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Lemma 7.31. Let \( n \in \mathbb{N}, n \geq 2 \), and suppose \( q \in {\mathbb{N}}_{0} \) is such that \( n + q \) is an even number. Then the function \[ {E}_{q}\left( x\right) \mathrel{\text{:=}} - \frac{1}{{\left( 2\pi \mathrm{i}\right) }^{n}q!}{\int }_{{S}^{n - 1}}{\left( x \cdot \xi \right) }^{q}\log \left( \frac... | Proof. Fix \( n \) and \( q \) as in the hypotheses of the lemma. Fix \( x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) and use the fact that \( \log \left( \frac{x \cdot \xi }{\mathrm{i}}\right) = \ln \left| {x \cdot \xi }\right| - \mathrm{i}\frac{\pi }{2}\operatorname{sgn}\left( {x \cdot \xi }\right) \) for every \(... | Yes |
Proposition 7.32. Let \( k \in \left( {0,\infty }\right) \) and let \( n \in \mathbb{N} \) be such that \( n \geq 2 \) . Then for each \( x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) one has\n\n\[ \n{\Phi }_{k}\left( x\right) = \left\{ \begin{array}{ll} {\Psi }_{\left( {n - 2}\right) /2}\left( {k\left| x\right| }\ri... | Proof. Assume for now that \( n \geq 3 \) . Then, starting with the expression in (7.6.11), at each point \( x \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) we may write\n\n\[ \n{\Phi }_{k}\left( x\right) = \frac{{c}_{n}{2}^{\left( {n - 2}\right) /2}\Gamma \left( \frac{n - 2}{2}\right) }{\mathrm{i}\pi } \cdot \frac{{H}... | Yes |
Theorem 7.33. Suppose \( n \in \mathbb{N} \) and fix \( k \in \left( {0,\infty }\right) \) . Then the function \( {\Phi }_{k} \) defined in (7.6.11)-(7.6.12) satisfies \( {\Phi }_{k} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \cap {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) and is a fundamental soluti... | Proof. Consider first the case \( n \geq 2 \) . The fact that \( {\Phi }_{k} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) for \( n \geq 2 \) is a consequence of (7.6.18) in Proposition 7.32, formula (7.6.15), and the membership \( {E}_{\Delta } \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) . Also, item (9) i... | Yes |
Proposition 7.34. Fix \( k \in \left( {0,\infty }\right) \) along with \( n \in \mathbb{N}, n \geq 2 \), and set \[ {b}_{n, k} \mathrel{\text{:=}} \frac{{k}^{\left( {n - 3}\right) /2}}{4\mathrm{i}{\left( 2\pi \right) }^{\left( {n - 2}\right) /2}}{\left( \frac{2}{\pi }\right) }^{1/2}{\mathrm{e}}^{-\mathrm{i}\pi \left( {... | Proof. Fix \( \alpha \in {\mathbb{N}}_{0}^{n} \) arbitrary. As a first step we shall prove that \[ \left( {{\partial }^{\alpha }{\Phi }_{k}}\right) \left( x\right) = {b}_{n, k}\frac{{\mathrm{e}}^{\mathrm{i}k\left| x\right| }}{{\left| x\right| }^{\left( {n - 1}\right) /2}}{\left( \mathrm{i}k\widehat{x}\right) }^{\alpha ... | Yes |
Theorem 7.40. Let \( n \in \mathbb{N}, n \geq 2 \), and \( k \in \left( {0,\infty }\right) \) . For \( N \in \mathbb{N} \) recall the function \( {\Phi }_{k}^{\left( N\right) } \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) from (7.7.9), which is explicitly given by\n\n\[ \n{\Phi }_{k}^{\left( N\right) }\left( x\ri... | Proof. We shall reason by induction over \( N \) . The case \( N = 1 \) has been dealt with in Theorem 7.33 (recall that \( {\Phi }_{k}^{\left( 1\right) } = {\Phi }_{k} \) ). Suppose next that for some positive integer \( N \) the distribution \( {\Phi }_{k}^{\left( N\right) } \) is known to be a fundamental solution f... | Yes |
Theorem 7.43. Consider the functions \[ E\left( {{x}_{1},{x}_{2}}\right) \mathrel{\text{:=}} \frac{1}{\pi } \cdot \frac{1}{{x}_{1} - \mathrm{i}{x}_{2}},\;F\left( {{x}_{1},{x}_{2}}\right) \mathrel{\text{:=}} \frac{1}{\pi } \cdot \frac{1}{{x}_{1} + \mathrm{i}{x}_{2}}, \] defined for \( x = \left( {{x}_{1},{x}_{2}}\right)... | Moreover, \[ \left\{ {u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{2}\right) : \frac{\partial }{\partial z}u = \delta \text{ in }{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{2}\right) }\right\} = \left\{ {E + P : P\text{ polynomial in }{\mathbb{R}}^{2}\text{ satisfying }\frac{\partial }{\partial z}P = 0\text{ in }{... | Yes |
Proposition 7.44. Let \( u\left( z\right) \mathrel{\text{:=}} \frac{1}{z} \) for all \( z \in \mathbb{C} \smallsetminus \{ 0\} \) . Then \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{2}\right) \) and \[ \widehat{u}\left( \xi \right) = \frac{2\pi }{\mathrm{i}\xi }\;\text{ in }{\mathcal{S}}^{\prime }\left( {\mathb... | Proof. That \( u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{2}\right) \) follows by observing that \( u \) is locally integrable near the origin, while \( \left| {u\left( z\right) }\right| \) decays like \( {\left| z\right| }^{-1} \) for \( \left| z\right| > 1 \) . To prove (7.8.13), first note that, since \( \bar... | Yes |
Proposition 7.48. If \( \varphi \in \mathcal{S}\left( \mathbb{R}\right) \) is a complex-valued function, define the Cauchy operator \( {by} \n\n\[ \n\left( {\mathcal{C}\varphi }\right) \left( z\right) \mathrel{\text{:=}} \frac{1}{{2\pi }\mathrm{i}}{\int }_{\mathbb{R}}\frac{\varphi \left( x\right) }{x - z}\mathrm{\;d}x,... | Proof. Apply Corollary 4.81 to the function \( \Phi : {\mathbb{R}}^{2} \smallsetminus \{ \left( {0,0}\right) \} \rightarrow \mathbb{C} \) defined by \n\n\[ \n\Phi \left( {x, y}\right) \mathrel{\text{:=}} \frac{-1}{{2\pi }\mathrm{i}\left( {x + \mathrm{i}y}\right) }\text{ for all }\left( {x, y}\right) \in {\mathbb{R}}^{2... | Yes |
Proposition 7.51. Let \( \Omega \subseteq {\mathbb{R}}^{n} \) be an open set. Then the Dirac operator \( D \) satisfies\n\n\[ \n{D}^{2} = - \Delta \;\text{ in }\;{\mathcal{D}}^{\prime }\left( {\Omega, C{\ell }_{n}}\right) \n\]\n\nwhere \( \Delta \) is the Laplacian. | Proof. Pick an arbitrary \( u \in {\mathcal{D}}^{\prime }\left( {\Omega ,{\mathcal{C}}_{n}}\right) \), say \( u = \mathop{\sum }\limits_{{l = 0}}^{n}\mathop{\sum }\limits_{{\left| I\right| = l}}{}^{\prime }{u}_{I}{\mathbf{e}}_{I} \) with \( {u}_{I} \in {\mathcal{D}}^{\prime }\left( \Omega \right) \) for each \( I \) . ... | Yes |
Theorem 7.53. The Clifford algebra-valued function\n\n\[ E\\left( x\\right) \\mathrel{\\text{:=}} - \\frac{1}{{\\omega }_{n - 1}}\\frac{x}{{\\left| x\\right| }^{n}} \\in {L}_{loc}^{1}\\left( {{\\mathbb{R}}^{n}, C{\\ell }_{n}}\\right) \\cap {\\mathcal{S}}^{\\prime }\\left( {{\\mathbb{R}}^{n}, C{\\ell }_{n}}\\right) \]\n... | Proof. Let \( {E}_{\\Delta } \) be the fundamental solution for \( \\Delta \) as described in (7.1.12) for \( n \\geq 2 \) . From (7.9.13) and Remark 5.7, we may infer that \( - D{E}_{\\Delta } \) computed in \( {\\mathcal{D}}^{\\prime }\\left( {\\mathbb{R}}^{n}\\right) \) is a fundamental solution for the Dirac operat... | Yes |
For each \( {\mathcal{C}}_{n} \) -valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) define the Cauchy-Clifford operator\n\n\[ \n\left( {\mathcal{C}\varphi }\right) \left( x\right) \mathrel{\text{:=}} - \frac{1}{{\omega }_{n - 1}}{\int }_{{\mathbb{R}}^{n - 1}}\frac{\mathop{\sum }\limits_{{j... | Consider the Clifford algebra-value function \( \Phi : {\mathbb{R}}^{n} \smallsetminus \{ 0\} \rightarrow C{\ell }_{n} \) given by\n\n\[ \n\Phi \left( x\right) \mathrel{\text{:=}} - \frac{\mathop{\sum }\limits_{{j = 1}}^{n}{x}_{j}{\mathbf{e}}_{j}}{{\omega }_{n - 1}{\left| x\right| }^{n}} \odot {\mathbf{e}}_{n}\;\text{ ... | Yes |
Theorem 7.57. Let \( k \in \left( {0,\infty }\right) \) and suppose \( n \in \mathbb{N}, n \geq 2 \) . Then the Clifford algebra-valued function\n\n\[ \n{E}_{k}\left( x\right) \mathrel{\text{:=}} \frac{{c}_{n}{k}^{n/2}}{{\left| x\right| }^{\left( {n - 2}\right) /2}}\left\lbrack {{H}_{n/2}^{\left( 1\right) }\left( {k\le... | Proof. Recall \( {F}_{\lambda } \) from (7.6.78). Then \( {\Phi }_{k} \) from (7.6.11)-(7.6.12) may be written in terms of \( {F}_{\lambda } \) corresponding to \( \lambda \mathrel{\text{:=}} \left( {n - 2}\right) /2 \) as\n\n\[ \n{\Phi }_{k}\left( x\right) = {c}_{n}{k}^{\left( {n - 2}\right) /2}{F}_{\left( {n - 2}\rig... | Yes |
Proposition 7.38 (applied with \( \lambda = \left( {n - 2}\right) /2 \) ) ensures that \( {E}_{k} \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) . | In addition, (7.10.4) and Proposition 7.38 imply\n\n\[ \n- {D}_{k}{\Phi }_{k} = - \mathop{\sum }\limits_{{j = 1}}^{n}{\partial }_{j}{\Phi }_{k}{\mathbf{e}}_{j} - k{\Phi }_{k}{\mathbf{e}}_{n + 1} \n\] \n\n\[ \n= - {c}_{n}{k}^{\left( {n - 2}\right) /2}\mathop{\sum }\limits_{{j = 1}}^{n}{\partial }_{j}{F}_{\left( {n - 2}\... | Yes |
Theorem 7.60. Assume that \( n \geq 2 \) and consider\n\n\[ L = \mathop{\sum }\limits_{{j, k = 1}}^{n}{a}_{jk}{\partial }_{j}{\partial }_{k},\;{a}_{jk} \in \mathbb{C}. \]\n\nThen for a function \( E \in {C}^{2}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \cap {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) wit... | Proof of Theorem 7.60. Let \( E \in {C}^{2}\left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \cap {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) with the property that \( \nabla E \) is positive homogeneous of degree \( 1 - n \) in \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) . Exercise 4.53 then implies that \( \... | No |
Corollary 8.2. The heat operator \( {\partial }_{t} - {\Delta }_{x} \) is hypoelliptic in \( {\mathbb{R}}^{n + 1} \) . | Proof. This is a consequence of Theorem 6.8 and Theorem 8.1. | No |
Proposition 8.7. Let \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and assume that \( F \in {C}^{0}\left( {{\mathbb{R}}^{n}\times \lbrack 0,\infty }\right) ) \) is such that its extension \( \widetilde{F} \) by zero to \( {\mathbb{R}}^{n + 1} \) satisfies \( \widetilde{F} \in {C}_{0}^{\infty }\left( {\mat... | Then the generalized Cauchy problem for the heat operator for the data \( \widetilde{F} \) and \( f \) has a solution \( u \in {\mathcal{D}}_{ + }^{\prime }\left( {\mathbb{R}}^{n + 1}\right) \) that is of function type, whose restriction to the set \( {\mathbb{R}}^{n} \times \left( {0,\infty }\right) \) is of class \( ... | Yes |
Theorem 8.8. Suppose \( A = {\left( {a}_{jk}\right) }_{1 \leq j, k \leq n} \in {\mathcal{M}}_{n \times n}\left( \mathbb{C}\right) \) satisfies (8.3.2) and consider the operator \( {\mathcal{L}}_{A} \) associated to \( A \) as in (8.3.1). Then the function defined by\n\n\[ \n{E}_{A}\left( x\right) \mathrel{\text{:=}} \f... | Moreover,\n\n\[ \n\left\{ {u \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n + 1}\right) : {\mathcal{L}}_{A}u = \delta \text{ in }{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n + 1}\right) }\right\} \n\]\n\n(8.3.16)\n\n\[ \n= \left\{ {{E}_{A} + P : P\text{ polynomial in }{\mathbb{R}}^{n + 1}\text{ satisfying }{\mathca... | Yes |
Lemma 9.3. If \( h : \lbrack 0,\infty ) \rightarrow \mathbb{R} \) is continuous and bounded, then for each \( t \) in \( \left( {0,\infty }\right) \) we have\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}{\int }_{0}^{\infty }\left( {\frac{\varepsilon }{{\varepsilon }^{2} + {\left( t - r\right) }^{2}}... | Proof. If \( t \geq 0 \) is fixed, then via suitable changes of variables we obtain\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ + }}}{\int }_{0}^{\infty }\left( {\frac{\varepsilon }{{\varepsilon }^{2} + {\left( t - r\right) }^{2}} - \frac{\varepsilon }{{\varepsilon }^{2} + {\left( t + r\right) }^{2}}}\r... | Yes |
Proposition 9.6. If \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) is a function with the property that\n\n\[ \n{\int }_{\mathbb{R}}\left| {f\left( {{x}^{\prime },{x}_{n}}\right) }\right| \mathrm{d}{x}_{n} < \infty \;\text{ for }{\mathcal{L}}^{n - 1}\text{-almost every }{x}^{\prime } \in {\mathbb{R}}^{n - 1} \... | Proof. Since \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) we have that \( {u}_{f} \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) as in (2.1.6) is well defined. Fix \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n - 1}\right) \) . Since \( f \) is absolutely integrable on supp \( \varphi \... | Yes |
Let \( m \in {\mathbb{N}}_{0} \) and let \( P\left( \partial \right) = P\left( {{\partial }^{\prime },{\partial }_{n}}\right) \) be a constant coefficient, linear operator of order \( m \) in \( {\mathbb{R}}^{n} \) . Define the differential operator \( {P}_{0}\left( {\partial }^{\prime }\right) \mathrel{\text{:=}} P\le... | Proof. Fix a sequence \( {\left\{ {\psi }_{j}\right\} }_{j \in \mathbb{N}} \subset {C}_{0}^{\infty }\left( \mathbb{R}\right) \) that converges in a dominated fashion to 1 and let \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n - 1}\right) \) . Using the definition of \( {u}_{0} \) we may write \[ \left\langle {{... | Yes |
Theorem 9.11. Let \( \widetilde{F} \in {\mathcal{D}}_{ + }^{\prime }\left( {\mathbb{R}}^{n + 1}\right) \) and \( f, g \in \mathcal{D}\left( {\mathbb{R}}^{n}\right) \) be given. Then the generalized Cauchy problem (9.2.3) has the unique solution\n\n\[ \widetilde{u} = E * \widetilde{F} + E * \left( {f \otimes {\delta }^{... | Proof. Let \( E \) be as in the statement of the theorem. Then, by (9.1.91), by (9.1.95), and by (9.1.101), for any \( G \in {\mathcal{D}}_{ + }^{\prime }\left( {\mathbb{R}}^{n + 1}\right) \) we have that, whenever \( K \) is a compact subset of \( {\mathbb{R}}^{n + 1} \), the set\n\n\[ {M}_{K} \mathrel{\text{:=}} \{ \... | Yes |
Proposition 10.2. Suppose \( L \) is a constant coefficient \( M \times M \) system of order \( m \in \mathbb{N} \) , \[ L = L\left( \partial \right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{\left| \alpha \right| \leq m}}{A}_{\alpha }{\partial }^{\alpha },\;{A}_{\alpha } = {\left( {a}_{\alpha }^{jk}\right) }_{1 \leq... | Proof. If \( \mathbb{U} \in {\mathcal{M}}_{M \times K}\left( {{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) }\right) \) is such that \( L\mathbb{U} = 0 \) in \( {\mathcal{M}}_{M \times K}\left( {{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) }\right) \), taking the Fourier transform, we obtain \( L\left(... | No |
Theorem 10.3 (A general Liouville type theorem for systems). Assume \( L \) is an \( M \times M \) system of order \( m \in \mathbb{N} \) of the form (10.1.20) which satisfies (10.1.21). Also, suppose \( \mathbf{u} = \left( {{u}_{1},\ldots ,{u}_{M}}\right) \in {\left\lbrack {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \... | Proof. Since (5.1.10) implies that the locally integrable function \( {u}_{j} \) belongs to \( {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) (cf. Example 4.4) for each \( j \in \{ 1,\ldots, M\} \), Proposition 10.2 implies that all the entries of \( \mathbf{u} \) are polynomials in \( {\mathbb{R}}^{n} \) . M... | Yes |
Proposition 10.4. Assume that\n\n\[ \nL = \mathop{\sum }\limits_{{\left| \alpha \right| \leq m}}{A}_{\alpha }{\partial }^{\alpha },\;{A}_{\alpha } = {\left( {a}_{\alpha }^{jk}\right) }_{1 \leq j, k \leq M} \in {\mathcal{M}}_{M \times M}\left( \mathbb{C}\right) ), \n\]\n\n(10.1.26)\n\nis an \( M \times M \) system of or... | Proof. This follows from (10.1.15), (10.1.19), and (10.1.12). | Yes |
Proposition 10.7. Let \( \mathfrak{R} \) be a commutative associative algebra over a field \( \mathbb{K} \) with multiplicative unit e and additive neutral element 0 . Also, let \( M \in \mathbb{N} \) be arbitrary. Then the following statements are true:\n\n(i) \( {\left( A \cdot B\right) }^{\top } = {B}^{\top } \cdot ... | Proof. All properties are established much as in the standard case \( \mathfrak{R} \equiv \mathbb{C} \) . Here we only wish to mention that (vii) is a direct consequence of \( \left( v\right) - \left( {vi}\right) \) (complete proofs may be found in, e.g., [36]). | No |
Theorem 10.8. Let \( L\left( \partial \right) \) be an \( M \times M \) system with constant coefficients as in (10.2.16), and let \( {D}_{L}\left( \partial \right) \) be the scalar differential operator associated with \( L \) as in (10.2.18). Then if \( E \in {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) i... | Proof. Let \( E \) be a fundamental solution for \( {D}_{L}\left( \partial \right) \) and define \( \mathbb{E} \) as in (10.2.23). That \( \mathbb{E} \) is a fundamental solution for \( L\left( \partial \right) \) follows from (10.2.19). Also it is clear that \( E \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\righ... | Yes |
Assume that \( L = L\left( \partial \right) \) is an \( M \times M \) homogeneous system in \( {\mathbb{R}}^{n} \), with constant complex coefficients, such that \[ \det \left\lbrack {L\left( \xi \right) }\right\rbrack \neq 0,\;\forall \xi \in {\mathbb{R}}^{n} \smallsetminus \{ 0\} . \] Then, for each open set \( \Omeg... | Proof. Let \( L \) be a system as in the statement of the theorem, and define the operator \( {D}_{L}\left( \partial \right) \mathrel{\text{:=}} \det \left\lbrack {L\left( \partial \right) }\right\rbrack \) . Then (10.2.22) ensures that \( {D}_{L}\left( \partial \right) \) is an elliptic, scalar differential operator, ... | Yes |
Corollary 10.10. Let \( L = L\left( \partial \right) \) be an \( M \times M \) homogeneous differential system in \( {\mathbb{R}}^{n} \), with constant complex coefficients, with the property that \( \det \left\lbrack {L\left( \xi \right) }\right\rbrack \neq 0 \) for all \( \xi \in {\mathbb{R}}^{n} \smallsetminus \{ 0\... | Proof. This is an immediate consequence of Theorem 10.8 and Theorem 10.9. | No |
Proposition 10.14. Given any Lamé moduli \( \lambda ,\mu \in \mathbb{C} \), define the characteristic matrix for the Lamé system (10.3.1) as\n\n\[ L\left( \xi \right) \mathrel{\text{:=}} \mu {\left| \xi \right| }^{2}{I}_{n \times n} + \left( {\lambda + \mu }\right) \xi \otimes \xi ,\;\forall \xi \in {\mathbb{R}}^{n}. \... | Proof. This is a direct consequence of Exercise 10.13. | No |
Theorem 10.15. Assume \( n \geq 3 \) and let \( L \) be the Lamé operator from (10.3.1) such that the constants the \( \lambda ,\mu \in \mathbb{C} \) satisfy (10.3.2). Define the matrix \( \mathbb{E} = {\left( {E}_{jk}\right) }_{1 \leq j, k \leq n} \) with entries given by the \( {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\ri... | Proof. Since the entries of \( \mathbb{E} \) as given in (10.3.21) are the same as the expressions from (10.3.17), the earlier analysis shows that \( \mathbb{E} \) belongs to \( {\mathcal{M}}_{n \times n}\left( {{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) }\right) \) and is a fundamental solution for the Lamé... | Yes |
Lemma 10.17. Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0 \) and \( \lambda + {2\mu } \neq 0 \) . Assume that the vector distribution \( \mathbf{u} = \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) \in {\left\lbrack {\mathcal{D}}^{\prime }\left( \Omega \right) \right\rbrack }^{n} \) satisfies the La... | Proof. The fact that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{n} \) is a consequence of Theorem 10.9, Proposition 10.14, and our assumptions on \( \lambda ,\mu \) . To show \( \left( i\right) \), we apply div to (10.4.1). Since \( \operatorname{div}\nabla = \Delta \) and \( \o... | Yes |
Proposition 10.18. Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0 \) and \( \lambda + {2\mu } \neq 0 \) . Assume \( \mathbf{u} = \) \( \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n}}\right) \in {\left\lbrack {\mathcal{D}}^{\prime }\left( \Omega \right) \right\rbrack }^{n} \) satisfies the Lamé system (10.4.1... | Proof. The fact that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{n} \) follows from Lemma 10.17. To proceed, fix some \( x \in \Omega, r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \), and \( j \in \{ 1,\ldots, n\} \) . By (iii) in Lemma 10.17, we... | Yes |
Theorem 10.19. Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0,\lambda + {2\mu } \neq 0 \), and \( \left( {n + 1}\right) \mu + \lambda \neq 0 \) . Assume \( \mathbf{u} \in {\left\lbrack {\mathcal{D}}^{\prime }\left( \Omega \right) \right\rbrack }^{n} \) satisfies the Lamé system (10.4.1). Then \( \math... | Proof. Once again, that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{n} \) is contained in Lemma 10.17. Formula (10.4.10) follows by taking a suitable linear combination of (10.4.2) and (10.4.3) so that \( {\partial }_{j}\left( {\operatorname{div}\mathbf{u}}\right) \) cancels (her... | Yes |
Theorem 10.20 ( \( {L}^{1} \) -Interior estimates for the Lamé operator). Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0,\lambda + {2\mu } \neq 0 \), and \( \left( {n + 1}\right) \mu + \lambda \neq 0 \) . Assume \( \mathbf{u} \in {\left\lbrack {\mathcal{D}}^{\prime }\left( \Omega \right) \right\rbrack... | Proof. The fact that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{n} \) has been established in Lemma 10.17. Fix \( k \in \) \( \{ 1,\ldots, n\} \) and observe that since \( \mathbf{u} = \left( {{u}_{1},\ldots ,{u}_{n}}\right) \) satisfies the Lamé system in \( \Omega \) , then so... | Yes |
Theorem 10.23. Suppose \( \lambda ,\mu \in \mathbb{C} \) are such that\n\n\[ \mu \neq 0,\lambda + {2\mu } \neq 0,\text{ and }\left( {n + 1}\right) \mu + \lambda \neq 0.\]\n\n(10.4.24)\n\nThen any null solution of the Lamé system (10.4.1) has components that are real-analytic in \( \Omega \) . | Proof. This is an immediate consequence of Theorem 10.22 and Lemma 6.24. | No |
Theorem 10.24. Suppose \( \lambda ,\mu \in \mathbb{C} \) satisfy the conditions in (10.4.24) and \( \Omega \subseteq {\mathbb{R}}^{n} \) is open and connected. Assume \( \mathbf{u} \) is a null solution of the Lamé system (10.4.1) with the property that \( {\partial }^{\alpha }\mathbf{u}\left( {x}_{0}\right) = 0 \) for... | Proof. This follows from Theorem 10.23 and Theorem 6.25. | No |
Theorem 10.25 (Liouville’s Theorem for the Lamé system). Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0,\lambda + {2\mu } \neq 0 \) and suppose \( \mathbf{u} \in {\left\lbrack {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \right\rbrack }^{n} \) satisfies the Lamé system\n\n\[{\mu \Delta }\mathbf{u} + \l... | Proof. This is a particular case of Theorem 10.3 since based on the current assumptions on \( \lambda \) and \( \mu \), Proposition 10.14 ensures that condition (10.1.21) is satisfied. | Yes |
Proposition 10.27. Let \( L \) be the Lamé operator from (10.3.1) such that (10.3.2) is satisfied. Assume \( n \geq 2 \) and let \( \mathbb{E} = {\left( {E}_{jk}\right) }_{1 \leq j, k \leq n} \) be the fundamental solution for \( L \) in \( {\mathbb{R}}^{n} \) with entries as in (10.3.21) for \( n \geq 3 \) and as in (... | Proof. This is established by arguing along the lines of the proof of Proposition 7.8. | No |
Theorem 10.28. Assume \( n \geq 3 \), and let \( L \) be the Lamé operator from (10.3.1) with \( \lambda ,\mu \in \mathbb{C} \) satisfying \( \mu \neq 0 \) and \( \lambda + {2\mu } \neq 0 \) . Also, suppose a vector-valued function \( \mathbf{f} \in {\left\lbrack {L}_{\text{comp }}^{\infty }\left( {\mathbb{R}}^{n}\righ... | Proof. From Proposition 10.27 we have that \( \mathbf{u} \) defined as in (10.5.4) is of class \( {C}^{1} \) in \( {\mathbb{R}}^{n} \) and satisfies \( L\mathbf{u} = \mathbf{f} \) in \( {\left\lbrack {\mathcal{D}}^{\prime }\left( {\mathbb{R}}^{n}\right) \right\rbrack }^{n} \) . In addition, by reasoning as in the proof... | No |
Theorem 10.29. Let \( n \geq 3 \) and let \( {L}_{S} \) be the Stokes operator from (10.6.1). Consider the following functions in \( {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) :\n\n\[ \n{E}_{jk}\left( x\right) \mathrel{\text{:=}} - \frac{1}{2\left( {n - 2}\right) {\omega }_{n - 1}}\frac{{\delta }_{jk}}{{\left| x\ri... | Proof. From (10.6.12) and (10.6.9) we have \( \mathbb{E} \in {\mathcal{M}}_{n \times n}\left( {{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) }\right) ,\mathbf{p} \in {\left\lbrack {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \right\rbrack }^{n} \), and\n\n\[ \n\widehat{{E}_{jk}} = \frac{{\xi }_{j}{\xi ... | Yes |
Proposition 11.3. Let \( \lambda ,\mu \in \mathbb{C} \) be such that \( \mu \neq 0,\lambda + {2\mu } \neq 0 \) . A fundamental solution for the Lamé operator (10.3.1) in \( {\mathbb{R}}^{3} \) is\n\n\[ E\left( x\right) = - \frac{1}{8\pi }\frac{\lambda + {3\mu }}{\mu \left( {\lambda + {2\mu }}\right) }\frac{1}{\left| x\... | Proof. We start by recalling formula (11.3.42) that gives an expression for the fundamental solution of a homogeneous differential operator for \( n \) odd. In our case,\n\n\[ L = {\left( {L}_{jk}\right) }_{1 \leq j, k \leq 3},\;{L}_{jk} = \mu {\delta }_{jk}\Delta + \left( {\lambda + \mu }\right) {\partial }_{j}{\parti... | Yes |
Proposition 11.6. Fix \( n, m, M \in \mathbb{N} \) with \( n \geq 2 \), and assume that \( L \) is an \( M \times M \) system in \( {\mathbb{R}}^{n} \) of order \( {2m} \) of the form\n\n\[ L = \mathop{\sum }\limits_{{\left| \alpha \right| = {2m}}}{A}_{\alpha }{\partial }^{\alpha },\;{A}_{\alpha } = {\left( {a}_{\alpha... | Proof. Let \( \mathbf{u} = {\left( {u}_{\ell }\right) }_{1 \leq \ell \leq M} \) and \( \mathbb{E} = {\left( {E}_{jk}\right) }_{1 \leq j, k \leq M} \) be as specified in the statement of the proposition. In particular,\n\n\[ \mathbb{E} \in {\mathcal{M}}_{M \times M}\left( {{\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\... | Yes |
Theorem 11.7. Let \( n, m, M \in \mathbb{N} \) and suppose \( L \) is an \( M \times M \) constant (complex) coefficient system in \( {\mathbb{R}}^{n} \), homogeneous of order \( {2m} \), and with the property that \( \det \left\lbrack {L\left( \xi \right) }\right\rbrack \neq 0 \) for each \( \xi \in {\mathbb{R}}^{n} \... | Proof. Theorem 10.9 gives that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{M} \) . As far as (11.4.13) is concerned, consider first the case when either \( n \) is odd, or \( n > {2m} \) . In this scenario,(11.4.5) gives that there exists some \( C \in \left( {0,\infty }\right) \... | Yes |
Lemma 11.9. Assume that \( \mathbf{u} \) is a continuous (complex) vector-valued function defined in \( \Omega \), and suppose that \( 0 < p < \infty \) . Then \( \mathbf{u} \) is \( p \) -subaveraging if and only if there exists a finite constant \( C > 0 \) such that\n\n\[ \mathop{\sup }\limits_{{z \in B\left( {x,{\l... | Proof. The fact that (11.5.2) implies (11.5.1) is obvious. Conversely, suppose that \( u \) is \( p \) -subaveraging. Pick \( x \in \Omega, r \in \left( {0,\operatorname{dist}\left( {x,\partial \Omega }\right) }\right) \), and \( z \in B\left( {x,{\lambda r}}\right) \) . Then, if \( R \mathrel{\text{:=}} \left( {1 - \l... | Yes |
Lemma 11.11. Let \( \\mathbf{u} \) be a subaveraging function. Then for every \( p, q \\in \\left( {0,\\infty }\\right) \) and \( \\lambda \\in \\left( {0,1}\\right) \) the following reverse Hölder estimate holds\n\n\[ \n{\\left( {\\int }_{B\\left( {x,{\\lambda r}}\\right) }{\\left| \\mathbf{u}\\left( y\\right) \\right... | Proof. If \( x \\in \\Omega \) and \( 0 < r < \\operatorname{dist}\\left( {x,\\partial \\Omega }\\right) \), we write\n\n\[ \n{\\left( {\\int }_{B\\left( {x,{\\lambda r}}\\right) }{\\left| \\mathbf{u}\\left( y\\right) \\right| }^{q}\\mathrm{\\;d}y\\right) }^{\\frac{1}{q}} \\leq \\mathop{\\sup }\\limits_{{z \\in B\\left... | Yes |
Theorem 11.12. Let \( n, m, M \in \mathbb{N} \) and suppose \( L \) is an \( M \times M \) constant (complex) coefficient system in \( {\mathbb{R}}^{n} \), homogeneous of order \( {2m} \), and with the property that \( \det \left\lbrack {L\left( \xi \right) }\right\rbrack \neq 0 \) for each \( \xi \in {\mathbb{R}}^{n} ... | Proof. As in the past, Theorem 10.9 gives that \( \mathbf{u} \in {\left\lbrack {C}^{\infty }\left( \Omega \right) \right\rbrack }^{M} \) . By working with the system \( \widetilde{\mathcal{L}} \) defined as in (11.4.20) and the function \( \widetilde{\mathbf{u}} \) defined as in (11.4.24), the same type of reasoning as... | Yes |
Theorem 11.13. Assume that the system \( L \) is as in (11.6.1)-(11.6.3) and suppose that \( \mathbb{E} \) is the fundamental solution for \( L \) in \( {\mathbb{R}}^{n} \) constructed in Theorem 11.1. Then\n\nfor each \( j \in \{ 1,\ldots, n\} \) one has\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow {0}^{ \pm ... | Proof. Fix \( j \in \{ 1,\ldots, n\} \) . Then \( \Phi \mathrel{\text{:=}} {\partial }_{j}\mathbb{E} \) is of class \( {C}^{\infty } \), odd, and positive homogeneous of degree \( 1 - n \) in \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) (cf. (11.6.6)-(11.6.7)). Moreover, by virtue of (11.3.28) used with \( m = 1 \), we... | Yes |
Theorem 11.14. Suppose that the system \( L \) is as in (11.6.1)-(11.6.3) and assume that \( \mathbb{E} \) is the fundamental solution for \( L \) in \( {\mathbb{R}}^{n} \) constructed in Theorem 11.1. Given any \( {\mathbb{C}}^{M} \) -valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \), def... | Proof. Let \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) be an arbitrary \( {\mathbb{C}}^{M} \) -valued function. That \( \mathcal{S}\varphi \) is of class \( {C}^{\infty } \) in \( {\mathbb{R}}^{n} \smallsetminus \left\{ {{x}_{n} = 0}\right\} \) is clear from (11.6.11),(11.6.6), and estimates for the... | Yes |
Theorem 11.16. Let the system \( L \) be as in (11.6.1)-(11.6.3) and assume that \( \mathbb{E} \) is the fundamental solution for \( L \) in \( {\mathbb{R}}^{n} \) constructed in Theorem 11.1. Given any \( {\mathbb{C}}^{M} \) - valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \), define for ... | Proof. Fix some \( {\mathbb{C}}^{M} \) -valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \) . The fact that \( \mathcal{D}\varphi \) is of class \( {C}^{\infty } \) in \( {\mathbb{R}}^{n} \smallsetminus \left\{ {{x}_{n} = 0}\right\} \) is seen from (11.6.16),(11.6.6), and estimates for the d... | Yes |
Corollary 11.18. Assume that the system \( L \) is as in (11.6.1)-(11.6.3) and let \( \mathbb{E} \) be the fundamental solution for \( L \) in \( {\mathbb{R}}^{n} \) constructed in Theorem 11.1. Then for every \( {\mathbb{C}}^{M} \) -valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \), \[ \m... | Proof. For each \( {\mathbb{C}}^{M} \) -valued function \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right) \), formula (11.6.21) follows using Corollary 11.14 by writing \[ \mathop{\lim }\limits_{{t \rightarrow {0}^{ \pm }}}\mathop{\sum }\limits_{{k = 1}}^{n}{A}_{nk}{\partial }_{k}\left( {\mathcal{S}\varphi }... | Yes |
(1) If \( {s}_{1},{s}_{2} \in \mathbb{R} \) then \( \langle \xi {\rangle }^{{s}_{1} + {s}_{2}} = \langle \xi {\rangle }^{{s}_{1}}\langle \xi {\rangle }^{{s}_{2}} \) for every \( \xi \in {\mathbb{R}}^{n} \) . | Proof. The identity in item (1) is immediate. | No |
Lemma 12.4. Let \( s \in \mathbb{R} \) . Define the space\n\n\[ \n{L}_{s}^{2}\left( {\mathbb{R}}^{n}\right) \mathrel{\text{:=}} \left\{ {v : {\mathbb{R}}^{n} \rightarrow \mathbb{C} : \langle \xi {\rangle }^{s}v \in {L}^{2}\left( {\mathbb{R}}^{n}\right) }\right\} \n\]\n\nand consider\n\n\[ \n\parallel v{\parallel }_{{L}... | Proof. That (12.1.6) is a norm on \( {L}_{s}^{2}\left( {\mathbb{R}}^{n}\right) \) is easily seen from the properties of the norm on \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) . Also, it is immediate that the norm (12.1.6) is induced by the inner product (12.1.7). To prove that \( {L}_{s}^{2}\left( {\mathbb{R}}^{n}\rig... | Yes |
Lemma 12.5. If \( u \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \), with \( s \in \mathbb{R} \) arbitrary, then\n\n\[ \langle u,\varphi \rangle = {\left( 2\pi \right) }^{-n}{\int }_{{\mathbb{R}}^{n}}\widehat{u}\left( \xi \right) \widehat{\varphi }\left( {-\xi }\right) \mathrm{d}\xi ,\;\forall \varphi \in \mathcal{S}\left... | Proof. For \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) write\n\n\[ \langle u,\varphi \rangle = {\left( 2\pi \right) }^{-n}\langle {\widehat{\widetilde{u}}}^{ \vee },\varphi \rangle = {\left( 2\pi \right) }^{-n}\langle \widehat{u},\widehat{{\varphi }^{ \vee }}\rangle = {\left( 2\pi \right) }^{-n}\left\la... | Yes |
Let \( s \in \mathbb{R} \) and \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) . Then for every \( u \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \) one has \( {\varphi u} \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \) and \[ \parallel {\varphi u}{\parallel }_{{H}^{s}\left( {\mathbb{R}}^{n}\right) } \leq C\left(... | Proof. Let \( \varphi \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) and \( u \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \) . Then \( \widehat{\varphi } \in \mathcal{S}\left( {\mathbb{R}}^{n}\right) \) and \( \widehat{u} \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \) and \( \widehat{\varphi } * \widehat{... | Yes |
For each open subset \( \Omega \) of \( {\mathbb{R}}^{n} \) and each \( s \in \mathbb{R} \) the mapping\n\n\[ \n{H}^{s}\left( \Omega \right) \ni u \mapsto \parallel u{\parallel }_{{H}^{s}\left( \Omega \right) } \mathrel{\text{:=}} \mathop{\inf }\limits_{{U \in {H}^{s}\left( {\mathbb{R}}^{n}\right) ,{\left. U\right| }_{... | Proof. The homogeneity, subadditivity, and positivity are consequences of the fact that \( \parallel \cdot {\parallel }_{{H}^{s}\left( {\mathbb{R}}^{n}\right) } \) is a norm on \( {H}^{s}\left( {\mathbb{R}}^{n}\right) \) . There remains to prove that if \( u \in {H}^{s}\left( \Omega \right) \) is such that \( \parallel... | Yes |
Corresponding to \( s = 0 \) we have \( {H}^{0}\left( \Omega \right) = {L}^{2}\left( \Omega \right) \) as vector spaces, and \[ \parallel u{\parallel }_{{H}^{0}\left( \Omega \right) } = {\left( 2\pi \right) }^{n/2}\parallel u{\parallel }_{{L}^{2}\left( \Omega \right) },\;\forall u \in {H}^{0}\left( \Omega \right) . \] | This is a consequence of Example 12.6 and Remark 3.29. Specifically, if \( u \in {H}^{0}\left( \Omega \right) \) then there exists \( U \in {H}^{0}\left( {\mathbb{R}}^{n}\right) = {L}^{2}\left( {\mathbb{R}}^{n}\right) \) satisfying \( {\left. U\right| }_{\Omega } = u \), hence \( u \) belongs to \( {L}^{2}\left( \Omega... | Yes |
Theorem 12.14. Let \( \Omega \) be an open subset of \( {\mathbb{R}}^{n} \) and let \( s \in \mathbb{R} \) . Then the following are true.\n\n(1) The space \( {H}^{s}\left( \Omega \right) \) endowed with the norm \( \parallel \cdot {\parallel }_{{H}^{s}\left( \Omega \right) } \) is complete, hence Banach. | Proof. Let \( A \mathrel{\text{:=}} \left\{ {v \in {H}^{s}\left( {\mathbb{R}}^{n}\right) : {\left. v\right| }_{\Omega } = 0\text{in}{\mathcal{D}}^{\prime }\left( \Omega \right) }\right\} \) . Then \( A \) is a closed subspace of the Banach space \( \left( {{H}^{s}\left( {\mathbb{R}}^{n}\right) ,\parallel \cdot {\parall... | Yes |
(1) Let \( s \in \mathbb{R} \) and \( \varphi \in {C}_{0}^{\infty }\left( \bar{\Omega }\right) \) . Then for every \( u \in {H}^{s}\left( \Omega \right) \) we have \( {\varphi u} \in {H}^{s}\left( \Omega \right) \) and\n\n\[ \n\parallel {\varphi u}{\parallel }_{{H}^{s}\left( \Omega \right) } \leq C\parallel u{\parallel... | Proof. If \( u \in {H}^{s}\left( \Omega \right) \) and \( \varphi \in {C}_{0}^{\infty }\left( \bar{\Omega }\right) \) then there exist \( U \in {H}^{s}\left( {\mathbb{R}}^{n}\right) \) and \( \Phi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) such that \( {\left. U\right| }_{\Omega } = u \) in \( {\left. {\mat... | Yes |
Suppose \( \Omega \) is an open set in \( {\mathbb{R}}^{n} \) and \( u \in {\mathcal{H}}^{m}\left( \Omega \right) \) for some \( m \in {\mathbb{N}}_{0} \) . If \( \psi \in {C}_{0}^{\infty }\left( \Omega \right) \) then \( \widetilde{\psi u} \), which is the extension by zero of \( {\psi u} \) outside \( \Omega \) belon... | Indeed, if \( \varphi \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and \( \xi \in {C}_{0}^{\infty }\left( \Omega \right) \) satisfies \( \xi \equiv 1 \) near supp \( \psi \), then for each \( \alpha \in {\mathbb{N}}_{0}^{n} \) we have that \( {\partial }^{\alpha }\varphi - {\partial }^{\alpha }\left( {\varphi... | Yes |
Suppose \( {\Omega }_{0},{\Omega }_{1} \), and \( C \) are open sets in \( {\mathbb{R}}^{n} \) such that \( {\Omega }_{0} \cap C = \) \( {\Omega }_{1} \cap C \) . Fix \( m \in \mathbb{N} \) and suppose \( u \in {\mathcal{H}}^{m}\left( {\Omega }_{0}\right) \) is such that \( u = 0 \) a.e. in \( {\Omega }_{0} \smallsetmi... | To see why \( v \) has the aforementioned properties fix some \( \xi \in {C}_{0}^{\infty }\left( C\right) \) with the property that \( \xi \equiv 1 \) near \( K \) . Then for each \( \varphi \in {C}_{0}^{\infty }\left( {\Omega }_{1}\right) \) and each \( \alpha \in {\mathbb{N}}_{0}^{n} \) we may write\n\n\[ {\left( -1\... | Yes |
Lemma 12.21. Let \( \phi \) be a function as in (1.2.3) and for each \( \varepsilon > 0 \) define \( {\phi }_{\varepsilon } \) as in (1.2.4). Also, let \( u \in {\mathcal{H}}^{m}\left( {\mathbb{R}}^{n}\right) \) for some \( m \in \mathbb{N} \) and set \( {u}_{\varepsilon } \mathrel{\text{:=}} u * {\phi }_{\varepsilon }... | Proof. That \( {u}_{\varepsilon } \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) for each \( \varepsilon > 0 \) is a consequence of Proposition 2.102. Next, pick a multi-index \( \alpha \in {\mathbb{N}}_{0}^{n} \) such that \( \left| \alpha \right| \leq m \). By part \( \left( e\right) \) in Theorem 2.96 we have \(... | Yes |
Theorem 12.23. Suppose \( \Omega \subseteq {\mathbb{R}}^{n} \) is an open set and fix \( m \in \mathbb{N} \) . Then the set \( \left\{ {u \in {\mathcal{H}}^{m}\left( \Omega \right) }\right. \) : \( u \) vanishes outside of a bounded subset of \( \Omega \} \) is dense in the intrinsic Sobolev space \( {\mathcal{H}}^{m}\... | Proof. Pick some \( \theta \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) with the property that \( \theta \equiv 1 \) on \( B\left( {0,1}\right) \) and, for each \( R > 0 \), define \( {\theta }_{R}\left( x\right) \mathrel{\text{:=}} \theta \left( {x/R}\right), x \in {\mathbb{R}}^{n} \) . Having fixed a functi... | Yes |
Proposition 12.25. If \( \\Omega \) is an open set in \( {\\mathbb{R}}^{n} \) and \( m \\in {\\mathbb{N}}_{0} \) then \( {H}^{m}\\left( \\Omega \\right) \) embeds continuously into \( {\\mathcal{H}}^{m}\\left( \\Omega \\right) \) . | Proof. Let \( u \\in {H}^{m}\\left( \\Omega \\right) \) and pick \( U \\in {H}^{m}\\left( {\\mathbb{R}}^{n}\\right) \) with \( {\\left. U\\right| }_{\\Omega } = u \) in \( {\\mathcal{D}}^{\\prime }\\left( \\Omega \\right) \). By Theorem 12.24 we have \( U \\in {\\mathcal{H}}^{m}\\left( {\\mathbb{R}}^{n}\\right) \). Giv... | Yes |
Let \( \Omega \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{n} : 0 < x < 1,0 < y < {x}^{4}}\right\} \) . Note that if \( \lambda < 5/2 \) then \[ {\begin{Vmatrix}{x}^{-\lambda }\end{Vmatrix}}_{{L}^{2}\left( \Omega \right) } = {\int }_{0}^{1}{\int }_{0}^{{x}^{4}}{x}^{-{2\lambda }}\mathrm{d}y\mathrm... | Hence, if we define \( u\left( {x, y}\right) \mathrel{\text{:=}} {x}^{-1/4} \) for all \( \left( {x, y}\right) \in \Omega \) then we have \( u \in {C}^{\infty }\left( \Omega \right) \) , \( {\partial }_{1}u\left( {x, y}\right) = - \frac{1}{4}{x}^{-5/4},{\partial }_{1}^{2}u\left( {x, y}\right) = \frac{5}{16}{x}^{-9/4} \... | Yes |
Theorem 12.30. Let \( \Omega \) be a bounded Lipschitz domain in \( {\mathbb{R}}^{n} \) and fix \( m \in \mathbb{N} \). Then \( {\mathcal{H}}^{m}\left( \Omega \right) = {H}^{m}\left( \Omega \right) \) as vector spaces, with equivalent norms. In particular,\n\n\[ \parallel u{\parallel }_{{H}^{m}\left( \Omega \right) } \... | Proof. Since the inclusion \( {H}^{m}\left( \Omega \right) \subseteq {\mathcal{H}}^{m}\left( \Omega \right) \) and the corresponding norm inequality hold for arbitrary open sets \( \Omega \) (cf. Proposition 12.25) there remains to prove the opposite inclusion and the naturally accompanying norm inequality. To see this... | Yes |
Theorem 12.32. Suppose \( \Psi : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is a bijective and bi-Lipschitz function. Let O be an open set in \( {\mathbb{R}}^{n} \) and consider \( \mathcal{U} \mathrel{\text{:=}} \Psi \left( O\right) \) . Then \( \mathcal{U} \) is an open subset of \( {\mathbb{R}}^{n} \), for eve... | Proof. Since \( \Psi \) is bijective and bi-Lipschitz, it follows that both \( \Psi \) and its inverse \( {\Psi }^{-1} \) are Lipschitz functions (in particular, continuous). The fact that \( \mathcal{U} \) is the preimage of the open set \( O \) under the continuous function \( {\Psi }^{-1} \) then implies that \( \ma... | Yes |
Corollary 12.33. Let \( \Omega \) be an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \) and let \( \Phi \) be the bijective bi-Lipschitz flattening map associated with \( \Omega \) as in Remark 14.52. Then the following are true.\n\n(i) For a measurable function \( u \) in \( \Omega \) we have\n\n\[ u \in {L}^{2... | Proof. All equivalences are consequences of Theorem 12.32. For example, for the left-to-right implications in (i) apply Theorem 12.32 with \( u \mathrel{\text{:=}} u,\Psi \mathrel{\text{:=}} \Phi, O \mathrel{\text{:=}} {\mathbb{R}}_{ + }^{n} \) and \( \mathcal{U} \mathrel{\text{:=}} \Omega \) . | Yes |
Corollary 12.35. Suppose \( \Omega \) is an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \) . Then \( {\mathcal{H}}^{1}\left( \Omega \right) = {H}^{1}\left( \Omega \right) \) as vector spaces, with equivalent norms. | Proof. This is an immediate consequence of Proposition 12.25, Theorem 12.34, Theorem 12.24, and definitions. | No |
The vector space \( {H}^{1/2}\left( {\mathbb{R}}^{n}\right) \) is equal to the collection of all functions \( u \in {L}^{2}\left( {\mathbb{R}}^{n}\right) \) with the property that\n\n\[ \n{\int }_{{\mathbb{R}}^{n}}{\int }_{{\mathbb{R}}^{n}}\frac{{\left| u\left( x\right) - u\left( y\right) \right| }^{2}}{{\left| x - y\r... | Proof. Let us check that \( \parallel \parallel \cdot {\parallel }_{\frac{1}{2},{\mathbb{R}}^{n}} \) is indeed a norm on the vector space \( {H}^{1/2}\left( {\mathbb{R}}^{n}\right) \) . Start by considering the ambient set \( X \mathrel{\text{:=}} {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) endowed with the measure\n\n... | Yes |
Proposition 12.40. Let \( \Omega \) be an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \), or a bounded Lipschitz domain in \( {\mathbb{R}}^{n} \). For each function \( u \in {H}^{1/2}\left( {\partial \Omega }\right) \) set\n\n\[ \parallel u{\parallel }_{{H}^{1/2}\left( {\partial \Omega }\right) } \mathrel{\text... | Proof. The proof of the fact that (12.4.12) is a norm on \( {H}^{1/2}\left( {\partial \Omega }\right) \) is similar to the proof used in Proposition 12.37, this time considering the set \( X \mathrel{\text{:=}} \partial \Omega \times \partial \Omega \) endowed with the measure \( \mu \left( {x, y}\right) \mathrel{\text... | Yes |
Let \( \Omega \) be either a bounded Lipschitz domain, or an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \). Also, suppose \( {\Omega }_{1} \) is either a bounded Lipschitz domain, or an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \) and assume that there exist \( {x}_{0} \in \partial \Omega \cap \parti... | Proof. It is immediate that\n\n\[ \parallel v{\parallel }_{{L}^{2}\left( {\partial {\Omega }_{1}}\right) } = \parallel u{\parallel }_{{L}^{2}\left( {\partial {\Omega }_{1} \cap C}\right) } = \parallel u{\parallel }_{{L}^{2}\left( {\partial \Omega \cap C}\right) } = \parallel u{\parallel }_{{L}^{2}\left( {\partial \Omeg... | Yes |
Theorem 12.44. Let \( \Omega \) be either an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \), or a bounded Lipschitz domain in \( {\mathbb{R}}^{n} \). Then the following are true.\n\n(1) If \( \varphi \in {\operatorname{Lip}}_{\text{comp }}\left( {\partial \Omega }\right) \) then \( {\varphi u} \in {H}^{1/2}\lef... | Proof. Fix \( \varphi \in {\operatorname{Lip}}_{\text{comp }}\left( {\partial \Omega }\right) \) and \( u \in {H}^{1/2}\left( {\partial \Omega }\right) \). Then\n\n\[ \parallel {\varphi u}{\parallel }_{{L}^{2}\left( {\partial \Omega }\right) } \leq \parallel \varphi {\parallel }_{{L}^{\infty }\left( {\partial \Omega }\... | Yes |
Theorem 12.45. Let \( \Omega \) be a bounded Lipschitz domain in \( {\mathbb{R}}^{n} \) . Then the following are true.\n\n(1) The vector space \( \mathcal{V}\left( \Omega \right) \mathrel{\text{:=}} {C}^{0}\left( \bar{\Omega }\right) \cap {\operatorname{Lip}}_{loc}\left( \Omega \right) \cap {H}^{1}\left( \Omega \right)... | Proof. The density statement in (1) follows by observing that \( {C}_{0}^{\infty }\left( \bar{\Omega }\right) \subseteq \mathcal{V}\left( \Omega \right) \) and recalling that the set \( {C}_{0}^{\infty }\left( \bar{\Omega }\right) \) is dense in \( {H}^{1}\left( \Omega \right) \) (cf. Theorem 12.14). | Yes |
Proposition 14.2. Assume \( X \) and \( Y \) are two given topological vector spaces, and denote by \( {X}^{\prime },{Y}^{\prime } \) their duals, each endowed with the corresponding weak*-topology. Also, suppose \( T : X \rightarrow Y \) is a linear and continuous operator, and define its transpose \( {T}^{t} \) as th... | Proof. For each \( {y}^{\prime } \in {Y}^{\prime } \) it follows that \( {y}^{\prime } \circ T \) is a composition of two linear and continuous mappings. Hence, \( {y}^{\prime } \circ T \in {X}^{\prime } \) which proves that \( {T}^{t} : {Y}^{\prime } \rightarrow {X}^{\prime } \) is well-defined. It is also clear from ... | Yes |
Proposition 14.3. Suppose that \( X, Y, Z \) are topological vector spaces, and denote by \( {X}^{\prime },{Y}^{\prime },{Z}^{\prime } \) their duals, each endowed with the corresponding weak*-topology. In addition, assume that \( T : X \rightarrow Y \) and \( R : Y \rightarrow Z \) are two linear and continuous operat... | Proof. Formula (14.1.13) is immediate from definitions, while the claims in the last part of the statement are direct consequences of (14.1.13) and the fact that the transpose of the identity is also the identity. | No |
Proposition 14.4. Suppose \( X \) and \( Y \) are topological vector spaces such that \( X \subseteq Y \) densely and the inclusion map \( \iota : X \rightarrow Y,\iota \left( x\right) \mathrel{\text{:=}} x \) for each \( x \in X \), is continuous. Then \( {Y}^{\prime } \) endowed with the weak*-topology embeds continu... | Proof. The fact that \( {\iota }^{t} \) is well-defined, linear, and continuous follows directly from Proposition 14.2 and assumptions. Assume now that \( {y}^{\prime } \in {Y}^{\prime } \) is such that \( {\iota }^{t}\left( {y}^{\prime }\right) = 0 \) . Then from the fact that \( {y}^{\prime } \circ \iota : X \rightar... | Yes |
Theorem 14.12 (Taylor’s Formula). Assume \( U \subseteq {\mathbb{R}}^{n} \) is an open convex set, and that \( N \in \mathbb{N} \) . Also, suppose that \( f : U \rightarrow \mathbb{C} \) is a function of class \( {C}^{N + 1} \) on \( U \) . Then for every \( x, y \in U \) one has\n\n\[ f\left( x\right) = \mathop{\sum }... | In particular, for each \( x, y \in U \) there exists \( \theta \in \left( {0,1}\right) \) with the property that\n\n\[ f\left( x\right) = \mathop{\sum }\limits_{{\left| \alpha \right| \leq N}}\frac{1}{\alpha !}{\left( x - y\right) }^{\alpha }\left( {{\partial }^{\alpha }f}\right) \left( y\right) \]\n\n(14.2.10)\n\n\[ ... | Yes |
Theorem 14.16 ( \( {L}^{p} \) version of Lebesgue’s Dominated Convergence Theorem). Let \( \left( {X,\mu }\right) \) be a positive measure space and assume that \( g \in {L}^{p}\left( {X,\mu }\right) \) for some \( p \in \) \( \lbrack 1,\infty ) \) is a nonnegative function. If \( {\left\{ {f}_{j}\right\} }_{j \in \mat... | One can prove Theorem 14.16 by applying Fatou's lemma to the sequence of functions \( {\left\{ {2}^{p - 1}{g}^{p} - {\left| {f}_{j} - f\right| }^{p}\right\} }_{j \in \mathbb{N}} \) . | Yes |
Lemma 14.18. Let \( \theta \in {L}^{1}\left( {\mathbb{R}}^{n}\right) \) and for each \( \varepsilon > 0 \) define \( {\theta }_{\varepsilon }\left( x\right) \mathrel{\text{:=}} {\varepsilon }^{-n}\theta \left( {x/\varepsilon }\right) \) for \( x \in {\mathbb{R}}^{n} \). Then for each \( p \in \left\lbrack {1,\infty }\r... | \[ \mathop{\sup }\limits_{{\varepsilon > 0}}{\begin{Vmatrix}{f}_{\varepsilon }\end{Vmatrix}}_{{L}^{p}\left( {\mathbb{R}}^{n}\right) } \leq \parallel \theta {\parallel }_{{L}^{1}\left( {\mathbb{R}}^{n}\right) }\parallel f{\parallel }_{{L}^{p}\left( {\mathbb{R}}^{n}\right) }.\] | Yes |
Theorem 14.21 (Hardy’s Inequality). Suppose \( p \in \lbrack 1,\infty ), r \in \left( {0,\infty }\right) \), and consider a measurable function \( f : \left\lbrack {0,\infty }\right\rbrack \rightarrow \left\lbrack {0,\infty }\right\rbrack \) . Then\n\n\[ \n{\int }_{0}^{\infty }{\rho }^{-1 - r}{\left( {\int }_{0}^{\rho ... | See e.g., [68, p. 272, A.4]. | No |
Theorem 14.22 (Minkowski’s Inequality). Suppose \( \left( {X,\mathcal{M},\mu }\right) \) and \( \left( {Y,\mathcal{N}, v}\right) \) are \( \sigma \) -finite measure spaces, and let \( f \) be an \( \left( {\mathcal{M} \otimes \mathcal{N}}\right) \) -measurable function on \( X \times Y \) . If \( f \geq 0 \) and \( p \... | For a proof of Minkowski’s Inequality see [18, 6.19, p. 194]. | No |
Theorem 14.29 (Vitali’s Convergence Theorem). Let \( \left( {X,\mu }\right) \) be a positive measure space with \( \mu \left( X\right) < \infty \) . Suppose \( {\left\{ {f}_{k}\right\} }_{k \in \mathbb{N}} \) is a sequence of functions in \( {L}^{1}\left( {X,\mu }\right) \) and that \( f \) is a function on \( X \) (al... | (14.2.38)\n\nIn particular, \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{\int }_{X}{f}_{k}\mathrm{\;d}\mu = {\int }_{X}f\mathrm{\;d}\mu \) .\n\nSee, e.g., [64, p. 133]. | Yes |
Proposition 14.30. Let \( \\left( {X,\\mu }\\right) \) be a positive measure space and suppose \( f \\in {L}^{1}\\left( {X,\\mu }\\right) \). Then for every \( \\varepsilon > 0 \) there exists \( \\delta > 0 \) such that for every \( \\mu \)-measurable set \( A \\subseteq X \) satisfying \( \\mu \\left( A\\right) < \\d... | Proof. Consider the measure \( \\lambda \\mathrel{\\text{:=}} \\left| f\\right| \\mu \) on \( X \). Then \( \\lambda \) is absolutely continuous with respect to \( \\mu \) and the \( \\left( {\\varepsilon ,\\delta }\\right) \) characterization of absolute continuity of measures (see, e.g., [64, Theorem 6.11, p. 124]) y... | Yes |
Lemma 14.31. Let \( f : \mathbb{R} \rightarrow \mathbb{R} \) be the function defined by\n\n\[ f\left( x\right) \mathrel{\text{:=}} \left\{ {\begin{matrix} {\mathrm{e}}^{-1/x},\text{ if }x > 0, \\ 0,\text{ if }x \leq 0, \end{matrix}\;\forall x \in \mathbb{R}.}\right.\n\n(14.3.1)\n\nThen \( f \) is of class \( {C}^{\inft... | Proof. Denote by \( C \) the collection of functions \( g : \mathbb{R} \rightarrow \mathbb{R} \) for which there exists a polynomial \( P \) such that\n\n\[ g\left( x\right) \mathrel{\text{:=}} \left\{ {\begin{matrix} {\mathrm{e}}^{-1/x}P\left( {1/x}\right) , & \text{ if }x > 0, \\ 0, & \text{ if }x \leq 0, \end{matrix... | Yes |
Lemma 14.32. The function \( \phi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) defined by\n\n\[ \phi \left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} C{\mathrm{e}}^{\frac{1}{{\left| x\right| }^{2} - 1}} & \text{ if }x \in B\left( {0,1}\right) , \\ 0 & \text{ if }x \in {\mathbb{R}}^{n} \smallsetminus B\left... | Proof. That \( \phi \geq 0 \) and \( \operatorname{supp}\phi \subseteq \overline{B\left( {0,1}\right) } \) is immediate from its definition. Also, since \( \phi \left( x\right) = {Cf}\left( {1 - {\left| x\right| }^{2}}\right) \) for \( x \in {\mathbb{R}}^{n} \) where \( f \) is as in (14.3.1), invoking Lemma 14.31\n\ni... | Yes |
Proposition 14.33. Let \( {F}_{0},{F}_{1} \subset {\mathbb{R}}^{n} \) be two nonempty sets with the property that \( \operatorname{dist}\left( {{F}_{0},{F}_{1}}\right) > 0 \) . Then there exists a function \( \psi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) with the following properties:\n\n\[ \psi \in {C}^{\infty }\l... | Proof. Let \( r \mathrel{\text{:=}} \operatorname{dist}\left( {{F}_{0},{F}_{1}}\right) > 0 \) and set \( \widetilde{{F}_{1}} \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{n} : \operatorname{dist}\left( {x,{F}_{1}}\right) \leq r/4}\right\} \) . Also, with \( \phi \) as in Lemma 14.32, define the function \( \theta \l... | Yes |
Proposition 14.34. If \( U \subseteq {\mathbb{R}}^{n} \) is open and \( K \subset U \) is compact, then there exists a function \( \psi : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) that is of class \( {C}^{\infty } \), satisfies \( 0 \leq \psi \left( x\right) \leq 1 \) for every \( x \in {\mathbb{R}}^{n} \) and \( \psi... | Proof. Since \( K \) is a compact set contained in \( U \) we may define \( r \mathrel{\text{:=}} \operatorname{dist}\left( {K,{\mathbb{R}}^{n} \smallsetminus U}\right) \) which is a positive number. Now Proposition 14.33 applied with \( {F}_{1} \mathrel{\text{:=}} K \) and \( {F}_{0} \mathrel{\text{:=}} \) \( \left\{ ... | Yes |
Lemma 14.35. If \( C \subset {\mathbb{R}}^{n} \) is compact, and \( U \subseteq {\mathbb{R}}^{n} \) is an open set such that \( C \subset U \) , then there exists a compact set \( D \subseteq {\mathbb{R}}^{n} \) such that \( C \subset \mathring{D} \subset D \subset U \) . | Proof. Let \( V = {U}^{c} \cap \overline{B\left( {0, R}\right) } \), where \( R > 0 \) is large enough so that \( \bar{U} \subset B\left( {0, R}\right) \) . Then \( V \) is compact and disjoint from \( C \) so \( r \mathrel{\text{:=}} \operatorname{dist}\left( {V, C}\right) = \mathop{\inf }\limits_{\substack{{x \in C} ... | Yes |
Lemma 14.36. Suppose that \( K \subseteq {\mathbb{R}}^{n} \) is a compact set and that \( {\left\{ {O}_{j}\right\} }_{1 \leq j \leq k} \) is a finite open cover of \( K \) . Then there are compact sets \( {D}_{j} \subset {O}_{j},1 \leq j \leq k \), with the property that\n\n\[ K \subset \mathop{\bigcup }\limits_{{j = 1... | Proof. Set \( {C}_{1} \mathrel{\text{:=}} K \smallsetminus \mathop{\bigcup }\limits_{{j = 2}}^{k}{O}_{j} \subseteq {O}_{1} \) . Since \( {C}_{1} \) is compact, Lemma 14.35 shows that there exists a compact set \( {D}_{1} \) with the property that \( {C}_{1} \subset {\mathring{D}}_{1} \subset {D}_{1} \subset {O}_{1} \) ... | Yes |
Theorem 14.37 (Partition of Unity for Compact Sets). Let \( K \subset {\mathbb{R}}^{n} \) be a compact set, and let \( {\left\{ {O}_{j}\right\} }_{1 \leq j \leq N} \) be a finite open cover of \( K \) . Then there exists a finite collection of \( {C}^{\infty } \) functions \( {\varphi }_{j} : {\mathbb{R}}^{n} \rightarr... | Proof. Let \( {\left\{ {O}_{j}\right\} }_{1 \leq j \leq N} \) be any finite open cover for \( K \) . From Lemma 14.36 we know that there exist compact sets \( {D}_{j} \subseteq {O}_{j},1 \leq j \leq N \), such that \( K \subset \mathop{\bigcup }\limits_{{j = 1}}^{N}{\mathring{D}}_{j} \) . By\n\nProposition 14.34, for e... | Yes |
Theorem 14.42 (Partition of Unity for Arbitrary Open Covers). Let \( {\left( {O}_{k}\right) }_{k \in I} \) be an arbitrary family of open sets in \( {\mathbb{R}}^{n} \) and set \( \Omega \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{k \in I}}{O}_{k} \) . Then there exists an at most countable collection \( {\left( {\v... | Proof. Start by defining\n\n\[ {\Omega }_{j} \mathrel{\text{:=}} \{ x \in \Omega : \parallel x\parallel \leq j\text{ and }\operatorname{dist}\left( {x,\partial \Omega }\right) \geq \frac{1}{j}\} ,\;j \in \mathbb{N}, \]\n\n(14.4.5)\n\nThen \( \Omega = \mathop{\bigcup }\limits_{{j = 1}}^{\infty }{\Omega }_{j} \) and\n\n\... | Yes |
Theorem 14.43 (Partition of Unity with Preservation of Indexes). Let \( {\left( {O}_{k}\right) }_{k \in I} \) be an arbitrary family of open sets in \( {\mathbb{R}}^{n} \) and set \( \Omega \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{k \in I}}{O}_{k} \). Then there exists a collection \( {\left( {\psi }_{k}\right) }... | Proof. Let \( {\left( {\varphi }_{j}\right) }_{j \in J} \) be a partition of unity subordinate to the family \( {\left( {O}_{k}\right) }_{k \in I} \), and denote by \( f : J \rightarrow I \) a function with the property that, for every \( j \in J \), the function \( {\varphi }_{j} \) is compactly supported in \( {O}_{f... | Yes |
Assume \( \Omega \subseteq {\mathbb{R}}^{n} \) is an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \) and let \( \varphi : {\mathbb{R}}^{n - 1} \rightarrow \mathbb{R} \) be the Lipschitz function such that (14.7.4) holds. Denote by \( M \) the Lipschitz constant of \( \varphi \) and fix \( \theta \in \left( {0,2\... | Proof. As far as the first inclusion in (14.7.24) is concerned, it suffices to show that\n\nif \( {x}^{\prime },{y}^{\prime } \in {\mathbb{R}}^{n - 1},\;s \in \mathbb{R} \) are such that\n\n(14.7.25)\n\n\[ \n\left( {{y}^{\prime }, s}\right) \in {\Gamma }_{\theta }\left( {\left( {{x}^{\prime },\varphi \left( {x}^{\prime... | Yes |
Lemma 14.56. Let \( \alpha > 0 \) . Suppose \( \Omega \) is either an upper-graph Lipschitz domain in \( {\mathbb{R}}^{n} \), or a bounded Lipschitz domain in \( {\mathbb{R}}^{n} \) . Then there exists some constant \( C = \) \( C\left( {\Omega ,\alpha }\right) \in \left( {0,\infty }\right) \) such that for each \( x \... | Proof. Suppose first that \( \Omega \) is the upper-graph of a Lipschitz function \( \varphi : {\mathbb{R}}^{n - 1} \rightarrow \mathbb{R} \) with Lipschitz constant \( M \) . Pick arbitrary \( x \in \partial \Omega \) and \( r > 0 \) . Then, by (14.7.31) we have\n\n\[ \n{\int }_{\partial \Omega }\frac{{\mathbf{1}}_{\l... | Yes |
Lemma 14.57. Let \( \alpha > 0 \) . Assume that \( \Omega \subseteq {\mathbb{R}}^{n} \) is either an upper-graph Lipschitz domain, or a bounded Lipschitz domain. Then for each \( x \in \partial \Omega \) and each \( r > 0 \) there exists a constant \( C \in \left( {0,\infty }\right) \) depending on \( \Omega, n \), and... | Proof. Given \( x \in \partial \Omega \) and \( r > 0 \) arbitrary we have\n\n\[{\int }_{\partial \Omega }\frac{{\mathbf{1}}_{\left| {x - y}\right| > r}}{{\left| x - y\right| }^{n - 1 + \alpha }}\mathrm{d}\sigma \left( y\right) = \mathop{\sum }\limits_{{j = 0}}^{\infty }{\int }_{\partial \Omega }\frac{{\mathbf{1}}_{{2}... | Yes |
Theorem 14.58. Suppose \( \Psi : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) is a bijective and bi-Lipschitz function. Let O and \( \mathcal{U} \) be open sets in \( {\mathbb{R}}^{n} \) such that \( \mathcal{U} = \Psi \left( \mathcal{O}\right) \) . Then for every function \( u \in {L}^{1}\left( \mathcal{U}\right) ... | Proof. This is a particular case of [15, Theorem 2, p. 99] applied with \( n \mathrel{\text{:=}} m \) , \( f \mathrel{\text{:=}} {\Psi }^{-1} \), and \( g \mathrel{\text{:=}} \widetilde{u} \), the extension by zero of \( u \) outside \( \mathcal{U} \) . | Yes |
Theorem 14.63 (Spherical Fubini and Polar Coordinates). Let \( f \in {L}_{loc}^{1}\left( {\mathbb{R}}^{n}\right) \) , \( n \geq 2 \) . Then for each \( {x}_{ * } \in {\mathbb{R}}^{n} \) and each \( R > 0 \) the following formulas hold: | \[ {\int }_{B\left( {{x}_{ * }, R}\right) }f\mathrm{\;d}x = {\int }_{0}^{R}\left( {{\int }_{\partial B\left( {{x}_{ * },\rho }\right) }f\mathrm{\;d}\sigma }\right) \mathrm{d}\rho \] (14.9.5) \[ {\int }_{B\left( {{x}_{ * }, R}\right) }f\mathrm{\;d}x = {\int }_{0}^{R}{\int }_{{S}^{n - 1}}f\left( {{x}_{ * } + {\rho \omega... | Yes |
Proposition 14.65. Let \( v \in {\mathbb{R}}^{n} \smallsetminus \{ 0\}, n \geq 2 \), be fixed. Then for any measurable and nonnegative function \( f \) defined on the real line, there holds\n\n\[ \n{\int }_{{S}^{n - 1}}f\left( {v \cdot \theta }\right) \mathrm{d}\sigma \left( \theta \right) = {\omega }_{n - 2}{\int }_{-... | Proof. Since integrals over the unit sphere are invariant under orthogonal transformations, we may assume that \( v/\left| v\right| = {e}_{1} \) and, hence, using polar coordinates and (14.9.3), we have\n\n\[ \n{\int }_{{S}^{n - 1}}f\left( {v \cdot \theta }\right) \mathrm{d}\sigma \left( \theta \right) = {\int }_{{S}^{... | Yes |
Proposition 14.68. Consider \( f\left( t\right) \mathrel{\text{:=}} \left| t\right| \) for \( t \in \mathbb{R} \), and let \( \alpha ,\beta \) be as in (14.9.17) for this choice of \( f \) . Then\n\n\[ \alpha = \beta = \frac{2{\omega }_{n - 2}}{{n}^{2} - 1} \]\n\nwhere \( {\omega }_{n - 2} \) denotes the surface measur... | Proof. Using the standard parametrization of \( {S}^{n - 1} \) (see (14.9.1) with \( R = 1 \) ) we have\n\n\[ \alpha = \frac{1}{n - 1}{\int }_{0}^{\pi }{\int }_{0}^{\pi }\ldots {\int }_{0}^{\pi }{\int }_{0}^{2\pi }\left| {\cos {\varphi }_{1}}\right| \left( {1 - {\left( \cos {\varphi }_{1}\right) }^{2}}\right) {\left( \... | Yes |
Proposition 14.69. For each multi-index \( \alpha = \left( {{\alpha }_{1},\ldots ,{\alpha }_{n}}\right) \in {\mathbb{N}}_{0}^{n} \) , \[ {\int }_{{S}^{n - 1}}{z}^{\alpha }\mathrm{d}\sigma \left( z\right) = \left\{ \begin{array}{ll} 0 & \text{ if }\alpha \notin 2{\mathbb{N}}_{0}^{n}, \\ \frac{\left( \frac{\left| \alpha ... | Proof. Fix an arbitrary \( k \in \mathbb{N} \) and set \[ {q}_{\alpha } \mathrel{\text{:=}} {\int }_{{S}^{n - 1}}{z}^{\alpha }\mathrm{d}\sigma \left( z\right) ,\;\forall \alpha \in {\mathbb{N}}_{0}^{n}\text{ with }\left| \alpha \right| = k. \] Also, with \ | No |
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