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Theorem 24.1.3 Let \( H \) be a Hilbert space and let \( I : H \rightarrow \mathbb{R} \) be a \( {C}^{1} \) functional having \( {I}^{\prime } \) Lipschitz continuous and such that \( I \) satisfies the Palais Smale condition.\n\nSuppose \( I\left( 0\right) = 0 \) and \( I\left( u\right) \geq a > 0 \) for all \( \paral... | Proof: First note that \( c \geq a > 0 \) . Suppose \( c \) is not a critical value. Then by the deformation theorem, for \( \varepsilon > 0,\varepsilon \) sufficiently small, there is \( \eta : H \rightarrow H \) and a \( \delta < \varepsilon \) small enough that\n\n\[ \eta \left( \left\lbrack {I\left( u\right) \leq c... | Yes |
Example 24.1.4 Let \( I : {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) satisfy \( \mathop{\lim }\limits_{{\left| \mathbf{x}\right| \rightarrow \infty }}I\left( \mathbf{x}\right) = \infty \) . Then \( I \) satisfies the Palais Smale conditions. | The growth condition implies that if \( I\left( {\mathbf{x}}_{k}\right) \) is bounded, then so is \( \left\{ {\mathbf{x}}_{k}\right\} \) and so this sequence is precompact. Nothing needs to be said about \( {I}^{\prime }\left( {\mathbf{x}}_{k}\right) \) . | No |
Lemma 24.1.6 Let \( Y \) be a metric space and let \( X \) be a normed linear space. (We will want to add in \( X \) .) Let \( \Gamma : Y \rightarrow \mathcal{P}\left( X\right) \) such that \( \Gamma \left( y\right) \) is a nonempty convex set. Suppose that for each \( y \in Y \), there exists an open set \( U \) conta... | Proof: Let \( \mathcal{U} \) denote the collection of all open sets \( U \) such that the nonempty intersection described above holds. Let \( \mathcal{V} \) be a locally finite open refinement which also covers. Thus for any \( V \in \mathcal{V} \)\n\n\[ \varnothing \neq { \cap }_{\widehat{y} \in V}\Gamma \left( \wideh... | Yes |
Lemma 24.1.7 Let \( \phi \) be a \( {C}^{1} \) function defined on \( X \) a Banach space. Then there exists a pseudogradient field for \( \phi \) on the set of regular points. \( (V\left( x\right) \in G\left( x\right) \) and \( x \rightarrow V\left( x\right) \) is locally Lipschitz on the set of regular points.) | Proof: First consider whether \( G\left( x\right) \), the set of pseudogradients of \( \phi \) at \( x \) is nonempty for \( {\phi }^{\prime }\left( x\right) \neq 0 \) . From the definition of the operator norm, there exists \( u \) such that \( \parallel u{\parallel }_{X} = 1 \) and \( \left\langle {{\phi }^{\prime }\... | Yes |
Corollary 24.1.10 Let \( f : X \rightarrow X \) be locally Lipschitz where \( X \) is a Banach space. Then there exists a unique local solution to the IVP\n\n\[ {y}^{\prime } = f\left( y\right) ,\;y\left( 0\right) = {y}_{0} \]\n\nIf \( f \) is bounded, then in fact the solutions exists on \( \left\lbrack {0, T}\right\r... | Proof: Say \( \parallel f\left( x\right) \parallel \leq M \) for all \( M \) . Then letting \( \lbrack 0,\widehat{T}) \) be the maximal interval, it must be the case that \( {\int }_{0}^{\widehat{T}}\parallel f\left( {y\left( t\right) }\right) \parallel {dt} = \infty \), but this does not happen if \( f \) is bounded. ... | No |
Corollary 24.1.11 Suppose \( f : X \rightarrow X \) is continuous and \( f \) is locally Lipschitz on \( U \), an open subset of \( X \), a Banach space. Suppose also that \( f\left( x\right) = 0 \) for all \( x \notin U \) and that \( \parallel f\left( x\right) \parallel < M \) for all \( x \in X \) . Then there exist... | Proof: Let \( T \) be given. If \( {y}_{0} \notin U \), there is nothing to show. The solution is \( y\left( t\right) \equiv {y}_{0} \) . Suppose then that \( {y}_{0} \in U \) . Then by Theorem 24.1.8, there exists a unique solution to the initial value problem on an interval \( \lbrack 0,\widehat{T}) \) of maximal len... | Yes |
Theorem 24.1.13 Let \( I \) be \( {C}^{1}, I \) is non constant, satisfy the Palais Smale condition, and \( {I}^{\prime } \) is bounded on bounded sets. Also suppose that \( c \in \mathbb{R} \) is such that either \( {I}^{-1}\left( \left\lbrack {c - \delta, c + \delta }\right\rbrack \right) = \varnothing \) for some \(... | Proof: Suppose \( {I}^{-1}\left( \left\lbrack {c - \widehat{\delta }, c + \widehat{\delta }}\right\rbrack \right) = \varnothing \) for some \( \widehat{\delta } > 0 \) . Then\n\n\[ \n{I}^{-1}\left( {( - \infty, c + \frac{\widehat{\delta }}{2}\rbrack }\right) \subseteq {I}^{-1}\left( {( - \infty, c - \frac{\widehat{\del... | Yes |
Theorem 24.1.14 Let \( X \) be a Banach space and let \( I : X \rightarrow \mathbb{R} \) be a \( {C}^{1} \) functional having \( {I}^{\prime } \) bounded on bounded sets and such that \( I \) satisfies the Palais Smale condition. Suppose \( I\left( 0\right) = 0 \) and \( I\left( u\right) \geq a > 0 \) for all \( \paral... | Proof: First note that \( c \geq a > 0 \) . Suppose \( c \) is not a critical value. Then either \( {I}^{-1}\left( \left( {c - \delta, c + \delta }\right) \right) = \varnothing \) for some \( \delta > 0 \) in which case the conclusion of the deformation theorem,(Theorem 24.1.13) holds, or for all \( \delta > 0,{I}^{-1}... | Yes |
Proposition 25.1.2 Let \( V \) be finite dimensional and let \( A : V \rightarrow {V}^{\prime } \) be pseu-domonotone and bounded (meaning \( A \) maps bounded sets to bounded sets). Then \( A \) is continuous. Also, if \( A : V \rightarrow {V}^{\prime } \) is pseudomonotone and bounded, and if \( W \subseteq V \) is a... | Proof: Say \( {u}_{n} \rightarrow u \) . Does it follow that \( A{u}_{n} \rightarrow {Au} \) ? If not, then there is a subsequence such that \( A{u}_{n} \rightarrow \xi \neq {Au} \) thanks to \( \left\{ {A{u}_{n}}\right\} \) being bounded. Then the lim sup condition holds obviously. In fact the limit of \( \left\langle... | Yes |
Theorem 25.1.4 Let \( V \) be a Banach space and let \( A : V \rightarrow {V}^{\prime } \) be monotone and hemicontinuous. Then \( A \) is pseudomonotone. | Proof: Let \( A \) be monotone and Hemicontinuous. First here is a claim.\n\nClaim: If 25.1.1 and 25.1.2 hold, then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = 0 \) .\n\nProof of the claim: Since \( A \) is monotone,\n\n\[ \left\langle {A{u}_{n} - {Au},{u}_{n} - ... | Yes |
Example 25.1.5 Let \( H \) be any Hilbert space (complete inner product space, more on these later) and let \( A : H \rightarrow {H}^{\prime } \) be given by\n\n\[ \langle {Ax}, y\rangle \equiv {\left( -x, y\right) }_{H}. \]\n\nThen A fails to be pseudomonotone. | Proof: Let \( {\left\{ {x}_{n}\right\} }_{n = 1}^{\infty } \) be an orthonormal set of vectors in \( H \) . Then Parsevall’s inequality implies\n\n\[ \parallel x{\parallel }^{2} \geq \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left| \left( {x}_{n}, x\right) \right| }^{2} \]\n\nand so for any \( x \in H,\mathop{\lim }\li... | Yes |
Proposition 25.1.6 Suppose \( A : V \rightarrow {V}^{\prime } \) is pseudomonotone and bounded where \( V \) is separable. Then it must be demicontinuous. This means that if \( {u}_{n} \rightarrow u \), then \( A{u}_{n} \rightharpoonup {Au} \) . In case that \( V \) is reflexive, you don’t need the assumption that \( V... | Proof: Since \( {u}_{n} \rightarrow u \) is strong convergence and since \( A{u}_{n} \) is bounded, it follows\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n} - u}\right\rangle = 0.... | Yes |
Proposition 25.1.8 If \( A \) is pseudomonotone, then \( A \) is type \( M \) . | Proof: Suppose \( A \) is pseudomonotone and \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} \rightharpoonup \xi \), and\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nThen\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}... | Yes |
Proposition 25.1.9 Suppose \( A : V \rightarrow {V}^{\prime } \) is type \( M \) and suppose \( L : V \rightarrow {V}^{\prime } \) is monotone, bounded and linear. Then \( L + A \) is type \( M \) . Let \( V \) be separable or reflexive so that the weak convergences in the following argument are valid. | Proof: Suppose \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} + L{u}_{n} \rightharpoonup \xi \) and also that\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n} + L{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nDoes it follow that \( \xi = {Au} + {Lu} \) ? Suppose not. The... | Yes |
Corollary 25.1.10 Suppose \( A : V \rightarrow {V}^{\prime } \) is type \( M \) and suppose \( L : W \rightarrow {W}^{\prime } \) is monotone, bounded and linear where \( V \subseteq W \) and \( V \) is dense in \( W \) so that \( {W}^{\prime } \subseteq {V}^{\prime } \) . Then for \( {u}_{0} \in W \) define \( M\left(... | Proof: Suppose \( {u}_{n} \rightharpoonup u \) and \( A{u}_{n} + M{u}_{n} \rightharpoonup \xi \) and also that\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {A{u}_{n} + M{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle \]\n\nDoes it follow that \( \xi = {Au} + {Mu} \) ? Suppose not. By ... | Yes |
Lemma 25.1.11 (Browder) Let \( K \) be a convex closed and bounded set in \( {\mathbb{R}}^{n} \) and let \( A : K \rightarrow {\mathbb{R}}^{n} \) be continuous and \( \mathbf{f} \in {\mathbb{R}}^{n} \) . Then there exists \( \mathbf{x} \in K \) such that for all \( \mathbf{y} \in K \) , \[ {\left( \mathbf{f} - A\mathbf... | Proof: Let \( {P}_{K} \) denote the projection onto \( K \) . Thus \( {P}_{K} \) is Lipschitz continuous. \[ \mathbf{x} \rightarrow {P}_{K}\left( {\mathbf{f} - A\mathbf{x} + \mathbf{x}}\right) \] is a continuous map from \( K \) to \( K \) . By the Brouwer fixed point theorem, it has a fixed point \( \mathbf{x} \in K \... | Yes |
Proposition 25.1.12 Let \( A : V \rightarrow {V}^{\prime } \) be continuous and coercive,\n\n\[ \mathop{\lim }\limits_{{\parallel v\parallel \rightarrow \infty }}\frac{\left\langle A\left( v + {v}_{0}\right), v\right\rangle }{\parallel v{\parallel }_{V}} = \infty \]\n\nfor some \( {v}_{0} \) . Then for all \( f \in {V}... | Proof: Define the closed convex sets \( {B}_{n} \equiv \overline{B\left( {{v}_{0}, n}\right) } \) . By Browder’s lemma, there exists \( {\mathbf{x}}_{n} \) such that\n\n\[ \left( {f - A{v}_{n}, y - {v}_{n}}\right) \leq 0 \]\n\nfor all \( y \in {B}_{n} \) . Then taking \( y = {v}_{0} \) ,\n\n\[ \left\langle {A{v}_{n},{v... | Yes |
Lemma 25.1.13 Let \( A : V \rightarrow {V}^{\prime } \) be type \( M \) and bounded and suppose \( V \) is reflexive or \( V \) is separable. Then \( A \) is demicontinuous. | Proof: Suppose \( {u}_{n} \rightarrow u \) and \( A{u}_{n} \) fails to converge weakly to \( {Au} \) . Then there is a further subsequence, still denoted as \( {u}_{n} \) such that \( A{u}_{n} \rightharpoonup \zeta \neq {Au} \) . Then thanks to the strong convergence, you have\n\n\[ \lim \mathop{\sup }\limits_{{n \righ... | No |
Theorem 25.1.14 Let \( A : V \rightarrow {V}^{\prime } \) be type \( M \), bounded, and coercive\n\n\[ \mathop{\lim }\limits_{{\parallel u\parallel \rightarrow \infty }}\frac{\left\langle A\left( u + {u}_{0}\right), u\right\rangle }{\parallel u\parallel } = \infty \]\n\n\( \left( {25.1.4}\right) \)\n\nfor some \( {u}_{... | Proof: Since \( V \) is separable, there exists an increasing sequence of finite dimensional subspaces \( \left\{ {V}_{n}\right\} \) such that \( \overline{{ \cup }_{n}{V}_{n}} = V \) and each \( {V}_{n} \) contains \( {u}_{0} \). Say \( \operatorname{span}\left( {{v}_{1},\cdots ,{v}_{n}}\right) = {V}_{n} \). Then cons... | Yes |
Lemma 25.2.2 Let \( F \) be a duality map for \( p = 2 \) where \( X,{X}^{\prime } \) are reflexive and have strictly convex norms. (If \( X \) is reflexive, there is always an equivalent strictly convex norm [8].) Then \( F \) is demicontinuous. | Proof: Say \( {x}_{n} \rightarrow x \) . Then does it follow that \( F{x}_{n} \rightharpoonup {Fx} \) ? Suppose not. Then there is a subsequence, still denoted as \( {x}_{n} \) such that \( {x}_{n} \rightarrow x \) but \( F{x}_{n} \rightharpoonup y \neq {Fx} \) where here \( \rightharpoonup \) denotes weak convergence.... | Yes |
Theorem 25.2.3 Let \( X \) be a reflexive Banach space with \( {X}^{\prime } \) having strictly convex norm \( {}^{1} \) . Then for \( p > 1 \), there exists a mapping \( F : X \rightarrow {X}^{\prime } \) which is bounded, monotone, hemicontinuous, coercive in the sense that \( \mathop{\lim }\limits_{{\left| x\right| ... | Note that these conclusions about duality maps show that they map onto the dual space.\n\nThe duality map was onto and it was monotone. This was shown above. Consider the form of a duality map for the \( {L}^{p} \) spaces. Let \( F : {L}^{p} \rightarrow {\left( {L}^{p}\right) }^{\prime } \) be the one which satisfies\n... | No |
Lemma 25.2.4 Let \( p \geq 2 \) . Then for \( a, b \) real numbers,\n\n\[ \n\left( {{\left| a\right| }^{p - 2}a - {\left| b\right| }^{p - 2}b}\right) \left( {a - b}\right) \geq C{\left| a - b\right| }^{p}\n\]\n\nfor some constant \( C \) independent of \( a, b \) . | Proof: There is nothing to show if \( a = b \) . Without loss of generality, assume \( a > b \) . Also assume \( p > 2 \) . There is nothing to show if \( p = 2 \) . I want to show that there exists a constant \( C \) such that for \( a > b \) ,\n\n\[ \n\frac{{\left| a\right| }^{p - 2}a - {\left| b\right| }^{p - 2}b}{{... | Yes |
Theorem 25.2.5 Let \( X \) be a reflexive Banach space and \( X,{X}^{\prime } \) have strictly convex norms as discussed above. Let \( F \) be the duality map with \( p = 2 \) . Then \( F \) is strictly monotone. This means\n\n\[ \langle {Fu} - {Fv}, u - v\rangle \geq 0 \]\n\nand it equals 0 if and only if \( u - v \) ... | Proof: First why is it monotone? By definition of \( F,\langle F\left( u\right), u\rangle = \parallel u{\parallel }^{2} \) and \( \parallel F\left( u\right) \parallel = \parallel u\parallel \) . Then\n\n\[ \left| {\langle {Fu}, v\rangle }\right| = \left| \left\langle {{Fu},\frac{v}{\parallel v\parallel }}\right\rangle ... | Yes |
Lemma 25.3.1 Let \( K \) be closed and convex nonempty subset of \( X \) a reflexive Banach space which has strictly convex norm. Then there exists a projection map \( P \) such that \( {Px} \in K \) and for all \( y \in K \) , \[ \parallel y - x\parallel \geq \parallel x - {Px}\parallel \] | Proof: Let \( \left\{ {y}_{n}\right\} \) be a minimizing sequence for \( y \rightarrow \parallel y - x\parallel \) for \( y \in K \) . Thus \[ d \equiv \inf \{ \parallel y - x\parallel : y \in K\} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\begin{Vmatrix}{{y}_{n} - x}\end{Vmatrix} \] Then obviously \( \left\{ {y}... | Yes |
Proposition 25.3.2 Let \( F \) be the duality map just described. Let \( \phi \left( x\right) \equiv \frac{\parallel x{\parallel }^{2}}{2} \) . Then \( F\left( x\right) = \partial \phi \left( x\right) \) . | Proof: This follows from\n\n\[ \langle {Fx}, y - x\rangle \leq \langle {Fx}, y\rangle - \langle {Fx}, x\rangle \leq \langle {Fx}, x{\rangle }^{1/2}\langle {Fy}, y{\rangle }^{1/2} - \langle {Fx}, x\rangle \]\n\n\[ \leq \frac{\langle {Fy}, y\rangle }{2} - \frac{\langle {Fx}, x\rangle }{2} = \frac{\parallel y{\parallel }^... | Yes |
Proposition 25.3.3 Let \( K \) be a nonempty closed convex set in \( X \) a reflexive Banach space in which both \( X,{X}^{\prime } \) have strictly convex norms. Then \( w \in K \) is equal to \( {Px} \) if and only if\n\n\[ \langle F\left( {x - w}\right), y - w\rangle \leq 0 \] \n\nfor every \( y \in K \) . | Proof: First suppose the condition. Then for \( y \in K \), it follows from the above proposition about the subgradient,\n\n\[ \frac{1}{2}\parallel x - y{\parallel }^{2} - \frac{1}{2}\parallel x - w{\parallel }^{2} \geq \langle F\left( {x - w}\right), w - y\rangle \geq 0 \] \n\nand so since this holds for all \( y \) i... | Yes |
Lemma 25.4.2 Let \( A \) satisfy 25.4.12. Then \( {AK} \) is a subset of a compact set whenever \( K \) is compact. Also the graph of \( A \) is closed if \( A\mathbf{x} \) is closed. | Proof: Let \( \mathbf{x} \in K \) . Then \( A\mathbf{x} \) is compact and contained in some open set whose closure is compact, \( {U}_{\mathbf{x}} \) . By assumption 25.4.12 there exists an open set \( {V}_{\mathbf{x}} \) containing \( \mathbf{x} \) such that if \( \mathbf{y} \in {V}_{\mathbf{x}} \), then \( A\mathbf{y... | Yes |
Lemma 25.4.3 If \( \mathbf{f} \) is upper semicontinuous on some set \( K \) and \( \mathbf{g} \) is continuous and defined on \( \mathbf{f}\left( K\right) \), then \( \mathbf{g} \circ \mathbf{f} \) is also upper semicontinuous. | Proof: Let \( {\mathbf{x}}_{n} \rightarrow \mathbf{x} \) in \( K \) . Let \( U \supseteq \mathbf{g} \circ \mathbf{f}\left( \mathbf{x}\right) \) . Is \( \mathbf{g} \circ \mathbf{f}\left( {\mathbf{x}}_{n}\right) \in U \) for all \( n \) large enough? We have \( \mathbf{f}\left( \mathbf{x}\right) \in {\mathbf{g}}^{-1}\lef... | Yes |
Lemma 25.4.6 Suppose in addition to 25.4.11 and 25.4.12, (compact convex valued and upper semicontinuous) \( A \) is coercive,\n\n\[ \mathop{\lim }\limits_{{\left| \mathbf{x}\right| \rightarrow \infty }}\inf \left\{ {\frac{\operatorname{Re}\left( {\mathbf{y},\mathbf{x}}\right) }{\left| \mathbf{x}\right| } : \mathbf{y} ... | Proof: Let \( \mathbf{y} \in {\mathbb{C}}^{n} \) and let \( {K}_{r} \equiv \overline{B\left( {\mathbf{0}, r}\right) } \) . By Lemma 25.4.5 there exists \( {\mathbf{x}}_{r} \in {K}_{r} \) and \( {\mathbf{w}}_{r} \in A{\mathbf{x}}_{r} \) such that\n\n\[ \operatorname{Re}\left( {\mathbf{y} - {\mathbf{w}}_{r},\mathbf{z} - ... | Yes |
Lemma 25.4.7 Let \( F \) be a finite dimensional Banach space of dimension \( n \), and let \( T \) be a mapping from \( F \) to \( \mathcal{P}\left( {F}^{\prime }\right) \) such that 25.4.11 and 25.4.12 both hold for \( {F}^{\prime } \) in place of \( {\mathbb{C}}^{n} \) . Then if \( T \) is also coercive,\n\n\[ \math... | Proof: Let \( \left| \cdot \right| \) be an equivalent norm for \( F \) such that there is an isometry of \( {\mathbb{C}}^{n} \) and \( F,\theta \) . Now define \( A : {\mathbb{C}}^{n} \rightarrow \mathcal{P}\left( {\mathbb{C}}^{n}\right) \) by \( A\mathbf{x} \equiv {\theta }^{ * }{T\theta }\mathbf{x} \).\n\n\[ \begin{... | Yes |
Lemma 25.4.9 Let \( T : X \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) satisfy conditions 25.4.15 and 25.4.17 above and suppose \( T \) is bounded ( \( {Tx} \) for \( x \) in a bounded set is bounded). Then if \( {x}_{n} \rightarrow x \) in \( X \), and if \( U \) is a weakly open set containing \( {Tx} \), th... | Proof: If this is not true, there exists \( {x}_{n} \rightarrow x \), also a weakly open set \( U \) , containing \( {Tx} \) and \( {z}_{n} \in T{x}_{n} \), but \( {z}_{n} \notin U \) . Then, taking a further subsequence, we can assume \( {z}_{n} \rightarrow z \) weakly and \( z \notin U \) . Then the strong convergenc... | Yes |
Proposition 25.4.10 Let \( V \) be finite dimensional and let \( T : V \rightarrow \mathcal{P}\left( V\right) \) be upper semicontinuous with closed values. Then if \( {u}_{n} \rightarrow u \) and \( {z}_{n} \in T{u}_{n} \) with \( {z}_{n} \rightarrow z \) , then\n\n\[ \lim \mathop{\inf }\limits_{{n \rightarrow \infty ... | Proof: Consider the last claim and suppose the limit condition does not hold for some \( v \) . Then take a subsequence such that\n\n\[ \lim \mathop{\inf }\limits_{{n \rightarrow \infty }}\operatorname{Re}{z}_{n}\left( {{u}_{n} - v}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{Re}{z}_{n}\left( ... | Yes |
Proposition 25.4.13 A single valued bounded operator \( T : V \rightarrow {V}^{\prime }, V \) reflexive is generalized bounded pseudomonotone then it is bounded and type \( M \) . | Proof: Suppose that \( {u}_{n} \rightarrow u \) weakly and \( T{u}_{n} \rightarrow \xi \) weakly and\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {T{u}_{n},{u}_{n}}\right\rangle \leq \langle \xi, u\rangle . \]\n\nThen\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {T{u}... | Yes |
Lemma 25.4.14 Let \( T : X \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) satisfy conditions 25.4.15 and 25.4.17 above and suppose \( T \) is bounded. Then if \( {x}_{n} \rightarrow x \) in \( X \), and if \( U \) is a weakly open set containing \( {Tx} \), then \( T{x}_{n} \subseteq U \) for all \( n \) large e... | Proof: If this is not true, there exists \( {x}_{n} \rightarrow x \), and a weakly open set \( U \) , containing \( {Tx} \) and \( {z}_{n} \in T{x}_{n} \), but \( {z}_{n} \notin U \) . Then, taking a further subsequence, we can assume \( {z}_{n} \rightarrow z \) weakly and \( z \notin U \) . Then the strong convergence... | Yes |
Lemma 25.6.2 Let \( f \) be as described in the above definition. Then \( \partial f\left( x\right) \) is a closed, bounded, convex, and non empty subset of \( {V}^{\prime } \) . Furthermore, for \( {x}^{ * } \in \partial f\left( x\right) \) , \[ \begin{Vmatrix}{x}^{ * }\end{Vmatrix} \leq {\operatorname{Lip}}_{x}\left(... | Proof: It is left as an exercise to verify the assertions that \( \partial f\left( x\right) \) is closed, and convex. It follows directly from the definition. To verify this set is bounded, let \( {Li}{p}_{x}\left( f\right) \) denote a Lipschitz constant valid near \( x \in V \) and let \( {x}^{ * } \in \partial f\left... | No |
Lemma 25.6.3 Let \( U \) be weakly open in \( {V}^{\prime } \) and suppose \( \partial f\left( x\right) \subseteq U \) . Then \( \partial f\left( z\right) \subseteq \) \( U \) whenever \( z \) is close enough to \( x \) . | Proof: Suppose to the contrary there exists \( {z}_{n} \rightarrow x \) but \( {z}_{n}^{ * } \in \partial f\left( {z}_{n}\right) \smallsetminus U \) . From the first lemma, we may assert that \( \begin{Vmatrix}{z}_{n}^{ * }\end{Vmatrix} \leq 2\operatorname{Lip}\left( f\right) \) for all \( n \) large enough. Therefore,... | Yes |
Theorem 25.6.4 Let \( f : V \rightarrow {V}^{\prime } \) be locally Lipschitz and suppose it satisfies the condition that whenever\n\n\[ \n{x}_{n}\text{converges weakly to}x \n\]\n\nand\n\n\[ \n\lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{f}^{0}\left( {{x}_{n}, x - {x}_{n}}\right) \geq 0 \n\]\n\n it follows tha... | Proof: 25.4.15 and 25.4.16 both are satisfied thanks to Lemmas 25.6.1 and 25.6.2. It remains to verify 25.4.17. To do so, I will adopt the convention that \( {x}^{ * } \in \partial f\left( x\right) \) . Suppose\n\n\[ \n\lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{x}_{n}^{ * }\left( {{x}_{n} - x}\right) \leq 0. ... | Yes |
Theorem 25.7.2 Let \( X,{X}^{\prime } \) be reflexive and have strictly convex norms. Let \( A \) be a monotone set valued map as just described. Then if \( {\lambda F} + A \) is onto for some \( \lambda > 0 \), then whenever\n\n\[ \langle y - z, x - u\rangle \geq 0\text{ for all }\left\lbrack {x, y}\right\rbrack \in \... | Proof: Suppose that for all \( \left\lbrack {x, y}\right\rbrack \in \mathcal{G}\left( A\right) \) ,\n\n\[ \langle y - z, x - t\rangle \geq 0 \]\n\nDoes it follow that \( z \in {At} \) ? By assumption, \( z + {\lambda F}\left( t\right) = {\lambda F}\widehat{x} + \widehat{\xi },\widehat{\xi } \in A\widehat{x} \) . Then r... | Yes |
Lemma 25.7.4 If \( \mathcal{F} \) is a set of functions which are upper semicontinuous, then \( g\left( x\right) \equiv \inf \{ f\left( x\right) : f \in \mathcal{F}\} \) is also upper semicontinuous. Similarly, if \( \mathcal{F} \) is a set of functions which are lower semicontinuous, then if \( g\left( x\right) \equiv... | Proof: Let \( f \in \mathcal{F} \) where these functions are upper semicontinuous. Then if \( {x}_{n} \rightarrow x \), and \( g\left( x\right) \equiv \inf \{ f\left( x\right) : f \in \mathcal{F}\} \) ,\n\n\[ f\left( x\right) \geq \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}f\left( {x}_{n}\right) \geq \lim \mat... | Yes |
Lemma 25.7.5 Suppose \( H : A \times B \rightarrow \mathbb{R} \) is strictly convex in the first argument and concave in the second argument where \( A, B \) are compact convex nonempty subsets of Banach spaces \( E, F \) respectively and \( x \rightarrow H\left( {x, y}\right) \) is lower semicontinuous while \( y \rig... | Proof: First suppose both \( z, w \) yield the definition of \( g\left( y\right) \) . Then\n\n\[ H\left( {\frac{z + w}{2}, y}\right) < \frac{1}{2}H\left( {z, y}\right) + \frac{1}{2}H\left( {w, y}\right) \]\n\nwhich contradicts the definition of \( g\left( y\right) \) . As to the existence of \( g\left( y\right) \) this... | No |
Theorem 25.7.6 Let \( E, F \) be Banach spaces with \( E \) having a strictly convex norm. Also suppose that \( A \subseteq E, B \subseteq F \) are compact and convex sets and that \( H : A \times B \rightarrow \) \( \mathbb{R} \) is such that\n\n\[ x \rightarrow H\left( {x, y}\right) \text{ is convex } \]\n\n\[ y \rig... | Proof: One part of the main equality is obvious.\n\n\[ \mathop{\max }\limits_{{y \in B}}H\left( {x, y}\right) \geq H\left( {x, y}\right) \geq \mathop{\min }\limits_{{x \in A}}H\left( {x, y}\right) \]\n\nand so for each \( x \) ,\n\n\[ \mathop{\max }\limits_{{y \in B}}H\left( {x, y}\right) \geq \mathop{\max }\limits_{{y... | Yes |
Lemma 25.7.7 Let \( E \) be a finite dimensional Banach space and let \( K \) be a convex and compact subset of \( E \) . Let \( \mathcal{G}\left( A\right) \) be a monotone subset of \( E \times {E}^{\prime } \) such that \( D\left( A\right) \subseteq K \) and \( B \) is a single valued monotone and continuous operator... | Proof: Let \( T : E \rightarrow K \) be the multivalued operator defined by\n\n\[ {Ty} \equiv \left\{ {x \in K : \langle {By} + v, u - x{\rangle }_{{E}^{\prime }, E} \geq 0\text{ for all }\left\lbrack {u, v}\right\rbrack \in \mathcal{G}\left( A\right) }\right\} \]\n\nHere \( y \in E \) and it is desired to show that \(... | Yes |
Lemma 25.7.10 Let \( E \) be finite dimensional and let \( B : E \rightarrow {E}^{\prime } \) be monotone and hemicontinuous. Then \( B \) is continuous. | Proof: The space can be considered a finite dimensional Hilbert space \( \left( {\mathbb{R}}^{n}\right) \) and so weak and strong convergence are exactly the same. First it is desired to show that \( B \) is bounded. Suppose it is not. Then there exists \( {\begin{Vmatrix}{x}_{k}\end{Vmatrix}}_{E} = 1 \) but \( {\begin... | Yes |
Theorem 25.7.13 Let \( X \) be a strictly convex reflexive Banach space. Suppose the graph of \( A : X \rightarrow \mathcal{P}\left( X\right) \) is maximal monotone in the sense that it is monotone and no monotone graph can properly contain the graph of \( A \) . Then for all \( \lambda > 0,{\lambda F} + A \) is onto. ... | Proof: In Theorem 25.7.9, let \( {Bx} \equiv {\lambda F}\left( x\right) - {y}_{0} \) . Then from the properties of the duality map, Theorem 25.2.3 above, it follows that \( B \) satisfies the necessary conditions to use the result of Corollary 25.7.11 with \( K = X \) . This \( B \) is monotone hemicontinuous, and coer... | Yes |
Proposition 25.7.14 Let \( A,\widehat{A} \) be as just defined. Then \( \widehat{A} \) is also maximal monotone. | Proof: From Theorem 25.7.13 it suffices to show that graph of \( \widehat{A} \) is monotone and is maximal. Suppose then that \( {x}_{i}^{ * } \in \widehat{A}{x}_{i} \) . Then\n\n\[ \left\langle {{x}_{1}^{ * } - {x}_{2}^{ * },{x}_{1} - {x}_{2}}\right\rangle = \left\langle {{x}_{1}^{ * } - {x}_{2}^{ * },{x}_{1} - {x}_{0... | Yes |
Theorem 25.7.15 Let \( B : X \rightarrow {X}^{\prime } \) be monotone hemicontinuous. Then \( B \) is maximal monotone. If \( B \) is coercive, then \( B \) is also onto. Here \( X \) is a strictly convex reflexive Banach space. | Proof: Suppose \( B \) is not maximal monotone. Then there exists \( \left( {{x}_{0},{x}_{0}^{ * }}\right) \in \) \( X \times {X}^{\prime } \) such that for all \( x \) ,\n\n\[ \left\langle {{Bx} - {x}_{0}^{ * }, x - {x}_{0}}\right\rangle \geq 0 \]\n\nand yet \( {x}_{0}^{ * } \neq B{x}_{0} \) . This is going to be a co... | Yes |
Lemma 25.7.16 Let \( X \) be a Banach space and suppose that\n\n\[ \n{x}_{n} \rightarrow 0,\;\begin{Vmatrix}{x}_{n}^{ * }\end{Vmatrix} \rightarrow \infty \n\]\n\nThen denoting by \( {D}_{r} \) the closed disk centered at 0 with radius \( r \) . It follows that for every \( {D}_{r} \), there exists \( {y}_{0} \in {D}_{r... | Proof: Suppose this is not true. Then there exists \( {D}_{r} \) which has the property that for all \( u \in {D}_{r} \) ,\n\n\[ \n\left\langle {{x}_{n}^{ * },{x}_{n} - u}\right\rangle \geq {C}_{u} \n\]\n\nfor all \( n \) . Now let\n\n\[ \n{E}_{k} \equiv \left\{ {y \in {D}_{r} : \left\langle {{x}_{n}^{ * },{x}_{n} - y}... | Yes |
Corollary 25.7.17 Let \( X \) be a Banach space and suppose that\n\n\[ \n{x}_{n} \rightarrow x,\;\begin{Vmatrix}{x}_{n}^{ * }\end{Vmatrix} \rightarrow \infty \n\]\n\nThen denoting by \( {D}_{r} \) the closed disk centered at \( x \) with radius \( r \) . It follows that for every \( {D}_{r} \), there exists \( {y}_{0} ... | Proof: It follows that \( {x}_{n} - x \rightarrow 0 \) . Therefore, from Lemma 25.7.16, for every \( r > 0 \), there exists \( {\widehat{y}}_{0} \in \overline{B\left( {0, r}\right) } \) and a subsequence \( {x}_{{n}_{k}} \) such that\n\n\[ \n\left\langle {{x}_{{n}_{k}}^{ * },\left( {{x}_{{n}_{k}} - x}\right) - {\wideha... | Yes |
Lemma 25.7.19 A set valued operator \( A \) is locally bounded at \( x \in \overline{D\left( A\right) } \) if and only if there exists \( r > 0 \) such that \( A \) is bounded on \( \overline{B\left( {x, r}\right) } \cap D\left( A\right) \) . | Proof: Say the limit condition holds. Then if no such \( r \) exists, it follows that \( A \) is unbounded on every \( B\left( {x, r}\right) \cap D\left( A\right) \) . Hence, you can let \( {r}_{n} \rightarrow 0 \) and pick \( {x}_{n} \in B\left( {x,{r}_{n}}\right) \cap D\left( A\right) \) with \( {x}_{n}^{ * } \in A{x... | Yes |
Theorem 25.7.20 Let \( A : D\left( A\right) \rightarrow {X}^{\prime } \) be monotone. Then if \( x \) is an interior point of \( D\left( A\right) \), it follows that \( A \) is locally bounded at \( x \) . | Proof: You could use Corollary 25.7.17. If \( x \) is an interior point of \( D\left( A\right) \), and \( A \) is not locally bounded, then there exists \( {x}_{n} \rightarrow x \) and \( {x}_{n}^{ * } \in A{x}_{n} \) such that \( \begin{Vmatrix}{x}_{n}^{ * }\end{Vmatrix} \rightarrow \infty \) . Then by Corollary 25.7.... | Yes |
Theorem 25.7.21 Let \( A : X \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) be monotone and satisfies the following conditions:\n\n1. If \( {\lambda }_{n} \rightarrow \lambda ,{\lambda }_{n} \in \left\lbrack {0,1}\right\rbrack \) and \( {z}_{n} \in A\left( {u + {\lambda }_{n}\left( {v - u}\right) }\right) \), th... | Proof: Let \( \widehat{A} \) be a monotone extension of \( A \) . Let \( \left\lbrack {\widehat{u},\widehat{w}}\right\rbrack \) be such that \( \widehat{w} \in \widehat{A}\left( \widehat{u}\right) \) . Now also by assumption, \( A\left( x\right) \) is not just convex but also closed.\n\nIf \( \left\lbrack {\widehat{u},... | Yes |
Proposition 25.7.23 Let \( A : D\left( A\right) \subseteq X \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) be maximal monotone and suppose \( 0 \in \operatorname{int}\left( {D\left( A\right) }\right) \) . Then \( A \) is quasi-bounded. | Proof: From local boundedness, Theorem 25.7.20, there exists \( \delta, C > 0 \) such that\n\n\[ \sup \left\{ {\begin{Vmatrix}{x}^{ * }\end{Vmatrix} : {x}^{ * } \in A\left( x\right) \text{ for }\parallel x\parallel \leq \delta }\right\} < C \]\n\nNow suppose that \( \parallel x\parallel ,\left| \left\langle {{x}^{ * },... | Yes |
Proposition 25.7.26 Let \( X \) and \( Y \) be reflexive Banach spaces with \( Y \subseteq {X}^{\prime } \) . Let \( 1 < p < \infty \) and let \( q = \frac{p}{p - 1} = {p}^{\prime } \) so \( \frac{1}{p} + \frac{1}{q} = 1 \) . Let \( F : \left\lbrack {0, T}\right\rbrack \times X \rightarrow \mathcal{P}\left( Y\right) \)... | Proof: First note that \( {F}_{\tau }x \) is convex and nonempty. To see this, say \( z,\widehat{z} \) are in \( {F}_{\tau }x \) . and let \( y,\widehat{y} \) be the corresponding functions. Then for \( \lambda \in \left\lbrack {0,1}\right\rbrack \)\n\n\[ \n{\lambda z} + \left( {1 - \lambda }\right) \widehat{z} = \frac... | Yes |
Lemma 25.7.28 Let \( A \) be maximal monotone. Then for each \( \lambda > 0 \) , \[ x \rightarrow {\lambda F}\left( {x - {x}_{0}}\right) + {Ax} \] is onto. | Proof: Let \( \widehat{A}\left( x\right) \equiv A\left( {{x}_{0} + x}\right) \) so as earlier, \( \widehat{A} \) is maximal monotone. Then let \( {y}^{ * } \in {X}^{\prime } \) . Then there exists \( y \) such that \( \widehat{A}\left( y\right) + {\lambda F}\left( y\right) \ni {y}^{ * } \) . Now define \( x \equiv y + ... | Yes |
Corollary 25.7.32 Suppose \( A : D\left( A\right) \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) is maximal monotone and coercive. Then \( A \) is onto. | Proof: From Theorem 25.7.31 it suffics to show that \( {A}^{-1} \) is locally bounded at \( {y}^{ * } \in \overline{A\left( {D\left( A\right) }\right) } \) . The case of an interior point follows from Theorem 25.7.20. Assume then that \( {y}^{ * } \) is a limit point of \( A\left( {D\left( A\right) }\right) \) . Of cou... | Yes |
Lemma 25.7.33 Suppose \( \lim \mathop{\sup }\limits_{{n, n \rightarrow \infty }}{a}_{mn} \leq 0 \) . Then\n\n\[ \lim \mathop{\sup }\limits_{{m \rightarrow \infty }}\left( {\lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{a}_{mn}}\right) \leq 0. \] | Proof: There exists \( N \) such that if both \( m, n \geq N,{a}_{mn} \leq \varepsilon \) . Then\n\n\[ \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}{a}_{mn} = \lim \mathop{\sup }\limits_{{n \rightarrow \infty, n > N}}{a}_{mn} \leq \varepsilon \]\n\nThus also\n\n\[ \lim \mathop{\sup }\limits_{{m \rightarrow \inft... | Yes |
Lemma 25.7.34 Let \( A : D\left( A\right) \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) be maximal monotone and let \( {v}_{n} \in A{u}_{n} \) and\n\n\[ \n{u}_{n} \rightarrow u,{v}_{n} \rightarrow v\text{weakly.} \n\] \n\nAlso suppose that \n\n\[ \n\lim \mathop{\sup }\limits_{{m, n \rightarrow \infty }}\left\la... | Proof: By monotonicity, \n\n\[ \n\mathop{\lim }\limits_{{m, n \rightarrow \infty }}\left\langle {{v}_{n} - {v}_{m},{u}_{n} - {u}_{m}}\right\rangle = 0 \n\] \n\nSuppose then that \( \left\langle {{v}_{n},{u}_{n}}\right\rangle \) fails to converge to \( \langle v, u\rangle \) . Then there is a subsequence, still denoted ... | Yes |
Theorem 25.7.36 The following hold. Here \( X \) is a reflexive Banach space with strictly convex norm. \( A : D\left( A\right) \rightarrow \mathcal{P}\left( {X}^{\prime }\right) \) is maximal monotone. Then\n\n1. \( {J}_{\lambda } \) and \( {A}_{\lambda } \) are bounded single valued operators defined on \( X \) . Bou... | Proof: 1.) It is clear that these are single valued operators. What about the assertion that they are bounded? Let \( {y}^{ * } \in A{x}_{\lambda } \) such that the inclusion defining \( {x}_{\lambda } \) becomes an equality. Thus\n\n\[ F\left( {{x}_{\lambda } - x}\right) + {\lambda }^{p - 1}{y}^{ * } = 0 \]\n\nThen le... | Yes |
Corollary 25.7.37 Let \( A \) be maximal monotone. \( A : X \rightarrow {X}^{\prime } \) where \( X \) is a strictly convex reflexive Banach space. Then \( \overline{D\left( A\right) } \) is convex. | Proof: It is known that \( {J}_{\lambda } : X \rightarrow D\left( A\right) \) for any \( \lambda \) . Also, if \( x \in \overline{\operatorname{conv}\left( {D\left( A\right) }\right) } \) , then it was shown that \( {J}_{\lambda }x \rightarrow x \) . Clearly\n\n\[ \overline{\text{conv}\left( {D\left( A\right) }\right) ... | Yes |
Lemma 25.7.39 Let \( 0 \in D\left( A\right) \) and let \( A \) be maximal monotone and let \( B : X \rightarrow \) \( {X}^{\prime } \) be monotone hemicontinuous, bounded, and coercive. Then \( B + A \) is also maximal monotone. Also \( B + A \) is onto. | Proof: By Theorem 25.7.9, there exists \( x \in \overline{D\left( A\right) } \) such that for all \( \left\lbrack {u,{u}^{ * }}\right\rbrack \in \) \( \mathcal{G}\left( A\right) \)\n\n\[ \left\langle {{Bx} + {Fx} - {y}^{ * } + {u}^{ * }, u - x}\right\rangle \geq 0 \]\n\nHence for all \( \left\lbrack {u,{u}^{ * }}\right... | Yes |
Corollary 25.7.40 Suppose instead of \( 0 \in D\left( A\right) \), it is known that \( {x}_{0} \in D\left( A\right) \) and\n\n\[\n\mathop{\lim }\limits_{{\parallel x\parallel \rightarrow \infty }}\frac{\left\langle B\left( {x}_{0} + x\right), x\right\rangle }{\parallel x\parallel } = \infty\n\]\n\nThen if \( B \) is mo... | Proof: Let \( \widehat{A}\left( x\right) \equiv A\left( {{x}_{0} + x}\right) \) so in fact \( 0 \in D\left( \widehat{A}\right) \) . Then letting \( \widehat{B} \) be defined similarly, it follows from the above lemma that if \( {y}^{ * } \in {X}^{\prime } \), there exists \( x \) such that\n\n\[\n{y}^{ * } \in \widehat... | Yes |
Theorem 25.7.42 Suppose \( A, B \) are maximal monotone and the interior of \( D\left( A\right) \) has nonempty intersection with \( D\left( B\right) \) . Then \( A + B \) is maximal monotone. | Proof: Let \( {x}_{0} \) be on the interior of \( D\left( A\right) \) and also in \( D\left( B\right) \) . Let \( \widehat{A}\left( x\right) = \) \( A\left( {{x}_{0} + x}\right) - {x}_{0}^{ * } \) where \( {x}_{0}^{ * } \in A\left( {x}_{0}\right) \) . Thus \( 0 \in D\left( \widehat{A}\right) \) and \( 0 \in \widehat{A}... | Yes |
Theorem 25.7.43 Suppose \( A, B \) are maximal monotone operators. Then for each \( {x}^{ * } \in {X}^{\prime } \), there exists a solution \( {x}_{\lambda } \) to\n\n\[ \n{x}^{ * } \in F{x}_{\lambda } + {B}_{\lambda }{x}_{\lambda } + A{x}_{\lambda },\lambda > 0 \n\] | Proof: The existence of a solution to the inclusion 25.7.64 comes from the above discussion. The last claim follows from almost a repeat of the last part of the proof of the above theorem. Since \( \left\{ {{B}_{\lambda }{x}_{\lambda }}\right\} \) is given to be bounded for \( \lambda \in \left( {0,\delta }\right) \), ... | Yes |
Lemma 25.7.46 Let \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) . Then \( \phi \) is lower semicontinuous if and only if \( {\phi }^{-1}(\left( {a,\infty \rbrack }\right) \) is open for any \( a \in \mathbb{R} \) . | Proof: Suppose first that \( \operatorname{epi}\left( \phi \right) \) is closed. Consider \( x \in {\phi }^{-1}(\left( {a,\infty \rbrack }\right) \) . Thus \( \phi \left( x\right) > a \) . Thus \( \left( {x, a}\right) \in \operatorname{epi}{\left( \phi \right) }^{C} \) because \( a < \phi \left( x\right) \) . Since \( ... | Yes |
Theorem 25.7.49 Suppose \( X \) is a reflexive Banach space and suppose \( \phi : X \rightarrow \) \( ( - \infty ,\infty \rbrack \) is convex, proper, l.s.c., and for all \( {y}^{ * } \in {X}^{\prime }, x \rightarrow \phi \left( x\right) - \left\langle {{y}^{ * }, x}\right\rangle \) is coercive,\n\n\[ \mathop{\lim }\li... | 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\[ \lambda \equiv \inf \left\{ {\phi \left( x\right) - \left\langle {{y}^{ * }, x}\right\rangle : x \in X}\right\} \]\n\nand let \( \left\{ {x}_{n}\right\} ... | Yes |
Corollary 25.7.52 Let \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) be convex, proper, and lower semicontinuous. Here \( X \) is a Banach space. Then \( \partial \phi \) is maximal monotone. | Proof: Let \( \psi \left( x\right) = \frac{1}{2}\parallel x{\parallel }^{2} \) . There exists \( {x}^{ * } \) and some number \( b \) such that \( \phi \left( x\right) \geq \) \( b + \left\langle {{x}^{ * }, x}\right\rangle \) . Therefore, \( \psi + \phi \) is convex, lower semicontinuous, and bounded. It follows \( \p... | Yes |
Proposition 25.7.53 Let \( \phi : X \rightarrow ( - \infty ,\infty \rbrack \) be convex proper and lower semicontinuous. Then \( D\left( {\partial \phi }\right) \) is dense in \( D\left( \phi \right) \) and so \( \overline{D\left( {\partial \phi }\right) } = \overline{D\left( \phi \right) } \) . | Proof: Let \( {x}_{\lambda } \) be the solution to \( 0 \in F\left( {{x}_{\lambda } - x}\right) + \lambda \partial \phi \left( {x}_{\lambda }\right) \) . Here \( x \in D\left( \phi \right) \) . Say \( {u}_{\lambda }^{ * } \in \partial \phi \left( {x}_{\lambda }\right) \) such that the inclusion becomes an equality. The... | Yes |
Theorem 25.7.54 Let \( \phi \) be a convex lower semicontinuous proper function defined on \( X \) . Define \( A \equiv \partial \phi ,{A}_{\lambda } = {\left( \partial \phi \right) }_{\lambda } \)\n\n\[{\phi }_{\lambda }\left( x\right) \equiv \mathop{\min }\limits_{{y \in X}}\left( {\frac{1}{2\lambda }\parallel x - y{... | 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\[\frac{1}{2\lambda }\parallel x - y{\parallel }^{2} + \phi \left( y\right) \... | Yes |
Proposition 25.8.2 Let \( \tau : V \times {V}^{\prime } \rightarrow {V}^{\prime } \times V \) be given by \( \tau \left( {a, b}\right) \equiv \left( {-b, a}\right) \) . Also for \( S \subseteq X \) a reflexive Banach space,\n\n\[ \n{S}^{ \bot } \equiv \left\{ {{z}^{ * } \in {X}^{\prime } : \left\langle {{z}^{ * }, s}\r... | Proof: Let \( \left( {x,{L}^{ * }x}\right) \in \mathcal{G}\left( {L}^{ * }\right) \) . This means that\n\n\[ \n\left| {\langle {Ly}, x\rangle }\right| \leq C\parallel y\parallel \text{ for all }y \in D\left( L\right) \n\]\n\nand \( \langle {Ly}, x\rangle = \left\langle {{L}^{ * }x, y}\right\rangle \) for all \( y \in D... | Yes |
Lemma 25.8.7 Suppose \( X \) is the Banach space\n\n\[ \nX = D\left( L\right) ,\parallel u{\parallel }_{X} \equiv \parallel u{\parallel }_{V} + \parallel {Lu}{\parallel }_{{V}^{\prime }}\n\]\n\nwhere \( L \) is as described in the above definition. Also assume that \( A \) is bounded. Then if \( A \) is \( L \) pseudom... | Proof: Is \( A \) bounded? Of course, because the norm of \( X \) is stronger than the norm on \( V \) . Is \( {Au} \) convex and closed? This also follows because \( X \subseteq V \) . It is clear that \( {Au} \) is convex. If \( \left\{ {z}_{n}\right\} \subseteq {Au} \) and \( {z}_{n} \rightarrow z \) in \( {X}^{\pri... | Yes |
Theorem 25.8.8 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 \( A : V \rightarrow \mathcal{P}\left( {V}^{\prime }\right) \) be \( L \) pseudomonotone, bounded,... | Proof: Let \( F \) be the duality map for \( p = 2 \) . Consider the Banach space \( X \) given by\n\n\[ \nX = D\left( L\right) ,\parallel u{\parallel }_{X} \equiv \parallel u{\parallel }_{V} + \parallel {Lu}{\parallel }_{{V}^{\prime }}\n\]\n\nThis is isometric with the graph of \( L \) with the graph norm and so \( X ... | Yes |
Lemma 25.8.11 Suppose \( A \) is a set valued operator, \( A : X \rightarrow \mathcal{P}\left( X\right) \) and \( {u}_{n}^{ * } \in \) \( A{u}_{n} \) . Suppose also that \( {u}_{n} \rightarrow u \) weakly and \( {u}_{n}^{ * } \rightarrow {u}^{ * } \) weakly. Suppose also that\n\n\[ \lim \mathop{\sup }\limits_{{m, n \ri... | Proof: Let \( \alpha \equiv \lim \mathop{\sup }\limits_{{n \rightarrow \infty }}\left\langle {{u}_{n}^{ * },{u}_{n}}\right\rangle \) . It is a finite number because these sequences are bounded. Then using the weak convergence,\n\n\[ 0 \geq \lim \mathop{\sup }\limits_{{m \rightarrow \infty }}\left( {\lim \mathop{\sup }\... | Yes |
Lemma 25.8.12 Let \( A \) be pseudomonotone, bounded and coercive and let \( 0 \in \) \( D\left( B\right) \) . Then if \( {y}^{ * } \in {X}^{\prime } \), there exists a solution \( {x}_{\lambda } \) to\n\n\[ \n{y}^{ * } \in L{x}_{\lambda } + A{x}_{\lambda } + {B}_{\lambda }{x}_{\lambda }\n\] | Proof: From the inequality \( {25.8.94}, A + {B}_{\lambda } \) is coercive. It is also bounded and pseudomonotone. It is pseudomonotone from Theorem 25.7.27. Therefore, there exists a solution \( {x}_{\lambda } \) by Theorem 25.8.8. \( \blacksquare \) | No |
Lemma 25.8.14 In the above situation, suppose the maximal monotone operator \( B \) is quasi-bounded and \( \left| \left\langle {{B}_{\lambda }{x}_{\lambda },{x}_{\lambda }}\right\rangle \right| \leq M \) . Then the \( {B}_{\lambda }{x}_{\lambda } \) are bounded. Also\n\n\[ \n{\begin{Vmatrix}{J}_{\lambda }{x}_{\lambda ... | Proof: Now \( {B}_{\lambda }{x}_{\lambda } \in B{J}_{\lambda }{x}_{\lambda } \)\n\n\[ \n- \left| {B\left( 0\right) }\right| \begin{Vmatrix}{x}_{\lambda }\end{Vmatrix} \leq \left\langle {{B}_{\lambda }{x}_{\lambda },{x}_{\lambda }}\right\rangle = \left\langle {{B}_{\lambda }{x}_{\lambda },{J}_{\lambda }{x}_{\lambda }}\r... | Yes |
Theorem 26.1.2 If \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \), then for \( \alpha > 0 \), \[ \bar{m}\left( \left\lbrack {{Mf} > \alpha }\right\rbrack \right) \leq \frac{{5}^{n}}{\alpha }\parallel f{\parallel }_{1} \] (Here and elsewhere, \( \left\lbrack {{Mf} > \alpha }\right\rbrack \equiv \left\{ {\mathbf{x} \in... | Proof: Let \( S \equiv \left\lbrack {{Mf} > \alpha }\right\rbrack \) . For \( \mathbf{x} \in S \), choose \( {r}_{\mathbf{x}} > 0 \) with \[ \frac{1}{m\left( {B\left( {\mathbf{x},{r}_{\mathbf{x}}}\right) }\right) }{\int }_{B\left( {\mathbf{x},{r}_{\mathbf{x}}}\right) }\left| f\right| {dm} > \alpha . \] The \( {r}_{\mat... | Yes |
Lemma 26.1.3 Suppose \( g \) is a continuous function. Then for all \( \mathbf{x} \) , \[ \mathop{\lim }\limits_{{r \rightarrow 0}}\frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }g\left( \mathbf{y}\right) {dy} = g\left( \mathbf{x}\right) . \] | Proof: Note that \[ g\left( \mathbf{x}\right) = \frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }g\left( \mathbf{x}\right) {dy} \] and so \[ \left| {g\left( \mathbf{x}\right) - \frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\... | Yes |
Corollary 26.1.6 (Fundamental Theorem of Calculus) Let \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{k}\right) \) . Then there exists a set of measure \( 0, N \), such that if \( \mathbf{x} \notin N \), then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow 0}}\frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{... | Proof: Consider \( B\left( {\mathbf{0}, n}\right) \) where \( n \) is a positive integer. Then \( {f}_{n} \equiv f{\mathcal{X}}_{B\left( {\mathbf{0}, n}\right) } \in \) \( {L}^{1}\left( {\mathbb{R}}^{k}\right) \) and so there exists a set of measure \( 0,{N}_{n} \) such that if \( \mathbf{x} \in B\left( {\mathbf{0}, n}... | Yes |
Corollary 26.1.7 If \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \), then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow 0}}\frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }f\left( \mathbf{y}\right) {dy} = f\left( \mathbf{x}\right) \;\text{ a.e. }\mathbf{x}. \]\n\... | Proof:\n\n\[ \left| {\frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }f\left( \mathbf{y}\right) {dy} - f\left( \mathbf{x}\right) }\right| \]\n\n\[ \leq \frac{1}{m\left( {B\left( {\mathbf{x}, r}\right) }\right) }{\int }_{B\left( {\mathbf{x}, r}\right) }\left| {f\left( \... | Yes |
Corollary 26.1.9 Let \( f \in {L}^{1}\left( \mathbb{R}\right) \) and let\n\n\[ F\left( x\right) = {\int }_{-\infty }^{x}f\left( t\right) {dt} \]\n\nThen for a.e. \( x,{F}^{\prime }\left( x\right) = f\left( x\right) \) . | Proof: For \( h > 0 \)\n\n\[ \frac{1}{h}{\int }_{x}^{x + h}\left| {f\left( y\right) - f\left( x\right) }\right| {dy} \leq 2\left( \frac{1}{2h}\right) {\int }_{x - h}^{x + h}\left| {f\left( y\right) - f\left( x\right) }\right| {dy} \]\n\nBy Theorem 26.1.5, this converges to 0 a.e. Similarly\n\n\[ \frac{1}{h}{\int }_{x -... | Yes |
Lemma 26.2.3 Let \( f \) be an absolutely continuous function defined on \( \left\lbrack {a, b}\right\rbrack \) and let \( V \) be its total variation function as described above. Then \( V \) is an increasing bounded function. Also if \( P \) and \( Q \) are two partitions of \( \left\lbrack {x, y}\right\rbrack \) wit... | Proof: The claim that \( V \) is increasing is obvious as is the next claim about \( P \subseteq Q \) leading to \( {V}_{P}\left\lbrack {x, y}\right\rbrack \leq {V}_{Q}\left\lbrack {x, y}\right\rbrack \) . To verify this, simply add in one point at a time and verify that from the triangle inequality, the sum involved g... | Yes |
Theorem 26.2.5 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) be absolutely continuous if and only if \( {f}^{\prime }\left( x\right) \) exists a.e., \( {f}^{\prime } \in {L}^{1}\left( \left\lbrack {a, b}\right\rbrack \right) \) and\n\n\[ f\left( x\right) = f\left( a\right) + {\int }_{a}^{x}{f}^{... | Proof: Suppose first that \( f \) is absolutely continuous. By Lemma 26.2.3 the total variation function, \( V \) is absolutely continuous and \( f\left( x\right) = V\left( x\right) - \left( {V\left( x\right) - f\left( x\right) }\right) \) where both \( V \) and \( V - f \) are increasing and absolutely continuous. By ... | Yes |
Corollary 26.2.6 Suppose \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is Lipschitz continuous,\n\n\[ \left| {f\left( x\right) - f\left( y\right) }\right| \leq K\left| {x - y}\right| .\n\]\n\nThen \( {f}^{\prime }\left( x\right) \) exists a.e. and\n\n\[ f\left( x\right) = f\left( a\right) + {\int }_... | Proof: It is easy to see that \( f \) is absolutely continuous. Therefore, Theorem 26.2.5 applies. | No |
Lemma 26.3.1 Suppose \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) and suppose\n\n\[ \n\int {f\phi dx} = 0 \n\]\n\nfor all \( \phi \in {C}_{c}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Then \( f\left( \mathbf{x}\right) = 0 \) a.e. \( \mathbf{x} \) . | Proof: Without loss of generality \( f \) is real-valued. Let\n\n\[ \nE \equiv \{ \mathbf{x} : f\left( \mathbf{x}\right) > \epsilon \} \n\]\n\nand let\n\n\[ \n{E}_{m} \equiv E \cap B\left( {0, m}\right) . \n\]\n\nWe show that \( m\left( {E}_{m}\right) = 0 \) . If not, there exists an open set, \( V \), and a compact se... | Yes |
Theorem 26.3.3 Let \( \Omega = \left( {a, b}\right) \) and suppose that \( f \) and \( {Df} \) are both in \( {L}^{1}\left( {a, b}\right) \) . Then \( f \) is equal to a continuous function a.e., still denoted by \( f \) and\n\n\[ f\left( x\right) = f\left( a\right) + {\int }_{a}^{x}{Df}\left( t\right) {dt}. \]\n | The proof of Theorem 26.3.3 depends on the following lemma.\n\nLemma 26.3.4 Let \( T \n | No |
Lemma 26.3.4 Let \( T \in {\mathcal{D}}^{ * }\left( {a, b}\right) \) and suppose \( {DT} = 0 \) . Then there exists a constant \( C \) such that \[ T\left( \phi \right) = {\int }_{a}^{b}{C\phi dx} \] | 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 \[ {\phi }_{0} \in {C}_{c}^{\infty }\left( {a, b}\right) ,{\int }_{a}^{b}{\phi }_{0}\left( x\right) {dx} = 1, \] and let \[ {\psi }_{\phi }\left( x\right) = {\int }_{a}^{x}\l... | Yes |
Corollary 26.4.3 Suppose \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is Lipschitz continuous,\n\n\[ \left| {f\left( x\right) - f\left( y\right) }\right| \leq K\left| {x - y}\right| .\n\]\n\nThen \( {f}^{\prime }\left( x\right) \) exists a.e. and\n\n\[ f\left( x\right) = f\left( a\right) + {\int }_... | Proof: If \( f \) were increasing, this would follow from the above lemma. Let \( g\left( x\right) = {2Kx} - f\left( x\right) \) . Then \( g \) is Lipschitz with a different Lipschitz constant and also if \( x < y \) ,\n\n\[ g\left( y\right) - g\left( x\right) = {2Ky} - f\left( y\right) - \left( {{2Kx} - f\left( x\righ... | Yes |
Proposition 26.5.1 Let \( \\left\\{ {f}_{n}\\right\\} \) be measurable with values in a complete normed vector space. Let \( A \\equiv \\left\\{ {\\omega : \\left\\{ {{f}_{n}\\left( \\omega \\right) }\\right\\} }\\right. \) converges \\} . Then \( A \) is measurable. | Proof: The set \( A \) is the same as the set on which \( \\left\\{ {{f}_{n}\\left( \\omega \\right) }\\right\\} \) is a Cauchy sequence.\n\nThis set is\n\[ \n{ \\cap }_{n = 1}^{\\infty }{ \\cup }_{m = 1}^{\\infty }{ \\cap }_{p, q > m}\\left\\lbrack {\\begin{Vmatrix}{{f}_{p}\\left( \\omega \\right) - {f}_{q}\\left( \\o... | Yes |
Lemma 26.5.2 Let \( u : {\mathbb{R}}^{p} \rightarrow \mathbb{R} \) be Lipschitz with Lipschitz constant \( K \) . Let \( {u}_{n} \equiv \) \( u * {\phi }_{n} \) where \( \left\{ {\phi }_{n}\right\} \) is a mollifier,\n\n\[ \n{\phi }_{n}\left( \mathbf{y}\right) \equiv {n}^{p}\phi \left( {n\mathbf{y}}\right) ,\int \phi \... | Proof: To get the existence of the gradient satisfying the condition given in 26.5.11, apply the corollary to each variable. Now\n\n\[ \n\frac{{u}_{n}\left( {\mathbf{x} + h{\mathbf{e}}_{i}}\right) - {u}_{n}\left( \mathbf{x}\right) }{h} = {\int }_{{\mathbb{R}}^{p}}\left( \frac{u\left( {\mathbf{x} + h{\mathbf{e}}_{i} - \... | Yes |
Corollary 26.5.4 Let \( u \) be Lipschitz on \( {\mathbb{R}}^{p} \) with constant \( K \) . Then there is a constant \( C \) depending only on \( p \) such that\n\n\[ \left| {u\left( \mathbf{x}\right) - u\left( \mathbf{y}\right) }\right| \leq C{\left( {\int }_{B\left( {\mathbf{x},2\left| {\mathbf{x} - \mathbf{y}}\right... | Proof: Let \( {u}_{n} = u * {\phi }_{n} \) where \( \left\{ {\phi }_{n}\right\} \) is a mollifier as in Lemma 26.5.2. Then from Lemma 26.5.3, there is a constant depending only on \( p \) such that\n\n\[ \left| {{u}_{n}\left( \mathbf{x}\right) - {u}_{n}\left( \mathbf{y}\right) }\right| \leq C{\left( {\int }_{B\left( {\... | Yes |
Corollary 26.5.6 Let \( u \) be Lipschitz. Then for any \( \mathbf{x} \) and \( \mathbf{v} \in {S}^{p - 1} \smallsetminus {B}_{\mathbf{x}} \) where \( \sigma \left( {B}_{\mathbf{x}}\right) = 0 \), it follows that for all \( t \) , | \[ u\left( {\mathbf{x} + t\mathbf{v}}\right) - u\left( \mathbf{x}\right) = {\int }_{0}^{t}{D}_{\mathbf{v}}u\left( {\mathbf{x} + s\mathbf{v}}\right) {ds} = {\int }_{0}^{t}\nabla u\left( {\mathbf{x} + s\mathbf{v}}\right) \cdot \mathbf{v}{ds} \] | Yes |
Theorem 26.5.7 If \( \mathbf{h} : \Omega \rightarrow {\mathbb{R}}^{m} \) is Lipschitz, then there exists \( \overline{\mathbf{h}} : {\mathbb{R}}^{p} \rightarrow {\mathbb{R}}^{m} \) 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(26.5.16)\n\nDefine\n\n\[ \bar{... | Yes |
Corollary 26.6.2 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 |
Lemma 26.6.3 Let \( u \) be a Lipschitz continuous function which vanishes outside some compact set. Then there exists a unique \( {u}_{, i} \in {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \) such that \[ \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{u\left( {\cdot + h}\right) - u\left( \cdot \right) }{h} = {u}_{, i}\... | Proof: By the Lipschitz condition, the above difference quotient is bounded in \( {L}^{\infty } \) by \( K \) the Lipschitz constant of \( u \) . It follows from the Banach Aloglu theorem and Corollary 17.5.6 on Page 490 that there exists a subsequence \( {h}_{k} \rightarrow 0 \) and \( g \in {L}^{\infty }\left( {\math... | Yes |
Lemma 26.6.4 Let \( u \) be a Lipschitz continuous function which vanishes outside a compact set and let \( {u}_{, i} \) be described above. For \( {\phi }_{\varepsilon } \) a mollifier and \( {u}_{\varepsilon } \equiv u * {\phi }_{\varepsilon } \), \[ {u}_{\varepsilon, i} = {u}_{, i} * {\phi }_{\varepsilon } \] where ... | Proof: This follows from a computation and Lemma 26.6.3. \[ {u}_{\varepsilon, i}\left( \mathbf{x}\right) \equiv \mathop{\lim }\limits_{{h \rightarrow 0}}\int \frac{u\left( {\mathbf{x} - \mathbf{y} + h{\mathbf{e}}_{i}}\right) - u\left( {\mathbf{x} - \mathbf{y}}\right) }{h}{\phi }_{\varepsilon }\left( \mathbf{y}\right) {... | Yes |
Theorem 26.6.6 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\( \left( {26.6.20}\right) \)\n... | Yes |
Theorem 26.6.7 Let \( \mathbf{h} : \Omega \rightarrow {\mathbb{R}}^{m} \) be Lipschitz on \( \Omega \) where \( \Omega \) is some nonempty measurable set in \( {\mathbb{R}}^{n} \) . Then \( D\mathbf{h}\left( \mathbf{x}\right) \) exists for a.e. \( \mathbf{x} \in \Omega \) . If \( \Omega = {\mathbb{R}}^{n} \), then for ... | Proof: The last two claims follow from the above argument applied to the components of \( \mathbf{h} \) . By Theorem 26.6.6 the function can be extended to a Lipschitz function defined on all of \( {\mathbb{R}}^{n} \), still denoted as \( \mathbf{h} \) . Let \( {\Omega }_{r} \equiv \Omega \cap B\left( {\mathbf{0}, r}\r... | Yes |
Lemma 26.6.8 Let \( u \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) . There exists \( {u}_{, i} \in {L}^{p}\left( {\mathbb{R}}^{n}\right) \) such that\n\n\[ \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{u\left( {\cdot + h{\mathbf{e}}_{i}}\right) - u\left( \cdot \right) }{h} = {u}_{, i}\text{ weakly in }{L}^{p}\left( {\... | Proof: If the weak limit exists, then the difference quotients must be bounded. This follows from the uniform boundedness theorem, Theorem 17.1.8. Here is why. Denote the difference quotient by \( {D}_{h} \) to save space. Weak convergence requires \( \int {D}_{h}f \rightarrow \int {u}_{.i}f \) for all \( f \in {L}^{{p... | Yes |
Theorem 26.6.10 Let \( \mathbf{h} \) be in \( {L}^{p}\left( {{\mathbb{R}}^{n};{\mathbb{R}}^{m}}\right), p > n \), and suppose it has weak derivatives \( {\mathbf{h}}_{, i} \in {L}^{p}\left( {{\mathbb{R}}^{n};{\mathbb{R}}^{m}}\right) \) for \( i = 1,\cdots, n \) . Then \( D\mathbf{h}\left( \mathbf{x}\right) \) exists a.... | Proof: As before,\n\n\[{\left( \mathbf{h} * {\phi }_{\varepsilon }\right) }_{, i}\left( \mathbf{x}\right) \equiv \mathop{\lim }\limits_{{h \rightarrow 0}}\int \frac{\mathbf{h}\left( {\mathbf{x} + h{\mathbf{e}}_{i} - \mathbf{y}}\right) - \mathbf{h}\left( {\mathbf{x} - \mathbf{y}}\right) }{h}{\phi }_{\varepsilon }\left( ... | Yes |
Corollary 26.7.6 Let \( \mu \) be a complex Borel measure on \( {\mathbb{R}}^{n} \). Then \( \frac{d\mu }{dm}\left( \mathbf{x}\right) \) exists a.e. | Proof: Letting \( {\nu }_{i} \) be defined in Definition 26.7.5. By Theorem 26.7.4, for \( m \) a.e. \( \mathbf{x},\frac{d{\nu }_{i}}{dm}\left( \mathbf{x}\right) \) exists. This proves the corollary because \( \mu \) is just a finite sum of these \( {\nu }_{i} \). | Yes |
Theorem 26.7.7 Let \( \mu \) be a Radon measure on \( {\mathbb{R}}^{n} \) and suppose there exists a \( \mu \) measurable set, \( N \) such that for all Borel sets, \( E,\mu \left( E\right) = \mu \left( {E \cap N}\right) \) where \( \bar{m}\left( N\right) = 0 \) . Then\n\n\[ \frac{d\mu }{dm}\left( \mathbf{x}\right) = 0... | Proof: For \( k \in \mathbb{N} \), let\n\n\[ {B}_{k}\left( M\right) \equiv \left\{ {\mathbf{x} \in {N}^{C} : \lim \mathop{\sup }\limits_{{r \rightarrow 0 + }}\frac{\mu \left( {B\left( {\mathbf{x}, r}\right) }\right) }{m\left( {B\left( {\mathbf{x}, r}\right) }\right) } > \frac{1}{k}}\right\} \cap B\left( {\mathbf{0}, M}... | Yes |
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