let \( {\psi }_{n} \) be increasing and piecewise linear such that \( {\psi }_{n}\left( t\right) = 1 \) for \( t \geq 2/n \) and equals 0 on \( \left\lbrack {0,1/n}\right\rbrack \) . Then clearly \( {\psi }_{n}u \rightarrow u \) in \( \mathcal{V} \) . \n\n\[ \n{\left... | Yes |
Theorem 34.5.1 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and let \( Y \in \) \( {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \) and\n\n\[ \n{Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\text{ in }{V}^{\prime },{u}_{0} \in W,{Bu}\left( t\ri... | Proof: By Lemma 34.3.1, there exists a sequence of partitions \( {\left\{ {t}_{k}^{n}\right\} }_{k = 0}^{{m}_{n}} = \) \( {\mathcal{P}}_{n},{\mathcal{P}}_{n} \subseteq {\mathcal{P}}_{n + 1} \), of \( \left\lbrack {0, T}\right\rbrack \) such that the lengths of the sub intervals converge uniformly to 0 as \( n \rightarr... | Yes |
Lemma 34.5.2 Let \( s < t \) . Then for \( u, Y \) satisfying 34.5.28\n\n\[ \langle {Bu}\left( t\right), u\left( t\right) \rangle - \langle {Bu}\left( s\right), u\left( s\right) \rangle + \langle \left( {B\left( t\right) - B\left( s\right) }\right) u\left( s\right), u\left( t\right) \rangle \]\n\n\[ + \langle \left( {B... | Proof: It follows from the following computations\n\n\[ B\left( t\right) u\left( t\right) - B\left( s\right) u\left( s\right) = {\int }_{s}^{t}Y\left( r\right) {dr} \]\n\nand so\n\n\[ 2{\int }_{s}^{t}\langle Y\left( r\right), u\left( t\right) \rangle {dr} - \langle B\left( t\right) u\left( t\right) - B\left( s\right) u... | Yes |
Lemma 34.5.3 Let the partitions \( {\mathcal{P}}_{k} \) be as above such that 34.5.29, \( {\mathcal{P}}_{k} = {\left\{ {t}_{j}^{k}\right\} }_{j = 0}^{{m}_{k}} \). Then for any \( m \leq {m}_{k} \), \[ \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}\left\langle {B\left( {t}_{j + 1}^{k}\right) u\left( {t}_{j + 1}^{k}\right) - ... | Proof: From the above lemma, the absolute value of the left side is no larger than \[ \mathop{\sum }\limits_{{j = 0}}^{{m - 1}}\left| \left\langle {\left( {B\left( {t}_{j}^{k}\right) - B\left( {t}_{j + 1}^{k}\right) }\right) u\left( {t}_{j}^{k}\right), u\left( {t}_{j + 1}^{k}\right) - u\left( {t}_{j}^{k}\right) }\right... | Yes |
Corollary 34.5.5 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, and \( B\left( t\right) \in \) \( \mathcal{L}\left( {W,{W}^{\prime }}\right) \) is nonnegative and self adjoint, \( B \in {C}^{1}\left( {\left\lbrack {0, T}\right\rbrack ;{W}^{\prime }}\right) \) . Also suppose \(... | \[ {Bu}\left( t\right) = B{u}_{0} + {\int }_{0}^{t}{\left( Bu\right) }^{\prime }\left( s\right) {ds}\text{ in }{V}^{\prime } \] (34.5.38) Then \( t \rightarrow {Bu}\left( t\right) \) is in \( C\left( {{N}^{C},{W}^{\prime }}\right) \) and also for such \( t \) ,\[ \frac{1}{2}\langle {Bu}\left( t\right), u\left( t\right)... | Yes |
Theorem 34.5.6 In the above corollary, the map \( u \rightarrow {Bu}\left( t\right) \) is continuous as a map from \( X \) to \( {V}^{\prime } \) . Also if \( Y \) denotes those \( f \in {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ;V}\right) \) for which \( {f}^{\prime } \in \) \( {L}^{p}\left( {\left\lbrack {0, T}... | Proof: First, why is \( u \rightarrow {Bu}\left( 0\right) \) continuous? Say \( u, v \in X \) and say \( p \geq 2 \) first.\n\n\[ \n{Bu}\left( t\right) - {Bv}\left( t\right) = {Bu}\left( 0\right) - {Bv}\left( 0\right) + {\int }_{0}^{t}{\left( Bu\right) }^{\prime }\left( s\right) - {\left( Bv\right) }^{\prime }\left( s\... | Yes |
Lemma 34.6.3 \( L \) is a closed operator. | We define\n\n\[ \nX \equiv D\left( L\right) ,\parallel u{\parallel }_{X} \equiv \parallel {Lu}{\parallel }_{{\mathcal{V}}^{\prime }} + \parallel u{\parallel }_{\mathcal{V}}\n\]\n\nThen \( X \) is isometric to a closed subspace of a product of reflexive Banach spaces and so \( X \) is reflexive by Lemma 17.5.11. | No |
Corollary 34.6.5 If \( {Bu}\left( 0\right) = 0 \) for \( u \in X \), then \( \langle {Bu}, u\rangle \left( 0\right) = 0 \) . The converse is also true. An analogous result will hold with 0 replaced with \( T \) . | Proof: Let \( {u}_{n} \rightarrow u \) in \( X \) with \( {u}_{n}\left( t\right) = 0 \) for all \( t \) close enough to 0 . For \( t \) off a set of measure zero consisting of the union of sets of measure zero corresponding to \( {u}_{n} \) and \( u \) ,\n\n\[ \left\langle {B{u}_{n},{u}_{n}}\right\rangle \left( t\right... | Yes |
Theorem 34.7.2 Let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Then for every \( \varepsilon > 0 \) there exists a constant, \( {C}_{\varepsilon } \) such that for all \( u \in E \) , \[ \parallel u{\parallel }_{W} \leq \varepsilon ... | Proof: Suppose not. Then there exists \( \varepsilon > 0 \) and for each \( n \in \mathbb{N},{u}_{n} \) such that \[ {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{W} > \varepsilon {\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{E} + n{\begin{Vmatrix}{u}_{n}\end{Vmatrix}}_{X} \] Now let \( {v}_{n} = {u}_{n}/{\begin{Vmatrix}{u}_{n}\end{... | Yes |
Theorem 34.7.4 Let \( q > 1 \) and let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Let \( S \) be defined by\n\n\[ \left\{ {u\text{ such that }\parallel u\left( t\right) {\parallel }_{E} \leq R\text{ for all }t \in \left\lbrack {a, ... | Proof: First consider the issue of \( S \) being a subset of \( C\left( {\left\lbrack {a, b}\right\rbrack ;W}\right) \) . Let \( \varepsilon > 0 \) be given. Then by Theorem 34.7.2 there exists a constant, \( {C}_{\varepsilon } \) such that for all \( u \in W \)\n\n\[ \parallel u{\parallel }_{W} \leq \frac{\varepsilon ... | Yes |
Corollary 34.7.5 Let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Then if \( \gamma > \alpha \), the embedding of \( {C}^{0,\gamma }\left( {\left\lbrack {0, T}\right\rbrack, E}\right) \) into \( {C}^{0,\alpha }\left( {\left\lbrack {0... | Proof: Let \( \phi \in {C}^{0,\gamma }\left( {\left\lbrack {0, T}\right\rbrack, E}\right) \)\n\n\[ \frac{\parallel \phi \left( t\right) - \phi \left( s\right) {\parallel }_{X}}{{\left| t - s\right| }^{\alpha }} \leq {\left( \frac{\parallel \phi \left( t\right) - \phi \left( s\right) {\parallel }_{W}}{{\left| t - s\righ... | Yes |
Corollary 34.7.7 Let \( E \subseteq W \subseteq X \) where the injection map is continuous from \( W \) to \( X \) and compact from \( E \) to \( W \) . Let \( p \geq 1 \), let \( q > 1 \), and define\n\n\[ S \equiv \left\{ {u \in {L}^{p}\left( {\left\lbrack {a, b}\right\rbrack ;E}\right) : }\right. \text{ for some }C,... | Proof: The first part is Theorem 34.7.6. Therefore, we just prove the new stuff which involves a bound on the \( {L}^{1} \) norm of the derivative. By Proposition 7.6.5 on Page 151 it suffices to show \( S \) has an \( \eta \) net in \( {L}^{p}\left( {\left\lbrack {a, b}\right\rbrack ;W}\right) \) for each \( \eta > 0 ... | Yes |
Lemma 34.8.2 For \( L \) as just defined, \( L \) is maximal monotone \( L : \mathcal{H} \rightarrow \mathcal{H} \) . | Proof: To show it is maximal monotone, it suffices to verify that \( L + I \) is onto. This is by Theorem 25.7.13 on Page 929. Thus consider the equation\n\n\[ {u}^{\prime } + u = f, u\left( 0\right) = {u}_{0} \]\n\nIs there a solution? Of course there is and it equals\n\n\[ u\left( t\right) = {e}^{-t}{u}_{0} + {\int }... | Yes |
Lemma 35.1.2 Suppose \( A \) is maximal monotone. Then so is \( {\lambda A} \) . Also \( {J}_{\lambda } \equiv \) \( {\left( I + \lambda A\right) }^{-1} \) makes sense for each \( \lambda > 0 \) and is Lipschitz continuous. | Proof: To begin with consider \( {\left( I + A\right) }^{-1} \) . Suppose\n\n\[ \n{x}_{1},{x}_{2} \in {\left( I + A\right) }^{-1}\left( y\right) \n\]\n\nThen \( y \in \left( {I + A}\right) {x}_{i} \) and so \( y - {x}_{i} \in A{x}_{i} \) . By monotonicity\n\n\[ \n\left( {y - {x}_{1} - \left( {y - {x}_{2}}\right) ,{x}_{... | Yes |
Lemma 35.1.3 \( {A}_{\lambda }x \in A{J}_{\lambda }x \) and \( \left| {{A}_{\lambda }x}\right| \leq \left| y\right| \) for all \( y \in {Ax} \) whenever \( x \in D\left( A\right) \) . Also \( {A}_{\lambda } \) is monotone. | Proof: Consider the first claim. From the definition,\n\n\[ \n{A}_{\lambda }x \equiv \frac{1}{\lambda }x - \frac{1}{\lambda }{J}_{\lambda }x \n\]\n\nIs\n\n\[ \n\frac{1}{\lambda }x - \frac{1}{\lambda }{J}_{\lambda }x \in A{J}_{\lambda }x? \n\]\n\nIs\n\n\[ \nx - {J}_{\lambda }x \in {\lambda A}{J}_{\lambda }x? \n\]\n\nIs\... | Yes |
Proposition 35.1.4 Suppose \( D\left( A\right) \) is dense in \( H \) . Then for all \( x \in H \) , \[ \left| {{J}_{\lambda }x - x}\right| \rightarrow 0 \] | Proof: From the above, if \( u \in D\left( A\right) \) and \( y \in {Au} \), then \[ \left| {\frac{1}{\lambda }u - \frac{1}{\lambda }{J}_{\lambda }u}\right| \leq \left| y\right| \] Hence \( {J}_{\lambda }u \rightarrow u \) . Now for \( x \) arbitrary, \[ \left| {{J}_{\lambda }x - x}\right| \leq \left| {{J}_{\lambda }x ... | No |
Lemma 35.1.5 Suppose \( \left( {{y}_{1} - y,{x}_{1} - x}\right) \geq 0 \) for all \( \left\lbrack {x, y}\right\rbrack \in \mathcal{G}\left( A\right) \) where \( A \) is maximal monotone. Then \( {x}_{1} \in D\left( A\right) \) and \( {y}_{1} \in A{x}_{1} \) . Also if \( \left\lbrack {{x}_{k},{y}_{k}}\right\rbrack \in \... | Proof: I want to show \( {y}_{1} \in A{x}_{1} \) or in other words I want to show\n\n\[ {x}_{1} + \lambda {y}_{1} \in {x}_{1} + {\lambda A}{x}_{1} \]\n\nor in other words\n\n\[ {J}_{\lambda }\left( {{x}_{1} + \lambda {y}_{1}}\right) = {x}_{1} \]\n\nThis is the motivation for the following argument.\n\nFrom Lemma 35.1.3... | Yes |
Proposition 35.1.6 Let \( A \) be maximal monotone and let \( B \) be Lipschitz and monotone. Then \( A + B \) is maximal monotone. | Proof: First suppose \( B \) has a Lipschitz constant less than 1 . The monotonicity is obvious. I need to show that for any \( y \) there exists \( x \in D\left( A\right) \) such that\n\n\[ y \in x + {Bx} + {Ax} \]\n\nThis hapens if and only if\n\n\[ y - {Bx} \in \left( {I + A}\right) x \]\n\nif and only if \( x = {\l... | Yes |
Lemma 35.2.1 Let \( f : \left\lbrack {0, T}\right\rbrack \rightarrow \mathbb{R} \) be continuous and suppose\n\n\[ \n{D}^{ + }f\left( t\right) \equiv \lim \mathop{\sup }\limits_{{h \rightarrow 0 + }}\frac{f\left( {t + h}\right) - f\left( t\right) }{h} < g\left( t\right) \n\]\n\nwhere \( g \) is a continuous function. T... | Proof: Suppose this is not so. Then let\n\n\[ \nS \equiv \left\{ {t \in \left\lbrack {0, T}\right\rbrack : f\left( t\right) - f\left( 0\right) > {\int }_{0}^{t}g\left( s\right) {ds}}\right\} \n\]\n\nand it would follow that \( S \neq \varnothing \) . Let \( a = \inf S \) . Then there exists a decreasing sequence \( {h}... | Yes |
Theorem 35.3.2 Let \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) be convex and suppose for some \( u \in \) \( \operatorname{dom}\left( \phi \right) ,\phi \) is continuous. Then \( {\delta \phi }\left( x\right) \neq \varnothing \) for all \( x \in \operatorname{int}\left( {\operatorname{dom}\left( \phi \right) ... | Proof: Let \( {x}_{0} \in \operatorname{int}\left( {\operatorname{dom}\left( \phi \right) }\right) \) and let\n\n\[ A \equiv \left\{ \left( {{x}_{0},\phi \left( {x}_{0}\right) }\right) \right\}, B \equiv \operatorname{epi}\left( \phi \right) \cap X \times \mathbb{R}. \]\n\nThen \( A \) and \( B \) are both nonempty and... | Yes |
Theorem 35.3.4 Let \( X \) be a real Banach space. Then \( {\phi }^{ * } \) is convex and l.s.c. | Proof: Let \( \lambda \in \left\lbrack {0,1}\right\rbrack \) . Then\n\n\[ \n{\phi }^{ * }\left( {\lambda {x}^{ * } + \left( {1 - \lambda }\right) {y}^{ * }}\right) = \sup \left\{ {\left( {\lambda {x}^{ * } + \left( {1 - \lambda }\right) {y}^{ * }}\right) \left( y\right) - \phi \left( y\right) : y \in X}\right\} \n\]\n\... | Yes |
Lemma 35.3.7 Let \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) be convex and lower semicontinuous and \( \phi \left( x\right) < \infty \) for some \( x \) . (proper). Then if \( \beta < \phi \left( {x}_{0}\right) \) so that \( \left( {{x}_{0},\beta }\right) \) is not in \( \operatorname{epi}\left( \phi \right) ... | Proof: Let \( C = \operatorname{epi}\left( \phi \right) \cap \left( {X \times \mathbb{R}}\right) \) . Then \( C \) is a closed convex nonempty set and it does not contain the point \( \left( {{x}_{0},\beta }\right) \) . Let \( \widehat{\beta } > \beta \) be slightly larger so that also \( \left( {{x}_{0},\widehat{\beta... | Yes |
Theorem 35.3.8 \( {\phi }^{* * }\left( x\right) \leq \phi \left( x\right) \) for all \( x \) and if \( \phi \) is convex and l.s.c., \( {\phi }^{* * }\left( x\right) = \phi \left( x\right) \) for all \( x \in X \) . | Proof:\n\n\[{\phi }^{* * }\left( x\right) \equiv \sup \left\{ {{x}^{ * }\left( x\right) - \overset{{\phi }^{ * }\left( {x}^{ * }\right) }{\overbrace{\sup \left\{ {{x}^{ * }\left( y\right) - \phi \left( y\right) : y \in X}\right\} }} : {x}^{ * } \in {X}^{\prime }}\right\}\]\n\n\[\leq \sup \left\{ {{x}^{ * }\left( x\righ... | Yes |
Corollary 35.3.9 \( \operatorname{epi}\left( {\phi }^{* * }\right) \) is the smallest closed convex set containing \( \operatorname{epi}\left( \phi \right) \) . | Proof: \( \operatorname{epi}\left( {\phi }^{* * }\right) \supseteq \operatorname{epi}\left( \phi \right) \) from Theorem 35.3.8. Also \( \operatorname{epi}\left( {\phi }^{* * }\right) \) is closed by the proof of Theorem 35.3.4. Suppose \( \operatorname{epi}\left( \phi \right) \subseteq K \subseteq \operatorname{epi}\l... | Yes |
Theorem 35.3.10 Suppose \( \phi \) is convex, l.s.c. (lower semicontinuous or in other words having a closed epigraph), and proper. Then\n\n\[ \n{y}^{ * } \in {\delta \phi }\left( x\right) \text{ if and only if }x \in \delta {\phi }^{ * }\left( {y}^{ * }\right)\n\]\n\nwhere this last expression means\n\n\[ \n\left( {{z... | Proof: If \( {y}^{ * } \in {\delta \phi }\left( x\right) \) then \( {y}^{ * }\left( {z - x}\right) \leq \phi \left( z\right) - \phi \left( x\right) \) and so\n\n\[ \n{y}^{ * }\left( z\right) - \phi \left( z\right) \leq {y}^{ * }\left( x\right) - \phi \left( x\right)\n\]\n\nfor all \( z \in X \) . Therefore,\n\n\[ \n{\p... | Yes |
Lemma 35.3.16 With \( F\left( x\right) \) defined as above, it follows that\n\n\[ F\left( x\right) = \left\{ {{x}^{ * } \in {X}^{\prime } : {x}^{ * }\left( x\right) = \parallel x{\parallel }^{2},\begin{Vmatrix}{x}^{ * }\end{Vmatrix} = \parallel x\parallel }\right\} \]\n\nand \( F\left( x\right) \) is a closed, nonempty... | Proof: If \( {x}^{ * } \) is in the set described in 35.3.28,\n\n\[ {x}^{ * }\left( \frac{x}{\parallel x\parallel }\right) = \parallel x\parallel \]\n\nand so \( \begin{Vmatrix}{x}^{ * }\end{Vmatrix} \geq \parallel x\parallel \) . Therefore\n\n\[ {x}^{ * } \in \left\{ {{x}^{ * } \in {X}^{\prime } : {x}^{ * }\left( x\ri... | Yes |
Theorem 35.3.18 Suppose \( X \) is a reflexive Banach space and suppose \( \phi : X \rightarrow \) \( ( - \infty ,\infty \rbrack \) is convex, proper, l.s.c., and for all \( {y}^{ * } \in {X}^{\prime }, x \rightarrow \phi \left( x\right) - {y}^{ * }\left( x\right) \) is coercive. Then \( {\delta \phi } \) is onto. | Proof: The function \( x \rightarrow \phi \left( x\right) - {y}^{ * }\left( x\right) \equiv \psi \left( x\right) \) is convex, proper, l.s.c., and coercive. Let\n\n\[ \lambda \equiv \inf \left\{ {\phi \left( x\right) - {y}^{ * }\left( x\right) : x \in X}\right\} \]\n\nand let \( \left\{ {x}_{n}\right\} \) be a minimizi... | Yes |
Corollary 35.3.19 Suppose \( X \) is a reflexive Banach space and \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) is convex, proper, and l.s.c. Then for each \( {y}^{ * } \in {X}^{\prime } \) there exist \( x \in X,{x}_{1}^{ * } \in F\left( x\right) \) , and \( {x}_{2}^{ * } \in {\delta \phi }\left( x\right) \) s... | Proof: Apply Theorem 35.3.18 to the convex function \( \frac{1}{2}\parallel x{\parallel }^{2} + \phi \left( x\right) \) and use Theorems 35.3.14 and 35.3.17. | No |
Lemma 35.3.22 If \( \phi \) is a convex, proper, l.s.c. function defined on a Hilbert space, then \( \partial \phi \) is maximal monotone and \( {\left( I + \partial \phi \right) }^{-1} \) is a Lipschitz continuous map from \( H \) to \( \operatorname{dom}\left( {\partial \phi }\right) \) having Lipschitz constant 1. | Proof: Let \( y \in H \) . Then \( {Ry} \in {H}^{\prime } \) and by Corollary 35.3.19, there exists \( x \in \operatorname{dom}\left( {\delta \phi }\right) \) such that \( {Rx} + {\delta \phi }\left( x\right) \ni {Ry} \) . Multiplying by \( {R}^{-1} \) we see \( y \in x + \partial \phi \left( x\right) \) . This shows \... | Yes |
Lemma 35.3.23 Let \( \phi \) be convex, proper and lower semicontinuous on \( X \) a reflexive Banach space having strictly convex norm, then for each \( \alpha > 0 \), \[ I + \alpha \partial \phi \] is onto. | Proof: By separation theorems applied to the eipgraph of \( \phi \), and since \( \phi \) is proper, there exists \( {w}^{ * } \) such that \[ \left( {{w}^{ * }, x}\right) + b \leq {\alpha \phi }\left( x\right) \] for all \( x \). Pick \( y \in H \). Then consider \[ \frac{1}{2}{\left| y - x\right| }^{2} + {\alpha \phi... | Yes |
Theorem 35.3.24 Let \( \phi \) be a convex lower semicontinuous proper function defined on \( H \) . Define\n\n\[ \n{\phi }_{\lambda }\left( x\right) \equiv \mathop{\min }\limits_{{y \in H}}\left( {\frac{1}{2\lambda }{\left| x - y\right| }^{2} + \phi \left( y\right) }\right)\n\]\n\nThen the function is well defined, co... | Proof: First of all, why does the minimum take place? By the convexity, closed epigraph, and assumption that \( \phi \) is proper, separation theorems apply and one can say that there exists \( {z}^{ * } \) such that for all \( y \in H \) ,\n\n\[ \n\frac{1}{2\lambda }{\left| x - y\right| }^{2} + \phi \left( y\right) \g... | Yes |
Theorem 35.4.1 Let \( A \) and \( B \) be maximal monotone operators and let \( {x}_{\lambda } \) be the solution to\n\n\[ y \in {x}_{\lambda } + {B}_{\lambda }{x}_{\lambda } + A{x}_{\lambda } \]\n\nThen \( y \in x + {Bx} + {Ax} \) for some \( x \in D\left( A\right) \cap D\left( B\right) \) if \( {B}_{\lambda }{x}_{\la... | The following is the perturbation theorem of this section. See [24] and [115]. | No |
Lemma 35.5.1 For \( x \in {L}^{2}\left( {0, T;H}\right), t \rightarrow \phi \left( x\right) \) is measurable. | Proof: This follows because \( \phi \) is Borel measurable and so \( \phi \circ x \) is also measurable. | Yes |
Lemma 35.5.2 \( \Phi \) is convex, nonnegative, and lower semicontinuous on \( {L}^{2}\left( {0, T;H}\right) \) . | Proof: Since \( \phi \) is nonnegative and convex, it follows that \( \Phi \) is also nonnegative and convex. It remains to verify lower semicontinuity. Suppose, \( {x}_{n} \rightarrow x \) in \( {L}^{2}\left( {0, T;H}\right) \) and let\n\n\[ \lambda = \lim \mathop{\inf }\limits_{{n \rightarrow \infty }}\Phi \left( {x}... | Yes |
Lemma 35.5.3 \( L \) is maximal monotone and if \( z \in {L}^{2}\left( {0, T;H}\right) \), then \( {J}_{\lambda }z \) is given \( {by} \) | \[ {J}_{\lambda }\left\lbrack z\right\rbrack \left( t\right) \equiv {\left( I + \lambda L\right) }^{-1}\left( \left\lbrack z\right\rbrack \right) \left( t\right) = {e}^{\frac{-t}{\lambda }}{x}_{0} + \frac{1}{\lambda }{e}^{\frac{-t}{\lambda }}{\int }_{0}^{t}{e}^{\frac{1}{\lambda }s}z\left( s\right) {ds}. \] | Yes |
Theorem 35.5.4 Let \( {x}_{0} \in D \equiv D\left( \phi \right) \). Then \( L + \partial \Phi \) is maximal monotone so there exists a unique solution to\n\n\[ \n{Lx} + x + \partial \Phi \left( x\right) \ni f \n\]\n\nfor every \( f \in {L}^{2}\left( {0, T;H}\right) \). Thus there exists \( x \in {L}^{2}\left( {0, T;H}\... | Proof: This is from Theorem 35.4.2. Since \( {x}_{0} \in D \), it follows that \( \phi \left( {x}_{0}\right) < \infty \).\n\nLet \( z \in D\left( \Phi \right) \), the effective domain of \( \Phi \). Then \( {\int }_{0}^{T}\phi \left( {z\left( t\right) }\right) {dt} < \infty \), so by convexity of \( \phi \) and 35.5.47... | Yes |
Theorem 35.5.5 Let \( f \in {L}^{2}\left( {0, T;H}\right) \) and \( {x}_{0} \in D \) . Let \( \phi \) be as described above, a lower semicontinuous convex proper function defined on \( H \) . Then there exists a unique solution \( x \in {L}^{2}\left( {0, T;H}\right) ,{x}^{\prime } \in {L}^{2}\left( {0, T;H}\right) \), ... | Proof: From Theorem 35.5.4, there exists a unique solution to\n\n\[ \n{x}_{v}^{\prime } + \partial \Phi \left( {x}_{v}\right) + {x}_{v} \ni f + v\text{ in }{L}^{2}\left( {0, T;H}\right) ,{x}_{v}\left( 0\right) = {x}_{0} \n\]\n\nwhenever \( v \in {L}^{2}\left( {0, T;H}\right) \) . Then a simple argument based on fundame... | Yes |
Lemma 35.7.1 Under the conditions,35.7.56 - 35.7.58, \( \phi : H \times \left\lbrack {0, T}\right\rbrack \rightarrow \left\lbrack {0,\infty }\right\rbrack \) is lower semicontinuous. | Proof: Let \( \left( {{x}_{n},{t}_{n}}\right) \rightarrow \left( {x, t}\right) \) and let \( \lambda \equiv \mathop{\liminf }\limits_{{n \rightarrow \infty }}\phi \left( {{t}_{n},{x}_{n}}\right) \) . Is\n\n\[ \phi \left( {t, x}\right) \leq \lambda ? \]\n\nIt suffices to assume \( \lambda < \infty \) and by taking a sub... | Yes |
Corollary 35.7.2 For \( \left\lbrack x\right\rbrack \in {L}^{2}\left( {0, T;H}\right), t \rightarrow \phi \left( {t, x\left( t\right) }\right) \) is measurable. | Proof: This follows because, due to Lemma 35.7.1, \( \phi \) is Borel measurable and so \( \phi \circ x \) is also measurable. | Yes |
Lemma 35.7.3 \( \Phi \) is convex, nonnegative, and lower semicontinuous on \( {L}^{2}\left( {0, T;H}\right) \) . | Proof: Since each \( \phi \left( {t, \cdot }\right) \) is nonnegative and convex, it follows that \( \Phi \) is also nonnegative and convex. It remains to verify lower semicontinuity. Suppose, \( \left\lbrack {x}_{n}\right\rbrack \rightarrow \) \( \left\lbrack x\right\rbrack \) in \( {L}^{2}\left( {0, T;H}\right) \) an... | Yes |
Lemma 35.7.4 \( L \) is maximal monotone and if \( \left\lbrack z\right\rbrack \in {L}^{2}\left( {0, T;H}\right) \), then the equivalence class, \( \left\lbrack {{J}_{\lambda }\left\lbrack z\right\rbrack }\right\rbrack \) is determined by the function, | \[ {J}_{\lambda }\left\lbrack z\right\rbrack \left( t\right) \equiv {\left( I + \lambda L\right) }^{-1}\left( \left\lbrack z\right\rbrack \right) \left( t\right) = {e}^{\frac{-t}{\lambda }}{x}_{0} + \frac{1}{\lambda }{e}^{\frac{-t}{\lambda }}{\int }_{0}^{t}{e}^{\frac{1}{\lambda }s}z\left( s\right) {ds}. \] | Yes |
Lemma 35.7.6 If 35.7.69 and 35.7.70 hold, and if \( \left\lbrack y\right\rbrack \in {L}^{2}\left( {0, T;H}\right) \), then \( \left\lbrack y\right\rbrack \in \) \( \partial \Phi \left( \left\lbrack x\right\rbrack \right) \) if and only if there exists \( x \in \left\lbrack x\right\rbrack \) such that \( {\partial }_{2}... | Proof: First suppose \( y\left( t\right) \in {\partial }_{2}\phi \left( {t, x\left( t\right) }\right) \) a.e. and \( {\partial }_{2}\phi \left( {t, x\left( t\right) }\right) \neq \varnothing \) for all \( t \) where \( x \in \left\lbrack x\right\rbrack \) . Then for all \( \left\lbrack w\right\rbrack \in {L}^{2}\left( ... | Yes |
Lemma 35.7.7 Suppose there exists \( \left\lbrack \xi \right\rbrack \in {L}^{2}\left( {0, T;H}\right) \) such that\n\n\[ \n{J}_{1}\left( t\right) \xi \left( t\right) ,\phi \left( {t,{J}_{1}\left( t\right) \xi \left( t\right) }\right) \n\]\n\nare bounded independent of \( t \in \left\lbrack {0, T}\right\rbrack \) and \(... | Proof: Let \( y\left( t\right) = {J}_{1}\left( t\right) \xi \left( t\right) \) . Thus\n\n\[ \ny\left( t\right) + {\partial }_{2}\phi \left( {t, y\left( t\right) }\right) \ni \xi \left( t\right) .\n\]\n\nNow suppose \( x \in H \) and let\n\n\[ \nx\left( s\right) + z\left( s\right) = x \n\]\n\n(35.7.71)\n\nwhere \( z\lef... | Yes |
Lemma 36.1.2 Let \( {X}^{\prime } \) be the dual of a Banach space, \( X \) and suppose \( X \) is separable. Then if \( \left\{ {x}_{n}^{ * }\right\} \) is a bounded sequence in \( {X}^{\prime } \), there exists a weak \( * \) convergent subsequence. | Proof: Let \( D \) be a dense countable set in \( X \) . Then the sequence, \( \left\{ {{x}_{n}^{ * }\left( x\right) }\right\} \) is bounded for all \( x \) and in particular for all \( x \in D \) . Use the Cantor diagonal process to obtain a subsequence, still denoted by \( n \) such that \( {x}_{n}^{ * }\left( d\righ... | Yes |
Theorem 36.1.3 Let \( \Omega \) be a measurable subset of \( {\mathbb{R}}^{n} \) and let \( \left\{ {f}_{k}\right\} \) be a bounded sequence in \( {L}^{p}\left( \Omega \right) \) where \( 1 < p \leq \infty \) . Then there exists a weak \( * \) convergent subsequence. | Proof: Since \( {L}^{{p}^{\prime }}\left( \Omega \right) \) is separable, this follows from the Riesz representation theorem. | No |
Lemma 36.2.3 Let \( T \in {\mathcal{D}}^{ * }\left( {a, b}\right) \) and suppose \( {DT} = 0 \) . Then there exists a constant \( C \) such that\n\[ T\left( \phi \right) = {\int }_{a}^{b}{C\phi dx} \] | Proof: \( T\left( {D\phi }\right) = 0 \) for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) from the definition of \( {DT} = 0 \) . Let\n\[ {\phi }_{0} \in {C}_{c}^{\infty }\left( {a, b}\right) ,{\int }_{a}^{b}{\phi }_{0}\left( x\right) {dx} = 1, \]\nand let\n\[ {\psi }_{\phi }\left( x\right) = {\int }_{a}^{x... | Yes |
Lemma 36.3.3 Suppose \( f \in {L}_{loc}^{1}\left( U\right) \) and suppose\n\n\[ \int {f\phi dx} = 0 \]\n\nfor all \( \phi \in {C}_{c}^{\infty }\left( U\right) \) . Then \( f\left( \mathbf{x}\right) = 0 \) a.e. \( \mathbf{x} \) . | Proof: Without loss of generality \( f \) is real valued. Let\n\n\[ E \equiv \{ \mathbf{x} : f\left( \mathbf{x}\right) > \varepsilon \} \]\n\nand let\n\n\[ {E}_{m} \equiv E \cap B\left( {0, m}\right) . \]\n\nIs \( m\left( {E}_{m}\right) = 0 \) ? If not, there exists an open set, \( V \), and a compact set \( K \) satis... | Yes |
Lemma 36.3.5 Let \( u \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) and suppose \( {u}_{, i} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \), where the subscript on the \( u \) following the comma denotes the \( {i}^{\text{th }} \) weak partial derivative. Then if \( {\phi }_{\varepsilon } \) is a mollifier and... | Proof: If \( \psi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \), then\n\n\[ \n\int u\left( {\mathbf{x} - \mathbf{y}}\right) {\psi }_{, i}\left( \mathbf{x}\right) {dx} = \int u\left( \mathbf{z}\right) {\psi }_{, i}\left( {\mathbf{z} + \mathbf{y}}\right) {dz} \n\]\n\n\[ \n= - \int {u}_{, i}\left( \mathbf{z}\righ... | Yes |
Lemma 36.3.6 Let \( U \) be an open set, \( \psi \in {C}^{\infty }\left( U\right) \) and suppose \( u,{u}_{, i} \in {L}_{loc}^{p}\left( U\right) \) . Then \( {\left( u\psi \right) }_{, i} \) and \( {u\psi } \) are in \( {L}_{loc}^{p}\left( U\right) \) and\n\n\[ \n{\left( u\psi \right) }_{, i} = {u}_{, i}\psi + u{\psi }... | Proof: Let \( \phi \in {C}_{c}^{\infty }\left( U\right) \) then\n\n\[ \n{\left( u\psi \right) }_{, i}\left( \phi \right) \equiv - {\int }_{U}{u\psi }{\phi }_{, i}{dx} \n\]\n\n\[ \n= - {\int }_{U}u\left\lbrack {{\left( \psi \phi \right) }_{, i} - \phi {\psi }_{, i}}\right\rbrack {dx} \n\]\n\n\[ \n= {\int }_{U}\left( {{u... | Yes |
Corollary 36.4.2 Suppose \( u \in {C}^{1}\left( {\mathbb{R}}^{n}\right) \) . Then\n\n\[ \left| {u\left( \mathbf{y}\right) - u\left( \mathbf{x}\right) - \nabla u\left( \mathbf{x}\right) \cdot \left( {\mathbf{y} - \mathbf{x}}\right) }\right| \]\n\n\[ \leq C{\left( \frac{1}{m\left( {B\left( {\mathbf{x},2\left| {\mathbf{x}... | Proof: This follows easily from letting \( g\left( \mathbf{y}\right) \equiv u\left( \mathbf{y}\right) - u\left( \mathbf{x}\right) - \nabla u\left( \mathbf{x}\right) \cdot \left( {\mathbf{y} - \mathbf{x}}\right) \) . Then \( g \in {C}^{1}\left( {\mathbb{R}}^{n}\right), g\left( \mathbf{x}\right) = 0 \), and \( \nabla g\l... | Yes |
Corollary 36.5.2 Let \( u,{u}_{, i} \in {L}_{loc}^{p}\left( {\mathbb{R}}^{n}\right) \) for \( i = 1,\cdots, n \) and \( p > n \) . Then the representative of \( u \) described in Theorem 36.5.1 is differentiable a.e. | Proof: From Theorem 36.5.1\n\n\[ \left| {u\left( \mathbf{y}\right) - u\left( \mathbf{x}\right) - \nabla u\left( \mathbf{x}\right) \cdot \left( {\mathbf{y} - \mathbf{x}}\right) }\right| \]\n\n\[ \leq C{\left( \frac{1}{m\left( {B\left( {\mathbf{x},2\left| {\mathbf{x} - \mathbf{y}}\right| }\right) }\right) }{\int }_{B\lef... | Yes |
Corollary 36.5.4 If \( u \) is Lipschitz continuous then \( u \) is differentiable a.e. and \( {\begin{Vmatrix}{u}_{, i}\end{Vmatrix}}_{\infty } \leq \operatorname{Lip}\left( u\right) \) | Proof: This is done by showing that Lipschitz continuous functions have weak derivatives in \( {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and then using the previous results. Let\n\n\[ \n{D}_{{\mathbf{e}}_{i}}^{h}u\left( \mathbf{x}\right) \equiv {h}^{-1}\left\lbrack {u\left( {\mathbf{x} + h{\mathbf{e}}_{i}}\right) ... | Yes |
Lemma 36.6.1 Suppose \( V \) is an \( n - 1 \) dimensional subspace of \( {\mathbb{R}}^{n} \) and \( K \) is a compact subset of \( V \) . Then letting\n\n\[ \n{K}_{\varepsilon } \equiv { \cup }_{\mathbf{x} \in K}B\left( {\mathbf{x},\varepsilon }\right) = K + B\left( {\mathbf{0},\varepsilon }\right) , \n\]\n\n it follo... | Proof: Let an orthonormal basis for \( V \) be \( \left\{ {{\mathbf{v}}_{1},\cdots ,{\mathbf{v}}_{n - 1}}\right\} \) and let\n\n\[ \n\left\{ {{\mathbf{v}}_{1},\cdots ,{\mathbf{v}}_{n - 1},{\mathbf{v}}_{n}}\right\} \n\]\n\nbe an orthonormal basis for \( {\mathbb{R}}^{n} \) . Now define a linear transformation, \( Q \) b... | Yes |
Lemma 36.6.3 Let \( E \) be a Lebesgue measurable set. Then there exists a set of measure zero, \( N \), such that if \( \mathbf{x} \in E \smallsetminus N \), then \( \mathbf{x} \) is a point of density of \( E \) . | Proof: Consider the function, \( f\left( \mathbf{x}\right) = {\mathcal{X}}_{E}\left( \mathbf{x}\right) \) . This function is in \( {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) . Let \( {N}^{C} \) denote the Lebesgue points of \( f \) . Then for \( \mathbf{x} \in E \smallsetminus N \) ,\n\n\[1 = {\mathcal{X}}_{E}\left... | Yes |
Lemma 36.6.7 If \( {m}_{n}\left( T\right) = 0 \) then \( {m}_{n}\left( {\mathbf{h}\left( T\right) }\right) = 0 \) . | Proof: Let \( V \) be an open set containing \( T \) whose measure is less than \( \varepsilon \) . Now using the Vitali covering theorem, there exists a sequence of disjoint balls \( \left\{ {B}_{i}\right\} \) , \( {B}_{i} = B\left( {{\mathbf{x}}_{i},{r}_{i}}\right) \) which are contained in \( V \) such that the sequ... | Yes |
Lemma 36.6.9 Let \( B = B\left( {\mathbf{0}, r}\right) \), a ball in \( {\mathbb{R}}^{k} \) and let \( \mathbf{F} : \bar{B} \rightarrow {\mathbb{R}}^{k} \) be continuous and suppose for some \( \varepsilon < 1 \) ,\n\n\[ \left| {\mathbf{F}\left( \mathbf{v}\right) - \mathbf{v}}\right| < {\varepsilon r} \]\n\nfor all \( ... | Proof: Suppose \( \mathbf{a} \in \overline{B\left( {\mathbf{0}, r\left( {1 - \varepsilon }\right) }\right) } \smallsetminus \mathbf{F}\left( \bar{B}\right) \) and let\n\n\[ \mathbf{G}\left( \mathbf{v}\right) \equiv \frac{r\left( {\mathbf{a} - \mathbf{F}\left( \mathbf{v}\right) }\right) }{\left| \mathbf{a} - \mathbf{F}\... | Yes |
Lemma 36.6.10 Let \( \mathbf{x} \in \Omega \smallsetminus \left( {S \cup N}\right) \) . Then if \( \varepsilon \in \left( {0,1}\right) \) the following hold for all \( r \) small enough.\n\n\[ \n{m}_{n}\left( {\mathbf{h}\left( \overline{B\left( {\mathbf{x}, r}\right) }\right) }\right) \geq {m}_{n}\left( {D\mathbf{h}\le... | Proof: Since \( D\mathbf{h}{\left( \mathbf{x}\right) }^{-1} \) exists,\n\n\[ \n\mathbf{h}\left( {\mathbf{x} + \mathbf{v}}\right) = \mathbf{h}\left( \mathbf{x}\right) + D\mathbf{h}\left( \mathbf{x}\right) \mathbf{v} + o\left( \left| \mathbf{v}\right| \right) \n\]\n\n\[ \n= \mathbf{h}\left( \mathbf{x}\right) + D\mathbf{h... | Yes |
Lemma 36.6.11 The function, \( J\left( \mathbf{x}\right) \) equals \( \left| {\det D\mathbf{h}\left( \mathbf{x}\right) }\right| \) a.e. | Proof: Define\n\n\[ Q \equiv \{ \mathbf{x} \in \Omega : \mathbf{x}\text{ is not a point of density of }\Omega \} \cup N \cup \]\n\n\[ \{ \mathbf{x} \in \Omega : \mathbf{x}\text{is not a Lebesgue point of}J\} \text{.} \]\n\nThen \( Q \) is a set of measure zero and if \( \mathbf{x} \notin Q \), then by 36.6.17, and 36.6... | Yes |
Theorem 36.6.12 Let \( \Omega \) be a Lebesgue measurable set, let \( f \geq 0 \) be Lebesgue measurable. Then for \( \mathbf{h} \) a Lipschitz mapping defined on \( {\mathbb{R}}^{n} \) which is one to one on \( \Omega \) , \[ {\int }_{\mathbf{h}\left( \Omega \right) }f\left( \mathbf{y}\right) d{m}_{n} = {\int }_{\Omeg... | Proof: Let \( F \) be a Borel set. It follows that \( {\mathbf{h}}^{-1}\left( F\right) \) is a Lebesgue measurable set. Therefore, by 36.6.22, \[ {m}_{n}\left( {\mathbf{h}\left( {{\mathbf{h}}^{-1}\left( F\right) \cap \Omega }\right) }\right) \] \[ = {\int }_{\mathbf{h}\left( \Omega \right) }{\mathcal{X}}_{F}\left( \mat... | Yes |
Theorem 36.6.13 If \( \mathbf{h} : \Omega \rightarrow {\mathbb{R}}^{m} \) is Lipschitz, then there exists \( \overline{\mathbf{h}} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{m} \) which extends \( \mathbf{h} \) and is also Lipschitz. | Proof: It suffices to assume \( m = 1 \) because if this is shown, it may be applied to the components of \( \mathbf{h} \) to get the desired result. Suppose\n\n\[ \left| {h\left( \mathbf{x}\right) - h\left( \mathbf{y}\right) }\right| \leq K\left| {\mathbf{x} - \mathbf{y}}\right| . \]\n\n\( \left( {36.6.26}\right) \)\n... | Yes |
Corollary 36.6.14 Let \( \mathbf{h} : \Omega \rightarrow {\mathbb{R}}^{n} \) be Lipschitz continuous and one to one where \( \Omega \) is a Lebesgue measurable set. Then if \( f \geq 0 \) is Lebesgue measurable, | \[ {\int }_{\mathbf{h}\left( \Omega \right) }f\left( \mathbf{y}\right) d{m}_{n} = {\int }_{\Omega }f\left( {\mathbf{h}\left( \mathbf{x}\right) }\right) \left| {\det D\overline{\mathbf{h}}\left( \mathbf{x}\right) }\right| d{m}_{n}. \] \( \left( {36.6.28}\right) \) where \( \overline{\mathbf{h}} \) denotes a Lipschitz ex... | Yes |
Lemma 37.1.2 Let \( \\mathfrak{C} \) be a set whose elements are open subsets of \( {\\mathbb{R}}^{n} \) and suppose \( \\cup \\mathfrak{C} \\supseteq H \), a closed set. Then there exists a countable list of open sets, \( {\\left\{ {U}_{i}\\right\} }_{i = 1}^{\\infty } \) such that each \( {U}_{i} \) is bounded, each ... | Proof: Let \( {W}_{k} \\equiv B\\left( {\\mathbf{0}, k}\\right) ,{W}_{0} = {W}_{-1} = \\varnothing \) . For each \( \\mathbf{x} \\in H \\cap \\overline{{W}_{k}} \) there exists an open set, \( {U}_{\\mathbf{x}} \) such that \( {U}_{\\mathbf{x}} \) is a subset of some set of \( \\mathfrak{C} \) and \( {U}_{\\mathbf{x}} ... | Yes |
Lemma 37.1.3 Let \( \mathfrak{C} \) be locally finite. Then\n\n\[ \overline{\cup \mathfrak{C}} = \cup \{ \bar{H} : H \in \mathfrak{C}\} \] | Proof: Let \( \mathbf{p} \) be a limit point of \( \cup \mathfrak{C} \) and let \( W \) be an open set which intersects only finitely many sets of \( \mathfrak{C} \) . Then \( \mathbf{p} \) must be a limit point of one of these sets. It follows \( \mathbf{p} \in \cup \{ \bar{H} : H \in \mathfrak{C}\} \) and so \( \wide... | Yes |
Lemma 37.1.5 Let \( U \) be a bounded open set and let \( K \) be a closed subset of \( U \) . Then there exist an open set, \( W \), such that \( W \subseteq \bar{W} \subseteq U \) and a function, \( f \in {C}_{c}^{\infty }\left( U\right) \) such that \( K \prec f \prec U \) . | Proof: The set, \( K \) is compact so is at a positive distance from \( {U}^{C} \) . Let\n\n\[ W \equiv \left\{ {\mathbf{x} : \operatorname{dist}\left( {\mathbf{x}, K}\right) < {3}^{-1}\operatorname{dist}\left( {K,{U}^{C}}\right) }\right\} \]\n\nAlso let\n\n\[ {W}_{1} \equiv \left\{ {\mathbf{x} : \operatorname{dist}\le... | Yes |
Corollary 37.1.7 If \( H \) is a compact subset of \( {V}_{i} \) for some \( {V}_{i} \) there exists a partition of unity such that \( {\psi }_{i}\left( x\right) = 1 \) for all \( x \in H \) in addition to the conclusion of Lemma 37.1.6. | Proof: Keep \( {V}_{i} \) the same but replace \( {V}_{j} \) with \( \widetilde{{V}_{j}} \equiv {V}_{j} \smallsetminus H \) . Now in the proof above, applied to this modified collection of open sets, if \( j \neq i,{\phi }_{j}\left( x\right) = 0 \) whenever \( x \in H \) . Therefore, \( {\psi }_{i}\left( x\right) = 1 \... | Yes |
Theorem 37.1.8 Let \( H \) be any closed set and let \( \mathfrak{C} \) be any open cover of \( H \) . Then there exist functions \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{\infty } \) such that \( \operatorname{spt}\left( {\psi }_{i}\right) \) is contained in some set of \( \mathfrak{C} \) and \( {\psi }_{i} \) is inf... | Proof: By Lemma 37.1.2 there exists an open cover of \( H \) composed of bounded open sets, \( {U}_{i} \) such that each \( {U}_{i} \) is a subset of some set of \( \mathfrak{C} \) and the collection, \( {\left\{ {U}_{i}\right\} }_{i = 1}^{\infty } \) is locally finite. Then the result follows from Lemma 37.1.6 and Lem... | Yes |
Corollary 37.1.9 Let \( H \) be any closed set and let \( {\left\{ {V}_{i}\right\} }_{i = 1}^{m} \) be a finite open cover of \( H \) . Then there exist functions \( {\left\{ {\phi }_{i}\right\} }_{i = 1}^{m} \) such that \( \operatorname{spt}\left( {\phi }_{i}\right) \subseteq {V}_{i} \) and \( {\phi }_{i} \) is infin... | Proof: By Theorem 37.1.8 there exists a set of functions, \( {\left\{ {\psi }_{i}\right\} }_{i = 1}^{\infty } \) having the properties listed in this theorem relative to the open covering, \( {\left\{ {V}_{i}\right\} }_{i = 1}^{m} \) . Let \( {\phi }_{1}\left( \mathbf{x}\right) \) equal the sum of all \( {\psi }_{j}\le... | Yes |
Theorem 37.2.4 Let \( \Gamma \) be a \( {C}^{m,1} \) manifold. Then there exists a unique Radon measure, \( \mu \), defined on \( \Gamma \) such that whenever \( f \) is a continuous function having compact support which is defined on \( \Gamma \) and \( \left( {{\Gamma }_{i},{\mathbf{g}}_{i}}\right) \) denotes an atla... | Proof: To begin, here is a claim.\n\nClaim : A set, \( S \subseteq {\Gamma }_{i} \), has \( \mu \) measure zero if and only if \( {\mathbf{g}}_{i}S \) has measure zero in \( {\mathbf{g}}_{i}{\Gamma }_{i} \) with respect to the measure, \( {\nu }_{i} \) .\n\nProof of the claim: Let \( \varepsilon > 0 \) be given. By out... | Yes |
Lemma 37.3.1 Let \( A = \left( {a}_{ij}\right) \) be a real \( p \times n \) matrix in which \( p \geq n \) . For \( I \in \Lambda \left( {p, n}\right) \) denote by \( {A}_{I} \) the \( n \times n \) matrix obtained by deleting from \( A \) all rows except for those corresponding to an element of \( I \) . Then\n\n\[ \... | Proof: For \( \left( {{j}_{1},\cdots ,{j}_{n}}\right) \in \Lambda \left( {p, n}\right) \), define \( \theta \left( {j}_{k}\right) \equiv k \) . Then let for \( \left\{ {{k}_{1},\cdots ,{k}_{n}}\right\} = \) \( \left\{ {{j}_{1},\cdots ,{j}_{n}}\right\} \) define\n\n\[ \operatorname{sgn}\left( {{k}_{1},\cdots ,{k}_{n}}\r... | Yes |
Lemma 37.3.2 \( \mu = {\mathcal{H}}^{n} \) on every \( \mu \) measurable set. | Proof: The Riesz representation theorem shows that\n\n\[ \n{\int }_{\Gamma }{fd\mu } = {\int }_{\Gamma }{fd}{\mathcal{H}}^{n} \n\]\n\nfor every continuous function having compact support. Therefore, since every open set is the countable union of compact sets, it follows \( \mu = {\mathcal{H}}^{n} \) on all open sets. S... | Yes |
Theorem 38.0.4 Suppose \( U \) is an open set and \( {U}_{0} \subseteq U \) is another open set. Suppose also \( {D}^{\alpha }u \in {L}^{p}\left( U\right) \) . Then for all \( \psi \in {C}_{c}^{\infty }\left( {U}_{0}\right) \) , | \[ {\int }_{{U}_{0}}\left( {{D}^{\alpha }u}\right) {\psi dx} = {\left( -1\right) }^{\left| \alpha \right| }{\int }_{{U}_{0}}u\left( {{D}^{\alpha }\psi }\right) . \] | No |
Theorem 38.0.8 (Meyer Serrin) Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \). Then if \( \delta > 0 \) and \( u \in {X}^{m, p}\left( U\right) \), there exists \( J \in {C}^{\infty }\left( U\right) \) such that \( \parallel J - u{\parallel }_{m, p, U} < \delta \). | Proof: Let \( \cdots {U}_{k} \subseteq \overline{{U}_{k}} \subseteq {U}_{k + 1}\cdots \) be a sequence of open subsets of \( U \) whose union equals \( U \) such that \( \overline{{U}_{k}} \) is compact for all \( k \). Also let \( {U}_{-3} = {U}_{-2} = {U}_{-1} = {U}_{0} = \varnothing \). Now define \( {V}_{k} \equiv ... | Yes |
Lemma 38.0.10 If \( u \in {W}^{m, p}\left( U\right) \) and \( \psi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \), then \( {u\psi } \in {W}^{m, p}\left( U\right) \) . | Proof: Let \( \left| \alpha \right| \leq m \) and let \( \phi \in {C}_{c}^{\infty }\left( U\right) \) . Then\n\n\[ \left( {{D}_{{x}_{i}}\left( {u\psi }\right) }\right) \left( \phi \right) \equiv - {\int }_{U}{u\psi }{\phi }_{,{x}_{i}}{dx} \]\n\n\[ = - {\int }_{U}u\left( {{\left( \psi \phi \right) }_{,{x}_{i}} - \phi {\... | Yes |
Corollary 38.0.12 Let \( U \) be an open set which has the segment property. Then \( {W}^{m, p}\left( U\right) = {X}^{m, p}\left( U\right) \) . | Proof: Start with an open covering of \( \bar{U} \) whose sets satisfy the segment condition and obtain a locally finite refinement consisting of bounded sets which are of the sort in the above theorem. | No |
Theorem 38.1.3 Let \( U \) be a bounded open set and for \( u \) a function defined on \( {\mathbb{R}}^{n} \) , let \( {r}_{U}u\left( \mathbf{x}\right) \equiv u\left( \mathbf{x}\right) \) for \( \mathbf{x} \in \bar{U} \) . Then if \( p > n,{r}_{U} : {W}^{1, p}\left( {\mathbb{R}}^{n}\right) \rightarrow C\left( \bar{U}\r... | Proof: First suppose \( {u}_{k} \rightarrow 0 \) in \( {W}^{1, p}\left( {\mathbb{R}}^{n}\right) \) . Then if \( {r}_{U}{u}_{k} \) does not converge to 0, it follows there exists a sequence, still denoted by \( k \) and \( \varepsilon > 0 \) such that \( {u}_{k} \rightarrow 0 \) in \( {W}^{1, p}\left( {\mathbb{R}}^{n}\r... | Yes |
Corollary 38.1.7 Let \( p > n, U \) and \( {r}_{U} \) be as in Theorem 38.1.3 and let \( m \) be a nonnegative integer. Then \( {r}_{U} : {W}^{m + 1, p}\left( {\mathbb{R}}^{n}\right) \rightarrow {C}^{m,\lambda }\left( \bar{U}\right) \) is continuous as a map into \( {C}^{m,\lambda }\left( \bar{U}\right) \) for all \( \... | Proof: Suppose \( {u}_{k} \rightarrow 0 \) in \( {W}^{m + 1, p}\left( {\mathbb{R}}^{n}\right) \) . Then from 38.1.13, if \( \lambda \leq 1 - \frac{n}{p} \) and \( \left| \alpha \right| = m \)\n\n\[{\rho }_{\lambda }\left( {{D}^{\alpha }{u}_{k}}\right) \leq C{\begin{Vmatrix}{D}^{\alpha }{u}_{k}\end{Vmatrix}}_{1, p}\oper... | Yes |
Lemma 38.1.9 If \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and \( n \geq 1 \), then\n\n\[ \parallel \phi {\parallel }_{n/\left( {n - 1}\right) } \leq \frac{1}{\sqrt[n]{n}}\mathop{\sum }\limits_{{j = 1}}^{n}{\begin{Vmatrix}\frac{\partial \phi }{\partial {x}_{j}}\end{Vmatrix}}_{1}. \]\n | Proof: The case where \( n = 1 \) is obvious if \( n/\left( {n - 1}\right) \) is interpreted as \( \infty \) . Assume then that \( n > 1 \) and note that for \( {a}_{i} \geq 0 \) ,\n\n\[ n\mathop{\prod }\limits_{{i = 1}}^{n}{a}_{i} \leq {\left( \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{i}\right) }^{n} \]\n\nIn fact, the ... | Yes |
Corollary 38.1.11 Suppose \( {mp} < n \) . Then \( {W}^{m, p}\left( {\mathbb{R}}^{n}\right) \subseteq {L}^{q}\left( {\mathbb{R}}^{n}\right) \) where \( q = \frac{np}{n - {mp}} \) and the identity map, id : \( {W}^{m, p}\left( {\mathbb{R}}^{n}\right) \rightarrow {L}^{q}\left( {\mathbb{R}}^{n}\right) \) is continuous. | Proof: This is true if \( m = 1 \) according to Theorem 38.1.10. Suppose it is true for \( m - 1 \) where \( m > 1 \) . If \( u \in {W}^{m, p}\left( {\mathbb{R}}^{n}\right) \) and \( \left| \alpha \right| \leq 1 \), then \( {D}^{\alpha }u \in {W}^{m - 1, p}\left( {\mathbb{R}}^{n}\right) \) so by induction, for all such... | Yes |
Corollary 38.1.12 Suppose \( m \geq 1 \) and \( j \) is a nonnegative integer satisfying \( {jp} < \) n. Then\n\n\[ \n{W}^{m + j, p}\left( {\mathbb{R}}^{n}\right) \subseteq {W}^{m, q}\left( {\mathbb{R}}^{n}\right)\n\]\n\nfor\n\n\[ \nq \equiv \frac{np}{n - {jp}}\n\]\n\n(38.1.16)\n\nand the identity map is continuous. | Proof: If \( \left| \alpha \right| \leq m \), then \( {D}^{\alpha }u \in {W}^{j, p}\left( {\mathbb{R}}^{n}\right) \) and so by Corollary 38.1.11, \( {D}^{\alpha }u \in \) \( {L}^{q}\left( {\mathbb{R}}^{n}\right) \) where \( q \) is given above. This means \( u \in {W}^{m, q}\left( {\mathbb{R}}^{n}\right) \) . | Yes |
Corollary 38.1.13 Suppose \( {jp} < n < \left( {j + 1}\right) p \) and let \( m \) be a positive integer. Let \( U \) be any bounded open set in \( {\mathbb{R}}^{n} \) . Then letting \( {r}_{U} \) denote the restriction to \( \bar{U} \) , \( {r}_{U} : {W}^{m + j, p}\left( {\mathbb{R}}^{n}\right) \rightarrow {C}^{m - 1,... | Proof: From Corollary 38.1.12 \( {W}^{m + j, p}\left( {\mathbb{R}}^{n}\right) \subseteq {W}^{m, q}\left( {\mathbb{R}}^{n}\right) \) where \( q \) is given by 38.1.16. Therefore,\n\n\[ \frac{np}{n - {jp}} > n \]\n\nand so by Corollary 38.1.7, \( {W}^{m, q}\left( {\mathbb{R}}^{n}\right) \subseteq {C}^{m - 1,\lambda }\lef... | Yes |
Lemma 38.1.15 Let \( u \in {W}^{1,1}\left( U\right) \) for \( U \) an open set and let \( \phi \in {C}_{c}^{\infty }\left( U\right) \) . Then there exists a constant, \[ C\left( {\phi ,\parallel u{\parallel }_{1,1, U}}\right) \] depending only on the indicated quantities such that whenever \( \mathbf{v} \in {\mathbb{R}... | Proof: First suppose \( u \in {C}^{\infty }\left( \bar{U}\right) \) . Then for any \( \mathbf{x} \in \operatorname{spt}\left( \phi \right) \cup \left( {\operatorname{spt}\left( \phi \right) - \mathbf{v}}\right) \equiv \) \( {G}_{\mathbf{v}} \), the chain rule implies \[ \left| {{\phi u}\left( {\mathbf{x} + \mathbf{v}}\... | Yes |
Lemma 38.2.2 If \( U \) is an open subset of \( {\mathbb{R}}^{n} \) which has a Lipschitz boundary, then it satisfies the segment condition and so \( {X}^{m, p}\left( U\right) = {W}^{m, p}\left( U\right) \) . | Proof: For \( \mathbf{x} \in \partial U \), simply look at a single open set, \( {Q}_{\mathbf{x}} \) described in the above which contains \( \mathbf{x} \) . Then consider an open set whose intersection with \( U \) is of the form \( {R}^{T}\left( \left\{ {\mathbf{y} : \widehat{\mathbf{y}} \in B, g\left( \widehat{\math... | No |
Lemma 38.2.3 Let \( B \times \left( {a, b}\right) \) be as described in Definition 38.2.1 and let\n\n\[ \n{V}^{ - } \equiv \left\{ {\left( {\widehat{\mathbf{y}},{y}_{n}}\right) : {y}_{n} < g\left( \widehat{\mathbf{y}}\right) }\right\} ,{V}^{ + } \equiv \left\{ {\left( {\widehat{\mathbf{y}},{y}_{n}}\right) : {y}_{n} > g... | Proof: Consider the following picture which is descriptive of the situation.\n\n\n\nNote first that \( u \) is Lipschitz continuous. To see this, consider \( \left| {u\left( {\mathbf{y}}_{1}\right) - u\left( {\math... | Yes |
In the situation of Lemma 38.2.3 let \( u \in {C}^{1}\left( \overline{{V}^{ - }}\right) \cap {C}_{c}^{1}{\left( B \times \left( a, b\right) \right) }^{3} \) and define \[ w\left( {\widehat{\mathbf{y}},{y}_{n}}\right) \equiv \left\{ \begin{array}{l} u\left( {\widehat{\mathbf{y}},{y}_{n}}\right) \text{ if }\widehat{\math... | Proof: As in the previous lemma, \( w \) is Lipschitz continuous and has compact support so it is clear \( w \in {W}^{1, p}\left( {\mathbb{R}}^{n}\right) \). The main task is to find \( {w}_{, i} \) for \( \widehat{\mathbf{y}} \in B \) and \( {y}_{n} > g\left( \widehat{\mathbf{y}}\right) \) and then to extract an estim... | Yes |
Lemma 38.2.6 In the situation of Definition 38.2.1 let \( u \in {C}^{1}\left( \bar{U}\right) \cap {C}_{c}^{1}\left( Q\right) \) and define\n\n\[ \n{Eu} \equiv {R}^{ * }{E}_{0}{\left( {R}^{T}\right) }^{ * }u.\n\]\n\nwhere \( {\left( {R}^{T}\right) }^{ * } \) maps \( {W}^{1, p}\left( {U \cap Q}\right) \) to \( {W}^{1, p}... | Proof: This follows from Theorem 38.0.14 and Lemma 38.2.4. | No |
Corollary 38.2.9 Let \( U \) be a bounded open set which has Lipschitz boundary and let \( W \) be an open set containing \( \bar{U} \). Then for each \( p \geq 1 \), there exists \( {E}_{W} \in \) \( \mathcal{L}\left( {{W}^{1, p}\left( U\right) ,{W}_{0}^{1, p}\left( W\right) }\right) \) such that \( {E}_{W}u\left( \ma... | Proof: Let \( \psi \in {C}_{c}^{\infty }\left( W\right) \) and \( \psi = 1 \) on \( U \). Then let \( {E}_{W}u \equiv {\psi Eu} \) where \( E \) is the extension operator of Theorem 38.2.7. | Yes |
Theorem 38.3.1 Let \( 1 \leq p < n \) and \( \frac{1}{q} = \frac{1}{p} - \frac{1}{n} \) and let \( U \) be any open set for which there exists a \( \left( {1, p}\right) \) extension operator. Then if \( u \in {W}^{1, p}\left( U\right) \), there exists a constant independent of \( u \) such that\n\n\[ \parallel u{\paral... | Proof: Let \( E \) be the \( \left( {1, p}\right) \) extension operator. Then by Theorem 38.1.10 on Page 1390\n\n\[ \parallel u{\parallel }_{{L}^{q}\left( U\right) } \leq \parallel {Eu}{\parallel }_{{L}^{q}\left( {\mathbb{R}}^{n}\right) } \leq \frac{1}{\sqrt[n]{n}}\frac{\left( {n - 1}\right) p}{\left( n - p\right) }\pa... | Yes |
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