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Corollary 38.3.2 Suppose \( {mp} < n \) and \( U \) is an open set satisfying the segment condition which has a \( \left( {1, p}\right) \) extension operator for all \( p \) . Then \( \mathrm{{id}} \in \mathcal{L}\left( {{W}^{m, p}\left( U\right) ,{L}^{q}\left( U\right) }\right) \) where \( q = \frac{np}{n - {mp}} \) . | Proof: This is true if \( m = 1 \) according to Theorem 38.3.1. Suppose it is true for \( m - 1 \) where \( m > 1 \) . If \( u \in {W}^{m, p}\left( U\right) \) and \( \left| \alpha \right| \leq 1 \), then \( {D}^{\alpha }u \in {W}^{m - 1, p}\left( U\right) \) so by induction, for all such \( \alpha \) ,\n\n\[ \n{D}^{\a... | Yes |
Corollary 38.3.3 Suppose \( m \geq 1 \) and \( j \) is a nonnegative integer satisfying \( {jp} < n \) . Also suppose \( U \) has a \( \left( {1, p}\right) \) extension operator for all \( p \geq 1 \) and satisfies the segment condition. Then\n\n\[ \mathrm{{id}} \in \mathcal{L}\left( {{W}^{m + j, p}\left( U\right) ,{W}... | Proof: If \( \left| \alpha \right| \leq m \), then \( {D}^{\alpha }u \in {W}^{j, p}\left( U\right) \) and so by Corollary 38.3.2, \( {D}^{\alpha }u \in \) \( {L}^{q}\left( U\right) \) where \( q \) is given above. Since \( U \) has the segment property, this means \( u \in {W}^{m, q}\left( U\right) \) . It remains to v... | Yes |
Theorem 38.3.4 Let \( U \) be a bounded open set having a \( \left( {1, p}\right) \) extension operator and let \( p > n \) . Then \( \operatorname{id} : {W}^{1, p}\left( U\right) \rightarrow C\left( \bar{U}\right) \) is continuous and compact. | Proof: Theorem 38.1.3 on Page 38.1.3 implies \( {r}_{U} : {W}^{1, p}\left( {\mathbb{R}}^{n}\right) \rightarrow C\left( \bar{U}\right) \) is continuous and compact. Thus\n\n\[ \parallel u{\parallel }_{\infty, U} = \parallel {Eu}{\parallel }_{\infty, U} \leq C\parallel {Eu}{\parallel }_{1, p,{\mathbb{R}}^{n}} \leq C\para... | Yes |
Corollary 38.3.5 Let \( p > n \), let \( U \) be a bounded open set having a \( \left( {1, p}\right) \) extension operator which also satisfies the segment condition, and let \( m \) be a nonnegative integer. Then \( \mathrm{{id}} : {W}^{m + 1, p}\left( U\right) \rightarrow {C}^{m,\lambda }\left( \bar{U}\right) \) is c... | Proof: Let \( {u}_{k} \rightarrow 0 \) in \( {W}^{m + 1, p}\left( U\right) \) . Then it follows that for each \( \left| \alpha \right| \leq m \) , \( {D}^{\alpha }{u}_{k} \rightarrow 0 \) in \( {W}^{1, p}\left( U\right) \) . Therefore,\n\n\[ E\left( {{D}^{\alpha }{u}_{k}}\right) \rightarrow 0\text{in}{W}^{1, p}\left( {... | Yes |
Theorem 38.3.6 Suppose \( {jp} < n < \left( {j + 1}\right) p \) and let \( m \) be a positive integer. Let \( U \) be any bounded open set in \( {\mathbb{R}}^{n} \) which has a \( \left( {1, p}\right) \) extension operator for each \( p \geq 1 \) and the segment property. Then \( \mathrm{{id}} \in \mathcal{L}\left( {{W... | Proof: From Corollary \( {38.3.3}{W}^{m + j, p}\left( U\right) \subseteq {W}^{m, q}\left( U\right) \) where \( q \) is given by 38.3.34. Therefore,\n\n\[ \n\frac{np}{n - {jp}} > n \n\]\n\nand so by Corollary 38.3.5, \( {W}^{m, q}\left( U\right) \subseteq {C}^{m - 1,\lambda }\left( \bar{U}\right) \) for all \( \lambda \... | Yes |
Lemma 38.4.1 Let \( {H}^{ - } \) be a half space as in 38.4.35. Let \( {H}^{ + } \) be the half space in which \( {y}_{n} < 0 \) is replaced with \( {y}_{n} > 0 \) . Also let \( \left( {{\mathbf{y}}^{\prime },{y}_{n}}\right) = \mathbf{y} \)\n\n\[ u\left( {{\mathbf{y}}^{\prime },{y}_{n}}\right) \equiv \left\{ {\begin{ar... | Proof: Consider the following for \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and \( \left| \alpha \right| \leq k \) .\n\n\[ {\left( -1\right) }^{\left| \alpha \right| }\left( {{\int }_{{\mathbb{R}}^{n - 1}}{\int }_{0}^{\infty }{u}^{ + }{D}^{\alpha }{\phi d}{y}_{n}d{y}^{\prime } + {\int }_{{\mathbb{R... | Yes |
Lemma 38.4.2 Let \( {H}^{ - } \) be the half space in 38.4.35 and let \( u \in {C}^{\infty }\left( \overline{{H}^{ - }}\right) \) . Then there exists a mapping,\n\n\[ E : {C}^{\infty }\left( \overline{{H}^{ - }}\right) \rightarrow {W}^{k, p}\left( {\mathbb{R}}^{n}\right) \]\n\nand a constant, \( C \) which is independe... | Proof: Define\n\n\[ {Eu}\left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) \equiv \left\{ \begin{array}{l} u\left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) \text{ if }{x}_{n} < 0 \\ \mathop{\sum }\limits_{{j = 1}}^{k}{\lambda }_{j}u\left( {{\mathbf{x}}^{\prime }, - j{x}_{n}}\right) \text{ if }{x}_{n} \geq 0 \end{array}\righ... | Yes |
Corollary 38.4.3 Let \( {H}^{ - } \) be the half space of 38.4.35. There exists \( E \) with the property that \( E : {W}^{l, p}\left( {H}^{ - }\right) \rightarrow {W}^{l, p}\left( {\mathbb{R}}^{n}\right) \) and is linear and continuous for each \( l \leq k \) . | Proof: This immediate from the density of \( {C}_{c}^{\infty }\left( \overline{{H}^{ - }}\right) \) in \( {W}^{k, p}\left( \overline{{H}^{ - }}\right) \) and Lemma 38.4.2. | Yes |
Corollary 38.4.4 Let \( \\left\\{ {{k}_{1},\\cdots ,{k}_{r}}\\right\\} \\subseteq \\{ 1,\\cdots, n\\} \) where the \( {k}_{i} \) are distinct and let\n\n\[ \n{H}_{{k}_{1}\\cdots {k}_{r}}^{ - } \\equiv {H}_{{k}_{1}}^{ - } \\cap {H}_{{k}_{2}}^{ - } \\cap \\cdots \\cap {H}_{{k}_{r}}^{ - }.\n\]\n\n(38.4.37)\n\nThen there e... | Proof: Follow the above argument with minor modifications to first extend from \( {H}_{{k}_{1}\\cdots {k}_{r}}^{ - } \) to \( {H}_{{k}_{1}\\cdots {k}_{r - 1}}^{ - } \) and then from from \( {H}_{{k}_{1}\\cdots {k}_{r - 1}}^{ - } \) to \( {H}_{{k}_{1}\\cdots {k}_{r - 2}}^{ - } \) etc.\n\nThis easily implies the ability ... | Yes |
Theorem 38.4.8 For \( u \in {W}_{0}^{m, p}\left( U\right) \), define\n\n\[ \n{Eu}\left( \mathbf{x}\right) \equiv \left\{ \begin{array}{l} u\left( \mathbf{x}\right) \;\text{ if }\mathbf{x} \in U \\ 0\;\text{ if }\mathbf{x} \notin U \end{array}\right. \n\]\n\nThen \( E \) is a strong \( \left( {k, p}\right) \) extension ... | Proof: Letting \( l \leq m \), it is clear that for \( \left| \alpha \right| \leq l \) ,\n\n\[ \n{D}^{\alpha }{Eu} = \left\{ {\begin{array}{l} {D}^{\alpha }u\text{ for }\mathbf{x} \in U \\ 0\text{ for }\mathbf{x} \notin U \end{array}.}\right. \n\]\n\nThis follows because, since \( {m}_{n}\left( {\partial U}\right) = 0 ... | Yes |
Lemma 39.1.3 The Schwartz class, \( \mathfrak{S} \), is dense in \( {W}^{m, p}\left( {\mathbb{R}}^{n}\right) \) . | Proof: The set, \( {\mathbb{R}}^{n} \) satisfies the segment condition and so \( {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) is dense in \( {W}^{m, p}\left( {\mathbb{R}}^{n}\right) \) . However, \( {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \subseteq \mathfrak{S} \) . This proves the lemma. | Yes |
Theorem 39.1.8 Suppose \( U \) satisfies Assumption 39.1.2. Then for \( m \) a nonnegative integer, \( {H}^{m}\left( U\right) = {W}^{m,2}\left( U\right) \) and the two norms are equivalent. | Proof: Let \( u \in {H}^{m}\left( U\right) \) . Then there exists \( v \in {H}^{m}\left( {\mathbb{R}}^{n}\right) \) such that \( {\left. v\right| }_{U} = u \) . Hence \( v \in {W}^{k,2}\left( {\mathbb{R}}^{n}\right) \) and so all its weak derivatives up to order \( m \) are in \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \... | Yes |
Lemma 39.2.2 \( {H}^{m + s}\left( U\right) \) is a Banach space. | Proof: Just repeat the proof of Lemma 39.1.7. | No |
Corollary 39.2.4 Let \( U \) be an open set and let \( {\left. \mathfrak{S}\right| }_{U} \) denote the restrictions of functions of \( \mathfrak{S} \) to \( U \) . Then \( {\left. \mathfrak{S}\right| }_{U} \) is dense in \( {H}^{t}\left( U\right) \) . | Proof: Let \( u \in {H}^{t}\left( U\right) \) and let \( v \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) such that \( {\left. v\right| }_{U} = u \) a.e. Then since \( \mathfrak{S} \) is dense in \( {H}^{t}\left( {\mathbb{R}}^{n}\right) \), there exists \( w \in \mathfrak{S} \) such that\n\n\[ \parallel w - v{\parallel }... | Yes |
Lemma 39.2.5 Let \( 0 \leq r < s < t \) . Then if \( u \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) , \[ \parallel u{\parallel }_{{H}^{s}\left( {\mathbb{R}}^{n}\right) } \leq \parallel u{\parallel }_{{H}^{r}\left( {\mathbb{R}}^{n}\right) }^{\theta }\parallel u{\parallel }_{{H}^{t}\left( {\mathbb{R}}^{n}\right) }^{1 - \... | Proof: This follows from Holder’s inequality applied to the measure \( \mu \) given by \[ \mu \left( E\right) = {\int }_{E}{\left| Fu\right| }^{2}{dx} \] Thus \[ \int {\left( 1 + {\left| \mathbf{x}\right| }^{2}\right) }^{s}{\left| Fu\right| }^{2}{dx} \] \[ = \int {\left( 1 + {\left| \mathbf{x}\right| }^{2}\right) }^{r\... | Yes |
Corollary 39.2.6 Let \( U \) be an open set satisfying Assumption 39.1.2 and let \( p < q \) where \( p, q \) are two nonnegative integers. Also let \( t \in \left( {p, q}\right) \) . Then exists a constant, \( C \) independent of \( u \in {H}^{q}\left( U\right) \) such that for all \( u \in {H}^{q}\left( U\right) \) ,... | Proof: Let \( E \in \mathcal{L}\left( {{H}^{q}\left( U\right) ,{H}^{q}\left( {\mathbb{R}}^{n}\right) }\right) \) such that for all positive integers, \( l \) less than or equal to \( q, E \in \mathcal{L}\left( {{H}^{l}\left( U\right) ,{H}^{l}\left( {\mathbb{R}}^{n}\right) }\right) \) . Then \( {\left. Eu\right| }_{U} =... | Yes |
Theorem 39.2.7 Let \( \mathbf{h} : U \rightarrow V \) be one to one and onto where \( U \) and \( V \) are two open sets. Also suppose that \( {D}^{\alpha }\mathbf{h} \) and \( {D}^{\alpha }\left( {\mathbf{h}}^{-1}\right) \) exist and are Lipschitz continuous if \( \left| \alpha \right| \leq m - 1 \) for \( m \) a posi... | \[ {\mathbf{h}}^{ * } : {W}^{m, p}\left( V\right) \rightarrow {W}^{m, p}\left( U\right) \] is continuous, linear, one to one, and has an inverse with the same properties, the inverse being \( {\left( {\mathbf{h}}^{-1}\right) }^{ * } \) . | Yes |
Theorem 39.3.1 Let \( s \in \left( {0,1}\right) \) and let \( m \) be a nonnegative integer. Then an equivalent norm for \( {H}^{m + s}\left( {\mathbb{R}}^{n}\right) \) is\n\n\[ \parallel \left| u\right| {\parallel }_{m + s}^{2} \equiv \parallel u{\parallel }_{m,2,{\mathbb{R}}^{n}}^{2} + \mathop{\sum }\limits_{{\left| ... | Proof: Let \( u \in \mathfrak{S} \) which is dense in \( {H}^{m + s}\left( {\mathbb{R}}^{n}\right) \) . The Fourier transform of the function, \( \mathbf{y} \rightarrow {D}^{\alpha }u\left( {\mathbf{x} + \mathbf{y}}\right) - {D}^{\alpha }u\left( \mathbf{y}\right) \) equals\n\n\[ \left( {{e}^{i\mathbf{x} \cdot \mathbf{z... | Yes |
Lemma 39.3.3 Let \( \\mathbf{h}\\left( U\\right) \\subseteq V \) where \( U \) and \( V \) are open subsets of \( {\\mathbb{R}}^{n} \) and suppose that \( \\mathbf{h},{\\mathbf{h}}^{-1} : {\\mathbb{R}}^{n} \\rightarrow {\\mathbb{R}}^{n} \) are both functions in \( {C}^{m,1}\\left( {\\mathbb{R}}^{n}\\right) \) . Recall ... | Proof: Let \( u \\in {H}^{m + s}\\left( V\\right) \) and let \( v \\in {H}^{m + s}\\left( {\\mathbb{R}}^{n}\\right) \) such that \( {\\left. v\\right| }_{V} = u \) . Then from the above, \( {\\mathbf{h}}^{ * }v \\in {H}^{m + s}\\left( {\\mathbb{R}}^{n}\\right) \) and so \( {\\mathbf{h}}^{ * }u \\in {H}^{m + s}\\left( U... | Yes |
Lemma 39.3.4 Let \( \phi \in {C}^{m,1}\left( {\mathbb{R}}^{n}\right) \) and suppose \( \operatorname{spt}\left( \phi \right) \) is compact. Then there exists a constant, \( {C}_{\phi } \) such that whenever \( u \in {H}^{m + s}\left( {\mathbb{R}}^{n}\right) \), \[ \parallel {\phi u}{\parallel }_{{H}^{m + s}\left( {\mat... | Proof: It is a routine exercise in the product rule to verify that \( \parallel {\phi u}{\parallel }_{{H}^{m}\left( {\mathbb{R}}^{n}\right) } \leq \) \( {C}_{\phi }\parallel u{\parallel }_{{H}^{m}\left( {\mathbb{R}}^{n}\right) } \). It only remains to consider the term involving the integral. A typical term is \[ \iint... | Yes |
Corollary 39.3.5 Let \( t = m + s \) for \( s \in \lbrack 0,1) \) and let \( U, V \) be open sets. Let \( \phi \in {C}_{c}^{m,1}\left( V\right) \) . This means \( \operatorname{spt}\left( \phi \right) \subseteq V \) and \( \phi \in {C}^{m,1}\left( {\mathbb{R}}^{n}\right) \) . Then if \( u \in {H}^{t}\left( U\right) \) ... | Proof: Let \( {\left. v\right| }_{U} = u \) a.e. where \( v \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) . Then by Lemma 39.3.4, \( {\phi v} \in \) \( {H}^{t}\left( {\mathbb{R}}^{n}\right) \) and \( {\left. \phi v\right| }_{U \cap V} = {\phi u} \) a.e. Therefore, \( {\phi u} \in {H}^{t}\left( {U \cap V}\right) \) and\n... | Yes |
Lemma 39.4.2 Let \( u \in \mathfrak{S} \) and let \( \frac{n}{2} + m < t \) . Then there exists \( C \) independent of \( u \) such that \[ \parallel u{\parallel }_{{C}_{b}^{m}\left( {\mathbb{R}}^{n}\right) } \leq C\parallel u{\parallel }_{{H}^{t}\left( {\mathbb{R}}^{n}\right) }.\] | Proof: Using the fact that the Fourier transform maps \( \mathfrak{S} \) to \( \mathfrak{S} \) and the definition of the Fourier transform, \[ \left| {{D}^{\alpha }u\left( \mathbf{x}\right) }\right| \leq C{\begin{Vmatrix}F{D}^{\alpha }u\end{Vmatrix}}_{{L}^{1}\left( {\mathbb{R}}^{n}\right) } \] \[ = C\int \left| {\mathb... | Yes |
Corollary 39.4.3 Let \( u \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) where \( t > m + \frac{n}{2} \) . Then \( u \) is a.e. equal to a function of \( {C}_{b}^{m}\left( {\mathbb{R}}^{n}\right) \) still denoted by \( u \) . Furthermore, there exists a constant, \( C \) independent of \( u \) such that \[ \parallel u{\p... | Proof: This follows from the above lemma. Let \( \left\{ {u}_{k}\right\} \) be a sequence of functions of \( \mathfrak{S} \) which converges to \( u \) in \( {H}^{t} \) and a.e. Then by the inequality of the above lemma, this sequence is also Cauchy in \( {C}_{b}^{m}\left( {\mathbb{R}}^{n}\right) \) and taking the limi... | Yes |
Corollary 39.4.4 Let \( t > m + \frac{n}{2} \) and let \( U \) be an open set with \( u \in {H}^{t}\left( U\right) \) . Then \( u \) is a.e. equal to a function of \( {C}^{m}\left( \bar{U}\right) \) still denoted by \( u \) . Furthermore, there exists a constant, \( C \) independent of \( u \) such that\n\n\[ \parallel... | Proof: Let \( u \in {H}^{t}\left( U\right) \) and let \( v \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) such that \( {\left. v\right| }_{U} = u \) . Then\n\n\[ \parallel u{\parallel }_{{C}^{m}\left( \bar{U}\right) } \leq \parallel v{\parallel }_{{C}_{b}^{m}\left( {\mathbb{R}}^{n}\right) } \leq C\parallel v{\parallel }_... | Yes |
Lemma 39.5.2 Consider the integral, \[ {\int }_{\mathbb{R}}{\left( {a}^{2} + {x}^{2}\right) }^{-t}{dx} \] for \( a > 0 \) and \( t > 1/2 \) . Then this integral is no more than \( {C}_{t}{a}^{-{2t} + 1} \) where \( {C}_{t} \) is some constant which depends on \( t \) . | Proof: If \( t > 1/2 \) the integrand is in \( {L}^{1}\left( \mathbb{R}\right) \) . This is easily seen because it is of the form \( \frac{1}{{\left( {a}^{2} + {x}^{2}\right) }^{t}} \) . Now change the variable letting \( x = {au} \) and the result is obtained. | No |
Lemma 39.5.3 Let \( u \in \mathfrak{S} \). Then there exists a constant, \( {C}_{n} \), depending on \( n \) but independent of \( u \in \mathfrak{S} \) such that\n\n\[ \n{F\gamma u}\left( {\mathbf{x}}^{\prime }\right) = {C}_{n}{\int }_{\mathbb{R}}{Fu}\left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) d{x}_{n} \n\] | Proof: Using the dominated convergence theorem,\n\n\[ \n{\int }_{\mathbb{R}}{Fu}\left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) d{x}_{n} \equiv \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{\int }_{\mathbb{R}}{e}^{-{\left( \varepsilon {x}_{n}\right) }^{2}}{Fu}\left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) d{x}_{n}... | Yes |
Lemma 39.5.6 Suppose \( U \) is an open subset of \( {\mathbb{R}}^{n} \) of the form\n\n\[ U \equiv \left\{ {\mathbf{u} \in {\mathbb{R}}^{n} : {\mathbf{u}}^{\prime } \in {U}^{\prime }\text{ and }0 < {u}_{n} < \phi \left( {\mathbf{u}}^{\prime }\right) }\right\} \]\n\nwhere \( {U}^{\prime } \) is an open subset of \( {\m... | Proof: First consider the second claim. Let \( v \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) and let \( {v}_{k} \rightarrow v \) in \( {H}^{t}\left( {\mathbb{R}}^{n}\right) \) where \( {v}_{k} \in \mathfrak{S} \) . Then from Lemma 39.3.4 and Theorem 39.5.4\n\n\[ \parallel \gamma \left( {\phi v}\right) - {\gamma \phi \... | Yes |
Theorem 39.5.7 Let \( t > 1/2 \) and let \( U \) be of the form\n\n\[ \left\{ {\mathbf{u} \in {\mathbb{R}}^{n} : {\mathbf{u}}^{\prime } \in {U}^{\prime }\text{ and }0 < {u}_{n} < \phi \left( {\mathbf{u}}^{\prime }\right) }\right\} \]\n\nwhere \( {U}^{\prime } \) is an open subset of \( {\mathbb{R}}^{n - 1} \) and \( \p... | Proof: Let \( u \in {H}^{t}\left( U\right) \) . Then \( {\left. u = v\right| }_{U} \) for some \( v \in {H}^{t}\left( {\mathbb{R}}^{n}\right) \) . Define\n\n\[ {\gamma u} \equiv {\left. \gamma v\right| }_{{U}^{\prime }} \]\n\nIs this well defined? The answer is yes because if \( {\left. {v}_{i}\right| }_{U} = u \) a.e.... | Yes |
Lemma 39.6.2 Let \( {\mathbf{g}}_{i},{\mathbf{h}}_{i},{U}_{i},{W}_{i} \), and \( {\Gamma }_{i} \) be as defined above. Then\n\n\[ \n{\mathbf{g}}_{i} \circ {\mathbf{h}}_{k} : {U}_{k} \cap {\mathbf{h}}_{k}^{-1}\left( {\Gamma }_{i}\right) \rightarrow {U}_{i} \cap {\mathbf{h}}_{i}^{-1}\left( {\Gamma }_{k}\right) \n\]\n\nis... | Proof: First it is well to show it does indeed map the given open sets. Let \( \mathbf{x} \in {U}_{k} \cap {\mathbf{h}}_{k}^{-1}\left( {\Gamma }_{i}\right) \) . Then \( {\mathbf{h}}_{k}\left( \mathbf{x}\right) \in {\Gamma }_{k} \cap {\Gamma }_{i} \) and so \( {\mathbf{g}}_{i}\left( {{\mathbf{h}}_{k}\left( \mathbf{x}\ri... | Yes |
Theorem 40.1.2 (Lax Milgram) Let \( A \in \mathcal{L}\left( {V,{V}^{\prime }}\right) \) be coercive. Then \( A \) maps one to one and onto. | Proof: The proof that \( A \) is onto involves showing \( A\left( V\right) \) is both dense and closed.\n\nConsider first the claim that \( A\left( V\right) \) is closed. Let \( A{x}_{n} \rightarrow {y}^{ * } \in {V}^{\prime } \) . Then\n\n\[ \delta {\begin{Vmatrix}{x}_{n} - {x}_{m}\end{Vmatrix}}_{V}^{2} \leq {\begin{V... | Yes |
Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \) and let \( V \) be a closed subspace of \( {H}^{1}\left( U\right) \) . Let \( {\alpha }^{ij} \in {L}^{\infty }\left( U\right) \) for \( i, j = 1,2,\cdots, n \) . Now define \( A : V \rightarrow {V}^{\prime } \) by\n\n\[ A\left( u\right) \left( v\right) \equiv {\in... | Here is why. It is obvious that \( A \) is in \( \mathcal{L}\left( {V,{V}^{\prime }}\right) \) . It only remains to verify that it is coercive.\n\n\[ A\left( u\right) \left( u\right) \equiv {\int }_{U}\left( {{\alpha }^{ij}\left( \mathbf{x}\right) {u}_{, i}\left( \mathbf{x}\right) {u}_{, j}\left( \mathbf{x}\right) + u\... | Yes |
Example 40.1.7 Let \( U \) be a bounded open connected subset of \( {\mathbb{R}}^{n} \) and let \( V \) be a closed subspace of \( {H}^{1}\left( U\right) \) defined by\n\n\[ V \equiv \left\{ {u \in {H}^{1}\left( U\right) : {\gamma u} = 0\text{ on }\Gamma }\right\} \]\n\nwhere the surface measure of \( \Gamma \) is posi... | This follows from Theorem 40.1.5 using the equivalent norm defined there. Define \( F \in {V}^{\prime } \) by\n\n\[ {\int }_{U}f\left( \mathbf{x}\right) v\left( \mathbf{x}\right) {dx} + {\int }_{\partial U \smallsetminus \Gamma }g\left( \mathbf{x}\right) {\gamma v}\left( \mathbf{x}\right) {dx} \]\n\nfor \( f \in {L}^{2... | Yes |
Theorem 40.2.2 Suppose the conditions 40.2.8 - 40.2.10 hold. Then there exists a nonzero \( u \in {H}_{0}^{1}\left( U\right) \) such that\n\n\[- {\Delta u} = f\left( u\right)\] | One can verify that an example of such a function \( f\left( u\right) \) is\n\n\[f\left( u\right) = {\left| u\right| }^{p - 2}u\]\n\nThis is very exciting to a large number of people because it gives an interesting example of non uniqueness of a boundary value problem. It is clear that \( u = 0 \) works. | No |
Lemma 41.1.2 Let \( {U}^{ - } \) denote the set,\n\n\[ \left\{ {\left( {\mathbf{x},{x}_{n}}\right) \in {\mathbb{R}}^{n} : {x}_{n} < g\left( \mathbf{x}\right) }\right\} \]\n\nwhere \( g : {\mathbb{R}}^{n - 1} \rightarrow \mathbb{R} \) is Lipschitz and denote by \( {U}^{ + } \) the set\n\n\[ \left\{ {\left( {\mathbf{x},{... | Proof: Let \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Then,\n\n\[ {\int }_{{\mathbb{R}}^{n}}f\frac{\partial \phi }{\partial {x}_{n}}{dx} = {\int }_{{U}^{ + }}\frac{\partial \phi }{\partial {x}_{n}}\left\lbrack {-{3f}\left( {\mathbf{x},{2g}\left( \mathbf{x}\right) - {x}_{n}}\right) + {4f}\left( {\m... | Yes |
Lemma 41.1.3 If \( f \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \), then the following formula holds. | Proof: For \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \)\n\n\[ \parallel \phi {\parallel }_{1,2,{\mathbb{R}}^{n}} \equiv {\left( {\int }_{{\mathbb{R}}^{n}}\left( 1 + {\left| \mathbf{t}\right| }^{2}\right) {\left| F\phi \right| }^{2}dt\right) }^{1/2} \]\n\nis an equivalent norm to the usual Sobolev spac... | Yes |
Theorem 41.2.1 (Korn’s second inequality) Let \( \Omega \) be any domain for which the conclusion of Theorem 41.1.1 holds. Then the two norms in 41.2.9 and 41.2.10 are equivalent. | Proof: Let \( \mathbf{u} \) be such that \( {u}_{i} \in {H}^{1}\left( \Omega \right) \) for each \( i = 1,\cdots, n \) . Note that\n\n\[ \frac{{\partial }^{2}{u}_{i}}{\partial {x}_{j},\partial {x}_{k}} = \frac{\partial }{\partial {x}_{j}}\left( {{\varepsilon }_{ik}\left( \mathbf{u}\right) }\right) + \frac{\partial }{\p... | Yes |
Lemma 42.2.2 Let \( W \) be one of the sets described in the above definition and let \( m \geq 1 \) . Let \( {W}_{1} \subseteq \overline{{W}_{1}} \subseteq W \) where \( {W}_{1} \) is an open set. Suppose also that\n\n\[ u \in {H}^{1}\left( \Omega \right) \]\n\n\[ {\alpha }^{rs} \in {C}^{0,1}\left( \overline{\Omega }\... | Proof: Let\n\n\[ E \equiv \left\{ {v \in {H}^{1}\left( {\Omega \cap W}\right) : \operatorname{spt}\left( v\right) \subseteq W}\right\} \]\n\n\( u \) restricted to \( W \cap \Omega \) is in \( {H}^{1}\left( {\Omega \cap W}\right) \) and\n\n\[ {\int }_{\Omega \cap W}{a}^{ij}\left( \mathbf{x}\right) {u}_{, i}{v}_{, j}{dx}... | Yes |
Theorem 42.2.3 Let \( \Omega \) be a bounded open set with \( {C}^{1,1} \) boundary as in Definition 42.2.1, let \( f \in {L}^{2}\left( \Omega \right) ,{h}_{k} \in {H}^{1}\left( \Omega \right) \), and suppose that for all \( \mathbf{x} \in \bar{\Omega } \) ,\n\n\[ \n{a}^{ij}\left( \mathbf{x}\right) {v}_{i}{v}_{j} \geq ... | Proof: Let the \( {W}_{i} \) for \( i = 1,\cdots, l \) be as described in Definition 42.2.1. Thus \( \partial \Omega \subseteq { \cup }_{j = 1}^{l}{W}_{j} \) . Then let \( {C}_{1} \equiv \partial \Omega \smallsetminus { \cup }_{i = 2}^{l}{W}_{i} \), a closed subset of \( {W}_{1} \) . Let \( {D}_{1} \) be an open set sa... | Yes |
Lemma 42.2.5 Let \( W \) be one of the sets described in Definition 42.2.1 and let \( m \geq k \) . Let \( {W}_{1} \subseteq \overline{{W}_{1}} \subseteq W \) where \( {W}_{1} \) is an open set. Suppose also that\n\n\[ u \in {H}^{k}\left( \Omega \right) \]\n\n\[ {\alpha }^{rs} \in {C}^{k - 1,1}\left( \bar{\Omega }\righ... | Proof: Let\n\n\[ E \equiv \left\{ {v \in {H}^{k}\left( {\Omega \cap W}\right) : \operatorname{spt}\left( v\right) \subseteq W}\right\} \]\n\n\( u \) restricted to \( W \cap \Omega \) is in \( {H}^{k}\left( {\Omega \cap W}\right) \) and\n\n\[ {\int }_{\Omega \cap W}{a}^{ij}\left( \mathbf{x}\right) {u}_{, i}{v}_{, j}{dx}... | Yes |
Theorem 42.2.6 Let \( \Omega \) be a bounded open set with \( {C}^{k,1} \) boundary as in Definition 42.2.1, let \( f \in {H}^{k - 1}\left( \Omega \right) ,{h}_{s} \in {H}^{k}\left( \Omega \right) \), and suppose that for all \( \mathbf{x} \in \bar{\Omega } \) ,\n\n\[ \n{a}^{ij}\left( \mathbf{x}\right) {v}_{i}{v}_{j} \... | Proof: Let the \( {W}_{i} \) for \( i = 1,\cdots, l \) be as described in Definition 42.2.1. Thus \( \partial \Omega \subseteq { \cup }_{j = 1}^{l}{W}_{j} \) . Then let \( {C}_{1} \equiv \partial \Omega \smallsetminus { \cup }_{i = 2}^{l}{W}_{i} \), a closed subset of \( {W}_{1} \) . Let \( {D}_{1} \) be an open set sa... | Yes |
Lemma 43.1.4 Let \( \bar{f} \) be as defined in Definition 69.2.2. Then for \( f \in {L}^{p}\left( {a, b;X}\right) \) for \( p \in \lbrack 1,\infty ) \) , \[ \mathop{\lim }\limits_{{\delta \rightarrow 0}}{\int }_{a}^{b}\parallel \bar{f}\left( {t - \delta }\right) - f\left( t\right) {\parallel }_{X}^{p}{dt} = 0. \] | Proof: Regarding the measure space as \( \left( {a, b}\right) \) with Lebesgue measure, by Lemma 21.5.9 there exists \( g \in {C}_{c}\left( {a, b;X}\right) \) such that \( \parallel f - g{\parallel }_{p} < \varepsilon \) . Here the norm is the norm in \( {L}^{p}\left( {a, b;X}\right) \) . Therefore, \[ {\begin{Vmatrix}... | Yes |
Lemma 43.1.6 The above definition is well defined. | Proof: Suppose both \( h \) and \( g \) work in the definition. Then for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) ,\n\n\[ \n{\int }_{a}^{b}\left( {h\left( t\right) - g\left( t\right) }\right) \phi \left( t\right) {dt} = 0. \n\]\n\nTherefore, by Lemma 43.1.1, \( h\left( t\right) - g\left( t\right) = 0 \)... | Yes |
Lemma 43.1.8 Suppose \( f \in {L}^{1}\left( {a, b;X}\right) \) and for all \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \) ,\n\n\[{\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} = 0\]\n\nThen there exists a constant, \( a \in X \) such that \( f\left( t\right) = a \) a.e. | Proof: Let \( {\phi }_{0} \in {C}_{c}^{\infty }\left( {a, b}\right) ,{\int }_{a}^{b}{\phi }_{0}\left( x\right) {dx} = 1 \) and define for \( \phi \in {C}_{c}^{\infty }\left( {a, b}\right) \)\n\n\[{\psi }_{\phi }\left( x\right) \equiv {\int }_{a}^{x}\left\lbrack {\phi \left( t\right) - \left( {{\int }_{a}^{b}\phi \left(... | Yes |
Theorem 43.1.9 Suppose \( f,{f}^{\prime } \) both are in \( {L}^{1}\left( {a, b;X}\right) \) where the derivative is taken in the sense of \( X \) valued distributions. Then there exists a unique point of \( X \) , denoted by \( f\left( a\right) \) such that the following formula holds a.e. \( t \) .\n\n\[ f\left( t\ri... | Proof:\n\n\[ {\int }_{a}^{b}\left( {f\left( t\right) - {\int }_{a}^{t}{f}^{\prime }\left( s\right) {ds}}\right) {\phi }^{\prime }\left( t\right) {dt} = {\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt} - {\int }_{a}^{b}{\int }_{a}^{t}{f}^{\prime }\left( s\right) {\phi }^{\prime }\left( t\right) {dsd... | Yes |
Corollary 43.1.10 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. | Proof: From Theorem 43.1.9\n\n\[{\int }_{a}^{b}f\left( t\right) {\phi }^{\prime }\left( t\right) {dt}\]\n\n\[= {\int }_{a}^{b}\left( {f\left( a\right) + {\int }_{a}^{t}{f}^{\prime }\left( s\right) {ds}}\right) {\phi }^{\prime }\left( t\right) {dt}\]\n\n\[= f\left( a\right) \left( {\phi \left( b\right) - \phi \left( a\r... | Yes |
Proposition 43.1.12 Let \( f \in {H}^{1}\left( {0, T, X}\right) \) . Then \( f \in {C}^{0,\left( {1/2}\right) }\left( {\left\lbrack {0, T}\right\rbrack, X}\right) \) and the inclusion map is continuous. | Proof: First note that\n\n\[ f\left( t\right) - f\left( s\right) = {\int }_{s}^{t}{f}^{\prime }\left( r\right) {dr} \]\n\nand so\n\n\[ \parallel f\left( t\right) - f\left( s\right) {\parallel }_{X} \leq {\int }_{s}^{t}{\begin{Vmatrix}{f}^{\prime }\left( r\right) \end{Vmatrix}}_{X}{dr} \leq \parallel f{\parallel }_{{H}^... | Yes |
Lemma 43.1.13 Let \( \bar{f} \) be given in Definition 43.1.2 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\( \left( {43.1.2}\right) \)\n... | Yes |
Lemma 43.2.1 Let \( Y : \left\lbrack {0, T}\right\rbrack \rightarrow E \), be \( \mathcal{B}\left( \left\lbrack {0, T}\right\rbrack \right) \) measurable and suppose\n\n\[ Y \in {L}^{p}\left( {0, T;E}\right) \equiv K, p \geq 1 \]\n\nThen there exists a sequence of nested partitions, \( {\mathcal{P}}_{k} \subseteq {\mat... | 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 \( t \in \left( {k/{2}^{n},\left( {k + 1}\right) /{2}^{n}}\right\rbrack \) , and \( {2}^{-n} < T/4 \) . Also suppose \( Y \) is defined to equal 0 on \( {\l... | Yes |
Theorem 43.2.2 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 43.2.1, 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 43.2.3 Let \( s < t \) . Then for \( X, Y \) satisfying 43.2.8\n\n\[{\left| X\left( t\right) \right| }^{2} = {\left| X\left( s\right) \right| }^{2} + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - {\left| X\left( t\right) - X\left( s\right) \right| }^{2}\]\n\n\( \left( {43.2.9}\right) \... | 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 43.2.4 In the above situation, \[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}{\left| X\left( t\right) \right| }_{H} \leq C\left( {\parallel Y{\parallel }_{{K}^{\prime }},\parallel X{\parallel }_{K}}\right) \] Also, \( t \rightarrow X\left( t\right) \) is weakly continuous with values in \( H... | Proof: From the above formula applied to the \( {k}^{\text{th }} \) partition of \( \left\lbrack {0, T}\right\rbrack \) described above, \[ {\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) \right| }^{2} - {\left| X\... | Yes |
Theorem 43.3.2 Let \( V \subseteq W,{W}^{\prime } \subseteq {V}^{\prime } \) be separable Banach spaces, \( W \) a separable Hilbert space, and let \( Y \in {L}^{{p}^{\prime }}\left( {0, T;{V}^{\prime }}\right) \equiv {K}^{\prime } \) and\n\n\[ \n{BX}\left( t\right) = B{X}_{0} + {\int }_{0}^{t}Y\left( s\right) {ds}\tex... | Proof: By Lemma 43.2.1, 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\[ \n\mathop{\sum }\limits_{{k = 0}}^{{{m}_{... | Yes |
Lemma 43.3.3 Let \( s < t \) . Then for \( X, Y \) satisfying 43.3.16\n\n\[ \langle {BX}\left( t\right), X\left( t\right) \rangle = \langle {BX}\left( s\right), X\left( s\right) \rangle \]\n\n\[ + 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - \langle B\left( {X\left( t\right) - X\left( s\rig... | Proof: It follows from the following computations\n\n\[ B\left( {X\left( t\right) - X\left( s\right) }\right) = {\int }_{s}^{t}Y\left( u\right) {du} \]\n\nand so\n\n\[ 2{\int }_{s}^{t}\langle Y\left( u\right), X\left( t\right) \rangle {du} - \langle B\left( {X\left( t\right) - X\left( s\right) }\right) ,\left( {X\left(... | Yes |
Theorem 43.4.2 Let \( L : D\left( L\right) \subseteq V \rightarrow {V}^{\prime } \) where \( D\left( L\right) \) is dense, \( L \) is monotone, \( L \) is closed, and \( {L}^{ * } \) is monotone, \( L \) a linear map. Let \( T : V \rightarrow \mathcal{P}\left( {V}^{\prime }\right) \) be \( L \) pseudomonotone, bounded,... | To apply this theorem, let \( B \) be as above and \( V \rightarrow \mathcal{V} \equiv {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack, V}\right) \) . Letting \( {u}_{0} \in V \), let\n\n\[ T\left( u\right) \equiv A\left( {u + {u}_{0}}\right) \]\n\nwhere \( A : \mathcal{V} \rightarrow \mathcal{P}\left( {\mathcal{V}}^{\... | Yes |
Theorem 43.5.2 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 43.6.4 \( {\left( {A}_{0},{A}_{1}\right) }_{\theta, q} \) is a normed linear space satisfying\n\n\[ \n{A}_{0} \cap {A}_{1} \subseteq {\left( {A}_{0},{A}_{1}\right) }_{\theta, q} \subseteq {A}_{0} + {A}_{1}, \]\n\n(43.6.26)\n\nwith the inclusion maps continuous, and\n\n\[ \n\left( {{\left( {A}_{0},{A}_{1}\right)... | Proof: Suppose first \( a \in {A}_{0} \cap {A}_{1} \) . Then\n\n\[ \n\parallel a{\parallel }_{\theta, q}^{q} \equiv {\int }_{0}^{r}{\left( {t}^{-\theta }K\left( t, a\right) \right) }^{q}\frac{dt}{t} + {\int }_{r}^{\infty }{\left( {t}^{-\theta }K\left( t, a\right) \right) }^{q}\frac{dt}{t} \]\n\n\( \left( {43.6.32}\righ... | Yes |
Lemma 43.7.2 Suppose \( a \in {\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J} \) and \( a = {\int }_{0}^{\infty }u\left( t\right) \frac{dt}{t} \) where \( u \) is described above. Then letting \( r > 1 \) , it follows that \( {\int }_{0}^{\infty }{u}_{r}\left( t\right) \frac{dt}{t} \in {A}_{0} \cap {A}_{1} \) | Proof: The integral equals \( {\int }_{1/r}^{r}u\left( t\right) \frac{dt}{t} \cdot {\int }_{1/r}^{r}\frac{1}{t}{dt} = 2\ln r < \infty \) . Now \( {u}_{r} \) is measurable in \( {A}_{0} \cap {A}_{1} \) and bounded. Therefore, there exists a sequence of measurable simple functions, \( \left\{ {s}_{n}\right\} \) having va... | Yes |
Lemma 43.7.3 Suppose for \( a \in {A}_{0} + {A}_{1},\mathop{\lim }\limits_{{t \rightarrow 0 + }}K\left( {t, a}\right) = 0 \) and \( \mathop{\lim }\limits_{{t \rightarrow \infty }}\frac{K\left( {t, a}\right) }{t} = \) 0 . Then for any \( \varepsilon > 0 \), there is a representation,\n\n\[ \na = \mathop{\sum }\limits_{{... | Proof: For each \( i \), there exist \( {a}_{0, i} \in {A}_{0} \) and \( {a}_{1, i} \in {A}_{1} \) such that\n\n\[ \na = {a}_{0, i} + {a}_{1, i} \n\]\n\nand\n\n\[ \n\left( {1 + \varepsilon }\right) K\left( {{2}^{i}, a}\right) \geq {\begin{Vmatrix}{a}_{0, i}\end{Vmatrix}}_{{A}_{0}} + {2}^{i}{\begin{Vmatrix}{a}_{1, i}\en... | Yes |
Lemma 43.7.4 If \( a \in {A}_{0} \cap {A}_{1} \), then \( K\left( {t, a}\right) \leq \min \left( {1,\frac{t}{s}}\right) J\left( {s, a}\right) \) . | Proof: If \( s \geq t \), then \( \min \left( {1,\frac{t}{s}}\right) = \frac{t}{s} \) and so\n\n\[ \min \left( {1,\frac{t}{s}}\right) J\left( {s, a}\right) = \frac{t}{s}\max \left( {\parallel a{\parallel }_{{A}_{0}}, s\parallel a{\parallel }_{{A}_{1}}}\right) \geq \left( \frac{t}{s}\right) s\parallel a{\parallel }_{{A}... | Yes |
Lemma 43.8.3 Let \( f\left( t\right) \geq 0 \), and let \( f\left( t\right) = {\alpha }_{i} \) for \( t \in \left\lbrack {{2}^{i},{2}^{i + 1}}\right) \) where \( \alpha \in {\lambda }^{\theta, q} \). Then there exists a constant, \( C \), such that \[ {\left| \left| {t}^{-\theta }f\right| \right| }_{{L}^{q}\left( {0,\i... | Proof: Consider 43.8.59. \[ {\int }_{0}^{\infty }{\left( {t}^{-\theta }f\left( t\right) \right) }^{q}\frac{dt}{t} = \mathop{\sum }\limits_{i}{\int }_{{2}^{i}}^{{2}^{i + 1}}{t}^{-{\theta q}}{\alpha }_{i}^{q}\frac{dt}{t} \] \[ \leq \mathop{\sum }\limits_{i}{\int }_{{2}^{i}}^{{2}^{i + 1}}{\left( {2}^{-{i\theta }}{\alpha }... | Yes |
Lemma 43.8.4 Let \( \theta \in \left( {0,1}\right) \) and let \( q \geq 1 \) . Then,\n\n\[{\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J}^{\prime } \subseteq {\left( {A}_{1}^{\prime },{A}_{0}^{\prime }\right) }_{1 - \theta ,{q}^{\prime }}\]\n\nand the inclusion map is continuous. | Proof: Let \( {a}^{\prime } \in {\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J}^{\prime } \) . Now\n\n\[{A}_{0} \cap {A}_{1} \subseteq {\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J}\]\n\nand if\n\n\[a \in {\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J},\]\n\nthen \( a \) has a representation of the form\n\n\[a = {\int... | Yes |
Theorem 43.8.6 Suppose \( {A}_{0} \cap {A}_{1} \) is dense in \( {A}_{i} \) and \( {A}_{i} \) is reflexive. Then\n\n\[ \n{\left( {A}_{1}^{\prime },{A}_{0}^{\prime }\right) }_{1 - \theta ,{q}^{\prime }} = {\left( {A}_{0},{A}_{1}\right) }_{\theta, q}^{\prime }\n\]\n\nand the norms are equivalent. | Proof: By Theorem 43.7.5, and the last two lemmas,\n\n\[ \n{\left( {A}_{0},{A}_{1}\right) }_{\theta, q}^{\prime } = {\left( {A}_{0},{A}_{1}\right) }_{\theta, q, J}^{\prime } \subseteq {\left( {A}_{1}^{\prime },{A}_{0}^{\prime }\right) }_{1 - \theta ,{q}^{\prime }}\n\]\n\n\[ \n= {\left( {A}_{1}^{\prime },{A}_{0}^{\prime... | Yes |
Lemma 44.1.2 \( {A}_{0} + {A}_{1} \) with the norm just described is a Banach space. | Proof: This was already explained in the treatment of the \( K \) method of interpolation. It is just \( K\left( {1, a}\right) \) . | No |
Lemma 44.1.8 T is a Banach space with norm given by\n\n\[ \parallel a{\parallel }_{T} \equiv \inf \left\{ {\parallel f{\parallel }_{W} : f\left( 0\right) = a}\right\} .\n\] | Proof: Define a mapping, \( \psi : W/Z \rightarrow T \) by\n\n\[ \psi \left( \left\lbrack f\right\rbrack \right) \equiv {\gamma f} \]\n\nThen \( \psi \) is one to one and onto. Also\n\n\[ \left| \right| \left\lbrack f\right\rbrack \left| \right| \equiv \inf \left\{ {\left| \right| f + g\left| \right| : g \in Z}\right\}... | Yes |
Theorem 44.1.10 Now suppose \( {A}_{0},{A}_{1} \) and \( {B}_{0},{B}_{1} \) are pairs of Banach spaces such that \( {A}_{i} \) embeds continuously into a topological vector space, \( X \) and \( {B}_{i} \) embeds continuously into a topological vector space, \( Y \) . Suppose also that \( L \in \mathcal{L}\left( {{A}_{... | Proof: To verify 44.1.11, let \( a \in {A}_{0} + {A}_{1} \) and pick \( {a}_{0} \in {A}_{0} \) and \( {a}_{1} \in {A}_{1} \) such that\n\n\[ \parallel a{\parallel }_{{A}_{0} + {A}_{1}} + \varepsilon > {\begin{Vmatrix}{a}_{0}\end{Vmatrix}}_{{A}_{0}} + {\begin{Vmatrix}{a}_{1}\end{Vmatrix}}_{{A}_{1}}. \]\n\nThen\n\n\[ \pa... | Yes |
Lemma 45.1.2 Let \( \phi \in {C}^{\infty }\left( \overline{{\mathbb{R}}_{ + }^{n}}\right) \) . Then \( {\gamma \phi }\left( {\mathbf{x}}^{\prime }\right) \equiv \phi \left( {{\mathbf{x}}^{\prime },0}\right) \) . Then \( \gamma : {C}^{\infty }\left( \overline{{\mathbb{R}}_{ + }^{n}}\right) \rightarrow \) \( {L}^{p}\left... | Proof: We know\n\n\[ \phi \left( {{\mathbf{x}}^{\prime },{x}_{n}}\right) = {\gamma \phi }\left( {\mathbf{x}}^{\prime }\right) + {\int }_{0}^{{x}_{n}}\frac{\partial \phi \left( {{\mathbf{x}}^{\prime }, t}\right) }{\partial t}{dt} \]\n\nThen by Jensen's inequality,\n\n\[ {\int }_{{\mathbb{R}}^{n - 1}}{\left| \gamma \phi ... | Yes |
Lemma 45.1.4 The trace map, \( \\gamma \), is a continuous map from \( {W}^{1, p}\\left( {\\mathbb{R}}_{ + }^{n}\\right) \) onto\n\n\[ {W}^{1 - \\frac{1}{p}, p}\\left( {\\mathbb{R}}^{n - 1}\\right) \\text{.} \]\n\nFurthermore, for \( f \\in {W}^{1, p}\\left( {\\mathbb{R}}_{ + }^{n}\\right) \) ,\n\n\[ {\\gamma f} = f\\l... | Proof: It remains to verify \( \\gamma \) is onto along with the displayed equation. But by definition, things in \( {W}^{1 - \\frac{1}{p}, p}\\left( {\\mathbb{R}}^{n - 1}\\right) \) are of the form \( \\mathop{\\lim }\\limits_{{t \\rightarrow 0 + }}f\\left( t\\right) \) where \( f \\in \) \( {L}^{p}\\left( {0,\\infty ... | Yes |
Lemma 45.3.1 Let \( g = f * h \) where \( f \in {L}^{1}\left( \mathbb{R}\right), h \in {L}^{p}\left( \mathbb{R}\right) \), and \( f, h \) are all Borel measurable, \( p \geq 1 \) . Then \( g \in {L}^{p}\left( \mathbb{R}\right) \) and \[ \parallel g{\parallel }_{{L}^{p}\left( \mathbb{R}\right) } \leq \parallel f{\parall... | Proof: First of all it is good to show \( g \) is well defined. Using Minkowski’s inequality \[ {\left( \int {\left( \int \left| h\left( t - s\right) f\left( s\right) \right| ds\right) }^{p}dt\right) }^{1/p} \] \[ \leq \int {\left( \int {\left| h\left( t - s\right) \right| }^{p}{\left| f\left( s\right) \right| }^{p}dt\... | Yes |
Lemma 45.3.2 Let \( f \) be a real valued function defined a.e. on \( \lbrack 0,\infty ) \) and let \( \alpha \in \left( {-\infty ,1}\right) \). For \( 1 \leq p < \infty \), prove the inequality:\n\n\[{\int }_{0}^{\infty }{t}^{\alpha p}{\left| g\left( t\right) \right| }^{p}\frac{dt}{t} \leq \frac{1}{{\left( 1 - \alpha ... | Proof: First it can be assumed the right side of 45.3.7 is finite since otherwise there is nothing to show. Changing the variables letting \( t = {e}^{\tau } \), the above inequality takes the form\n\n\[{\int }_{-\infty }^{\infty }{e}^{\tau p\alpha }{\left| g\left( {e}^{\tau }\right) \right| }^{p}{d\tau } \leq \frac{1}... | Yes |
Lemma 45.3.3 Let \( {A}_{0} = D\left( \Lambda \right) \) as just described. Then for \( u \in {A}_{1} \)\n\n\[ \parallel u{\parallel }_{{A}_{1} + {A}_{0}} = \parallel u{\parallel }_{{A}_{1}} \] | Proof: \( D\left( \Lambda \right) \subseteq {A}_{1} \) . Now let \( u \in {A}_{1} \) .\n\n\[ \parallel u{\parallel }_{{A}_{0} + {A}_{1}} \equiv \inf \left\{ {{\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{{A}_{1}} + {\begin{Vmatrix}\Lambda {u}_{0}\end{Vmatrix}}_{{A}_{1}} + {\begin{Vmatrix}{u}_{1}\end{Vmatrix}}_{{A}_{1}} : u = ... | Yes |
Lemma 45.3.9 Let \( t \neq 0 \) be a number. Then there is a constant \( C\left( {n,\theta, p}\right) \) depending on the indicated quantities such that\n\n\[{\int }_{{\mathbb{R}}^{n - 1}}\frac{1}{{\left( {t}^{2} + {\left| \mathbf{s}\right| }^{2}\right) }^{\frac{1}{2}\left( {n + {p\theta }}\right) }}{ds} = \frac{C\left... | Proof: Change the integral to polar coordinates. Thus the integral equals\n\n\[{\int }_{{S}^{n - 1}}{\int }_{0}^{\infty }\frac{{\rho }^{n - 2}}{{\left( {t}^{2} + {\rho }^{2}\right) }^{\frac{1}{2}\left( {n + {p\theta }}\right) }}{d\rho d\sigma }\]\n\nNow change the variables, \( \rho = \left| t\right| u \) . Then the ab... | Yes |
Theorem 45.3.10 An equivalent norm for \( {W}^{\theta, p}\left( {\mathbb{R}}^{n}\right) \) is\n\n\[ \parallel u\parallel = \]\n\n\[ {\left( \parallel u{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\right) }^{p} + {\int }_{{\mathbb{R}}^{n}}{\int }_{{\mathbb{R}}^{n}}\frac{{\left| u\left( \mathbf{y}\right) - u\left( \mathbf... | Proof: It only remains to verify this is a norm. Recall the \( {l}_{p} \) norm on \( {\mathbb{R}}^{2} \) given by\n\n\[ {\left| \left( x, y\right) \right| }_{{l}_{p}} \equiv {\left( {\left| x\right| }^{p} + {\left| y\right| }^{p}\right) }^{1/p} \]\n\nFor \( u, v \in {W}^{\theta, p} \) denote by \( \rho \left( u\right) ... | Yes |
Theorem 45.3.13 Let \( U \) be a bounded open set which has Lipschitz boundary and \( \theta \in \left( {0,1}\right) \) . Then \( {W}^{\theta, p}\left( U\right) = {W}^{\theta, p}\left( U\right) \) and the two norms are equivalent. | Proof: Let \( u \in \widetilde{{W}^{\theta, p}\left( U\right) } \) . Letting \( E \) be the extension operator of Lemma 45.3.12, there is a constant \( C \) such that\n\n\[ C\parallel u{\parallel }_{\widetilde{{W}^{\theta, p}\left( U\right) }} \geq \parallel {Eu}{\parallel }_{\widetilde{{W}^{\theta, p}\left( {\mathbb{R... | Yes |
Corollary 45.3.14 Let \( U \) be a bounded open set with Lipschitz boundary. Then \( {W}^{\theta, p}\left( U\right) \) is reflexive. | Proof: From Proposition 45.3.12 and Theorem 45.3.13, there exists an extension operator \( E : {W}^{\theta, p}\left( U\right) \rightarrow {W}^{\theta, p}\left( {\mathbb{R}}^{n}\right) \) which is continuous. This operator is one to one and continuous. Furthermore, \( \parallel {Eu}{\parallel }_{{W}^{\theta, p}\left( {\... | Yes |
Corollary 45.4.2 The space, \( {W}^{s, p}\left( \Omega \right) \) is a reflexive Banach space whenever \( p > 1 \) . | Proof: From the theory of interpolation spaces, \( {W}^{\sigma, p}\left( \Omega \right) \) is reflexive. This is because it is an iterpolation space for the two reflexive spaces, \( {L}^{p}\left( \Omega \right) \) and \( {W}^{1, p}\left( \Omega \right) \) . (Alternatively, you could use Corollary 45.3.14 in the case wh... | Yes |
Theorem 45.4.3 The trace map, \( \gamma : {W}^{m, p}\left( {\mathbb{R}}_{ + }^{n}\right) \rightarrow {W}^{m - \frac{1}{p}, p}\left( {\mathbb{R}}^{n - 1}\right) \) is continuous. | Proof: Let \( f \in \mathfrak{S} \), the Schwartz class. Let \( \sigma = 1 - \frac{1}{p} \) so that \( m - \left( \frac{1}{p}\right) = m - 1 + \sigma \) . Then from the definition and using \( f \in \mathfrak{S} \) ,\n\n\[ \n\parallel {\gamma f}{\parallel }_{m - \frac{1}{p}, p,{\mathbb{R}}^{n - 1}} = {\left( \parallel ... | Yes |
Theorem 45.4.4 Let \( \\mathbf{h} : U \rightarrow V \) where \( U \) and \( V \) are two open sets and suppose \( \\mathbf{h} \) is bilipschitz and that \( {D}^{\\alpha }\\mathbf{h} \) and \( {D}^{\\alpha }{\\mathbf{h}}^{-1} \) exist and are Lipschitz continuous if \( \\left| \\alpha \\right| \\leq m \) where \( m = 0,... | Proof: In case \( m = 0 \), the conclusion of the theorem is immediate from the general theory of trace spaces. Therefore, assume \( m \\geq 1 \) . It follows from the definition that\n\n\[ \n{\\begin{Vmatrix}{\\mathbf{h}}^{ * }u\\end{Vmatrix}}_{m + \\sigma, p, U} \\equiv {\\left\\lbrack {\\begin{Vmatrix}{\\mathbf{h}}^... | Yes |
Theorem 45.4.5 Let \( \Omega \) be an open set in \( {\mathbb{R}}^{n} \) which has the segment property and let \( f \in {W}^{m + 1, p}\left( \Omega \right) \) and \( \sigma \in \left( {0,1}\right) \) . Then for some constant, \( C \), independent of \( f \) , \[ \parallel f{\parallel }_{m + \sigma, p,\Omega } \leq C\p... | Proof: Recall from above, \( {W}^{1 - \theta, p}\left( \Omega \right) \equiv T\left( {{W}^{1, p}\left( \Omega \right) ,{L}^{p}\left( \Omega \right), p,\theta }\right) \) . Therefore, from Theorem 44.1.9, if \( f \in {W}^{1, p}\left( \Omega \right) \) , \[ \parallel f{\parallel }_{1 - \theta, p,\Omega } \leq K\parallel ... | Yes |
Lemma 46.1.2 \( {W}^{s, p}\left( \Gamma \right) \) as just described, is a Banach space. If \( p > 1 \) then it is reflexive. | Proof: Let \( L : {W}^{s, p}\left( \Gamma \right) \rightarrow \mathop{\prod }\limits_{{i = 1}}^{l}{W}^{s, p}\left( {U}_{i}\right) \) be defined by \( {\left( Lu\right) }_{i} \equiv {\mathbf{h}}_{i}^{ * }\left( {u{\psi }_{i}}\right) \) . Let \( {\left\{ {u}_{j}\right\} }_{j = 1}^{\infty } \) be a Cauchy sequence in \( {... | Yes |
Theorem 47.0.2 Let \( \\left( {{X}_{i},{d}_{i}}\\right) \) denote a complete metric space and let \( X \\equiv \\mathop{\\prod }\\limits_{{i = 1}}^{\\infty }{X}_{i} \) . Then \( X \) is also a complete metric space with the metric\n\n\[ \n\\rho \\left( {\\mathbf{x},\\mathbf{y}}\\right) \\equiv \\mathop{\\sum }\\limits_... | Proof: It is clear from the above lemma that \( \\rho \) is a metric on \( X \) . We need to verify \( X \) is complete with this metric. Let \( \\left\{ {\\mathbf{x}}^{n}\\right\} \) be a Cauchy sequence in \( X \) . Then it is clear from the definition that \( \\left\{ {x}_{i}^{n}\\right\} \) is a Cauchy sequence for... | Yes |
Theorem 47.0.4 Let \( X \) be a polish space. Then there exists \( f : {\mathbb{N}}^{\mathbb{N}} \rightarrow X \) which is onto and continuous. Here \( {\mathbb{N}}^{\mathbb{N}} \equiv \mathop{\prod }\limits_{{i = 1}}^{\infty }\mathbb{N} \) and a metric is given according to the above theorem. Thus for \( \mathbf{n},\m... | Proof: Since \( X \) is polish, there exists a countable covering of \( X \) by closed sets having diameters no larger than \( {2}^{-1},\{ B\left( i\right) {\} }_{i = 1}^{\infty } \) . Each of these closed sets is also a polish space and so there exists a countable covering of \( B\left( i\right) \) by a countable coll... | Yes |
Corollary 47.0.6 \( X \) is a Suslin space, if and only if there exists a continuous mapping from \( {\mathbb{N}}^{\mathbb{N}} \) onto \( X \) . | Proof: We know there exists a polish space \( Z \) and a continuous function, \( h \) : \( Z \rightarrow X \) which is onto. By the above theorem there exists a continuous map, \( g : {\mathbb{N}}^{\mathbb{N}} \rightarrow Z \) which is onto. Then \( h \circ g \) is a continuous map from \( {\mathbb{N}}^{\mathbb{N}} \) ... | Yes |
Lemma 47.0.8 Let \( \left( {\Omega ,\mathcal{F},\mu }\right) \) be a measure space and denote by \( {\mu }^{ * } \) the outer measure generated by \( \mu \) . Thus\n\n\[ \n{\mu }^{ * }\left( S\right) \equiv \inf \{ \mu \left( E\right) : E \supseteq S, E \in \mathcal{F}\} .\n\]\n\nThen \( {\mu }^{ * } \) is regular, mea... | Proof: First we verify that \( {\mu }^{ * } \) is regular. If \( {\mu }^{ * }\left( S\right) = \infty \), let \( E = \Omega \) . Then \( {\mu }^{ * }\left( S\right) = \mu \left( E\right) \) and \( E \supseteq S \) . On the other hand, if \( {\mu }^{ * }\left( S\right) < \infty \), then we can obtain \( {E}_{n} \in \mat... | Yes |
Corollary 47.0.11 Let \( \Omega \) be a compact metric space and let \( A \) be a Suslin subset of \( \Omega \) . Then \( A \in \widehat{B\left( \Omega \right) } \) . | Proof: Let \( \mu \) be a finite measure defined on \( B\left( \Omega \right) \) . By Theorem 47.0.9 \( A \in \) \( B{\left( \Omega \right) }_{\mu } \) . Since this is true for every finite measure, \( \mu \), it follows \( A \in \widehat{B\left( \Omega \right) } \) as claimed. This proves the corollary. | Yes |
Lemma 47.0.12 Let \( \mu \) be a finite measure on a \( \sigma \) algebra, \( \sum \) . Then \( A \in {\sum }_{\mu } \) if and only if there exists \( {A}_{1} \in \sum \) and \( {N}_{1} \) such that \( A = {A}_{1} \cup {N}_{1} \) where there exists \( N \in \sum \) such that \( \mu \left( N\right) = 0 \) and \( {N}_{1}... | Proof: Suppose first \( A = {A}_{1} \cup {N}_{1} \) where these sets are as described. Let \( S \in \mathcal{P}\left( \Omega \right) \) and let \( {\mu }^{ * } \) denote the outer measure determined by \( \mu \) . Then since \( {A}_{1} \in \) \( \sum \subseteq {\sum }_{\mu } \)\n\n\[{\mu }^{ * }\left( S\right) \leq {\m... | Yes |
Lemma 47.0.14 Let \( \\left( {\\Omega ,\\sum }\\right) \) be separable. Then there exists \( E \\in \\{ 0,1{\\} }^{\\mathbb{N}} \) such that \( \\left( {\\Omega ,\\sum }\\right) \) and \( \\left( {E, B\\left( E\\right) }\\right) \) are isomorphic. | Proof: First we show \( \\left\\{ {A}_{n}\\right\\} \) separates the points. Here \( \\sigma \\left( \\left\\{ {A}_{n}\\right\\} \\right) = \\sum \) . We already know \( \\sum \) separates the points but now we show the smaller set does so also.\n\nIf this is not so, there exists \( \\omega ,{\\omega }_{1} \\in \\Omega... | Yes |
Lemma 47.0.15 Let \( \phi : \left( {{\Omega }_{1},{\sum }_{1}}\right) \rightarrow \left( {{\Omega }_{2},{\sum }_{2}}\right) \) where \( {\phi }^{-1}\left( U\right) \in {\sum }_{1} \) for all \( U \in {\sum }_{2} \) . Then if \( F \in {\widehat{\sum }}_{2} \), it follows \( {\phi }^{-1}\left( F\right) \in {\widehat{\sum... | Proof: Let \( \mu \) be a finite measure on \( {\sum }_{1} \) and define a measure \( \phi \left( \mu \right) \) on \( {\sum }_{2} \) by the rule\n\n\[ \phi \left( \mu \right) \left( F\right) \equiv \mu \left( {{\phi }^{-1}\left( F\right) }\right) . \]\n\nNow let \( A \in {\sum }_{{2\phi }\left( \mu \right) } \) . Then... | Yes |
Lemma 47.0.18 Let \( X \) be a Hausdorff space and let \( G \in \sum \times B\left( X\right) \) where \( \sum \) is \( {a\sigma } \) algebra of sets of \( \Omega \) . Then there exists \( {\sum }_{0} \subseteq \sum \) a countably generated \( \sigma \) algebra such that \( G \in {\sum }_{0} \times B\left( X\right) \) . | Proof: First suppose \( G \) is a measurable rectangle, \( G = A \times B \) where \( A \in \sum \) and \( B \in B\left( X\right) \) . Letting \( {\sum }_{0} \) be the finite \( \sigma \) algebra, \( \left\{ {\varnothing, A,{A}^{C},\Omega }\right\} \), we see that \( G \in {\sum }_{0} \times B\left( X\right) \) . Simil... | Yes |
Theorem 48.1.2 The following are equivalent in case of a complete \( \sigma \) finite measure space. However 3 and 4 are equivalent for any measurable space consisting only of a set \( \Omega \) and a \( \sigma \) algebra \( \mathcal{C} \) . | Proof: It is obvious that 1.) \( \Rightarrow \) 2.). To see that 2.) \( \Rightarrow \) 3.) note that \( {\Gamma }^{ - }\left( {{ \cup }_{i = 1}^{\infty }{F}_{i}}\right) = \) \( { \cup }_{i = 1}^{\infty }{\Gamma }^{ - }\left( {F}_{i}\right) \) . Since any open set in \( X \) can be obtained as a countable union of close... | No |
Proposition 48.1.4 Let \( X \) be a Polish space and let \( \Gamma : X \rightarrow \mathcal{P}\left( X\right) \) have compact values. Then \( \Gamma \) is measurable if and only if it is strongly measurable, the latter being the statement that \( {\Gamma }^{ - }\left( C\right) \) is measurable whenever \( C \) is close... | Let \( \Gamma \) be strongly measurable. Let \( \mathcal{G} \) be the sets \( G \) such that \( {\Gamma }^{ - }\left( G\right) \) and \( {\Gamma }^{ - }\left( {G}^{C}\right) \) are both in \( \mathcal{C} \) . Then clearly \( \mathcal{G} \) is closed with respect to complements. If \( G \in \mathcal{G} \) is \( {G}^{C} ... | Yes |
Lemma 48.1.5 Suppose \( f : K\left( \omega \right) \times \Omega \rightarrow X, K \subseteq X \) . Here \( X \) is Polish space, separable complete metric space, and \( \left( {\Omega ,\mathcal{F}}\right) \) is a measurable space. Also \( \omega \rightarrow K\left( \omega \right) \) is a measurable multifunction as in ... | Proof: Let \( \left\{ {{x}_{n}\left( \omega \right) }\right\} \) be a countable dense subset of \( K\left( \omega \right) \), each \( {x}_{n} \) measurable. Then if \( U \) is open, \[ \{ \omega : \mathcal{K}\left( \omega \right) \cap U \neq \varnothing \} = { \cup }_{n = 1}^{\infty }f{\left( {x}_{n}\left( \cdot \right... | Yes |
Corollary 48.2.3 Let \( K\left( \omega \right) \) be a compact subset of a separable metric space \( X \) and suppose \( {\left\{ {u}_{j}\left( \omega \right) \right\} }_{j = 1}^{\infty } \subseteq K\left( \omega \right) \) with each \( \omega \rightarrow {u}_{j}\left( \omega \right) \) measurable into \( X \) . Then t... | Proof: Define\n\n\[ \n{\Gamma }_{n}\left( \omega \right) = \overline{{ \cup }_{k \geq n}{u}_{k}\left( \omega \right) } \n\] \n\nThis is a nonempty compact subset of \( K\left( \omega \right) \subseteq X \) . I claim that \( \omega \rightarrow {\Gamma }_{n}\left( \omega \right) \) is a measurable multifunction into \( X... | Yes |
Corollary 48.2.5 Let \( K \) be a closed convex bounded subset of \( {\mathbb{R}}^{n} \). Let \( \mathbf{f}\left( {\cdot ,\omega }\right) \) : \( K \rightarrow K \) be continuous for each \( \omega \) and \( \omega \rightarrow \mathbf{f}\left( {\mathbf{x},\omega }\right) \) is measurable, meaning inverse images of sets... | Proof: Let \( S \) be a large simplex containing \( K \) and let \( P \) be the projection map onto \( K \). Consider \( \mathbf{g}\left( {\mathbf{x},\omega }\right) \equiv \mathbf{f}\left( {P\left( \mathbf{x}\right) ,\omega }\right) \). Then \( \mathbf{g} \) satisfies the necessary conditions for Theorem 48.2.4 and so... | Yes |
Theorem 48.2.6 Let \( E \) be a compact metric space and let \( \left( {\Omega ,\mathcal{F}}\right) \) be a measure space. Suppose \( \psi : E \times \Omega \rightarrow \mathbb{R} \) has the property that \( x \rightarrow \psi \left( {x,\omega }\right) \) is continuous and \( \omega \rightarrow \psi \left( {x,\omega }\... | Proof: Let \( C = {\left\{ {e}_{i}\right\} }_{i = 1}^{\infty } \) be a countable dense subset of \( E \) . For example, take the union of \( 1/{2}^{n} \) nets for all \( n \) . Let \( {C}_{n} \equiv \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) . Let \( \omega \rightarrow {f}_{n}\left( \omega \right) \) be measurable an... | Yes |
Theorem 48.2.7 Let \( E\left( \omega \right) \) be a compact metric space in a separable metric space \( \left( {X, d}\right) \) and that \( \omega \rightarrow E\left( \omega \right) \) is a measurable multifunction where \( \left( {\Omega ,\mathcal{F}}\right) \) be a measure space. Suppose \( {\psi }_{\omega } : E\lef... | Proof: Let \( C\left( \omega \right) = {\left\{ {e}_{i}\left( \omega \right) \right\} }_{i = 1}^{\infty } \) be a countable dense subset of \( E\left( \omega \right) \) with each \( {e}_{i}\left( \omega \right) \) measurable. Since \( \omega \rightarrow E\left( \omega \right) \) is measurable, such a countable dense su... | Yes |
Theorem 48.2.8 Let \( K \) be a closed convex bounded subset of \( {\mathbb{R}}^{n} \). Let \( \mathbf{f}\left( {\cdot ,\omega }\right) \) : \( K \rightarrow K \) be continuous for each \( \omega \) and \( \omega \rightarrow \mathbf{f}\left( {\mathbf{x},\omega }\right) \) is measurable, meaning inverse images of sets o... | Proof: Simply consider \( E = K \) and \( \psi \left( {\mathbf{x},\omega }\right) \equiv - \left| {\mathbf{x} - \mathbf{f}\left( {\mathbf{x},\omega }\right) }\right| \). It has a maximum \( \mathbf{x}\left( \omega \right) \) for each \( \omega \) thanks to continuity of \( \mathbf{f}\left( {\cdot ,\omega }\right) \). T... | Yes |
Corollary 48.2.9 Let \( K\left( \omega \right) \) be a closed convex bounded subset of \( {\mathbb{R}}^{n} \) and let \( \omega \rightarrow \) \( K\left( \omega \right) \) be a measurable multifunction for \( \omega \in \Omega \) with \( \left( {\Omega ,\mathcal{F}}\right) \) a measurable space. Let \( {\mathbf{f}}_{\o... | Proof: Consider \( {\psi }_{\omega }\left( {\mathbf{x},\omega }\right) \equiv - \left| {{\mathbf{f}}_{\omega }\left( {\mathbf{x}\left( \omega \right) ,\omega }\right) - \mathbf{x}\left( \omega \right) }\right| \) . By the Brower fixed point theorem, the maximum for fixed \( \omega \) is 0 . Therefore, there exists such... | Yes |
Lemma 48.2.11 Let \( f\left( {\cdot ,\omega }\right) \) be as above and \( f\left( {K\left( \omega \right) ,\omega }\right) \subseteq K\left( \omega \right) \) for \( K\left( \omega \right) \) convex and closed and \( \omega \rightarrow K\left( \omega \right) \) a measurable mutifunction. Suppose also that \( f\left( {... | Proof: Using the compactness of \( \overline{f\left( {K\left( \omega \right) ,\omega }\right) } \), Proposition 48.1.6 says there exist measurable functions \( {y}_{i}\left( \omega \right) \)\n\n\[ \n\left\{ {{y}_{1}\left( \omega \right) ,\cdots ,{y}_{n\left( \omega \right) }\left( \omega \right) }\right\} \subseteq \o... | Yes |
Lemma 48.2.12 Let \( \overline{f\left( {K\left( \omega \right) ,\omega }\right) } \) be compact. For each \( r > 0 \), there exists \( {x}_{r}\left( \omega \right) \in \) convex hull of \( \overline{f\left( {K\left( \omega \right) ,\omega }\right) } \subseteq K\left( \omega \right) \) such that\n\n\[ \n{f}_{r}\left( {{... | Proof: The upper limit in the sum of the above lemma \( n\left( \omega \right) \) is a measurable function. One can partition the measure space according to the value of \( n\left( \omega \right) \) . This gives a countable set of disjoint measurable subsets \( {\left\{ {\Omega }_{n}\right\} }_{n = 1}^{\infty } \) in t... | Yes |
Theorem 48.2.13 Let \( \omega \rightarrow K\left( \omega \right) \) be a measurable multifunction which has convex and closed values in a separable Banach space. Let \( f\left( {\cdot ,\omega }\right) : K\left( \omega \right) \rightarrow K\left( \omega \right) \) be continuous and \( \omega \rightarrow f\left( {x,\omeg... | Proof: Recall that \( f\left( {{x}_{r}\left( \omega \right) ,\omega }\right) - {f}_{r}\left( {{x}_{r}\left( \omega \right) ,\omega }\right) \in B\left( {0, r}\right) \) and \( {f}_{r}\left( {{x}_{r}\left( \omega \right) ,\omega }\right) = \) \( {x}_{r}\left( \omega \right) \) with \( {x}_{r}\left( \omega \right) \in \)... | Yes |
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