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Lemma 10.2.2. Let \( {\Omega }_{0} \subset \subset \Omega, g \in {W}^{1,2}\left( \Omega \right), u \in {W}^{1,2}\left( {\Omega }_{0}\right), u - g \in {H}_{0}^{1,2}\left( {\Omega }_{0}\right) \) . Then\n\n\[ v\left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} u\left( x\right) & \text{ for }x \in {\Omega }_{0... | Proof. By Corollary 10.2.1, there exist \( {g}_{n} \in {C}^{\infty }\left( \Omega \right) ,{u}_{n} \in {C}^{\infty }\left( {\Omega }_{0}\right) \) with\n\n\[ {g}_{n} \rightarrow g\;\text{ in }{W}^{1,2}\left( \Omega \right) \]\n\n\[ {u}_{n} \rightarrow u\;\text{ in }{W}^{1,2}\left( {\Omega }_{0}\right) \]\n\n\[ {u}_{n} ... | Yes |
Lemma 10.2.3. For \( u \in {W}^{1,2}\left( \Omega \right), f \in {C}^{1}\left( \mathbb{R}\right) \), suppose\n\n\[ \mathop{\sup }\limits_{{y \in \mathbb{R}}}\left| {{f}^{\prime }\left( y\right) }\right| < \infty \]\n\nThen \( f \circ u \in {W}^{1,2}\left( \Omega \right) \), and the weak derivative satisfies \( D\left( ... | Proof. Let \( {u}_{n} \in {C}^{\infty }\left( \Omega \right) ,{u}_{n} \rightarrow u \) in \( {W}^{1,2}\left( \Omega \right) \) for \( n \rightarrow \infty \) . Then\n\n\[ {\int }_{\Omega }{\left| f\left( {u}_{n}\right) - f\left( u\right) \right| }^{2}\mathrm{\;d}x \leq \sup {\left| {f}^{\prime }\right| }^{2}{\int }_{\O... | Yes |
Corollary 10.2.2. If \( u \in {W}^{1,2}\left( \Omega \right) \), then also \( \left| u\right| \in {W}^{1,2}\left( \Omega \right) \), and \( D\left| u\right| = \operatorname{sign}u \cdot {Du} \) . | Proof. We consider \( {f}_{\varepsilon }\left( u\right) \mathrel{\text{:=}} {\left( {u}^{2} + {\varepsilon }^{2}\right) }^{\frac{1}{2}} - \varepsilon \), apply Lemma 10.2.3, and let \( \varepsilon \rightarrow 0 \) , using once more Lebesgue's theorem on dominated convergence to justify the limit as before. | Yes |
Theorem 10.2.2. For \( u \in {H}_{0}^{1,2}\left( \Omega \right) \), we have\n\n\[ \parallel u{\parallel }_{{L}^{2}\left( \Omega \right) } \leq {\left( \frac{\left| \Omega \right| }{{\omega }_{d}}\right) }^{\frac{1}{d}}\parallel {Du}{\parallel }_{{L}^{2}\left( \Omega \right) } \]\n\nwhere \( \left| \Omega \right| \) den... | Proof. Suppose first \( u \in {C}_{0}^{1}\left( \Omega \right) \) ; we put \( u\left( x\right) = 0 \) for \( x \in {\mathbb{R}}^{d} \smallsetminus \Omega \) . For \( \omega \in {\mathbb{R}}^{d} \) with \( \left| \omega \right| = 1 \), by the fundamental theorem of calculus, we obtain by integrating along the ray \( \{ ... | Yes |
Lemma 10.2.4. For \( f \in {L}^{1}\left( \Omega \right) ,0 < \mu \leq 1 \), let\n\n\[ \n\left( {{V}_{\mu }f}\right) \left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }{\left| x - y\right| }^{d\left( {\mu - 1}\right) }f\left( y\right) \mathrm{d}y.\n\]\n\nThen\n\n\[ \n{\begin{Vmatrix}{V}_{\mu }f\end{Vmatrix}}_{{L}^{2}\... | Proof. \( B\left( {x, R}\right) \mathrel{\text{:=}} \left\{ {y \in {\mathbb{R}}^{d} : \left| {x - y}\right| \leq R}\right\} \) . Let \( R \) be chosen such that \( \left| \Omega \right| = \) \( \left| {B\left( {x, R}\right) }\right| = {\omega }_{d}{R}^{d} \) . Since in that case\n\n\[ \n\left| {\Omega \smallsetminus \l... | Yes |
Theorem 10.2.3. Let \( \Omega \in {\mathbb{R}}^{d} \) be open and bounded. Then \( {H}_{0}^{1,2}\left( \Omega \right) \) is compactly embedded in \( {L}^{2}\left( \Omega \right) \) ; i.e., any sequence \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \subset {H}_{0}^{1,2}\left( \Omega \right) \) with | Proof. The strategy is to find functions \( {w}_{n,\varepsilon } \in {C}^{1}\left( \Omega \right) \), for every \( \varepsilon > 0 \), with | No |
Lemma 10.3.1 (Stability lemma). Let \( {u}_{i = 1,2} \) be a weak solution of \( \Delta {u}_{i} = {f}_{i} \) with \( {u}_{1} - {u}_{2} \in {H}_{0}^{1,2}\left( \Omega \right) \) . Then\n\n\[ \n{\begin{Vmatrix}{u}_{1} - {u}_{2}\end{Vmatrix}}_{{W}^{1,2}\left( \Omega \right) } \leq \text{ const }{\begin{Vmatrix}{f}_{1} - {... | Proof. We have\n\n\[ \n{\int }_{\Omega }D\left( {{u}_{1} - {u}_{2}}\right) {Dv} = - {\int }_{\Omega }\left( {{f}_{1} - {f}_{2}}\right) v\;\text{ for all }v \in {H}_{0}^{1,2}\left( \Omega \right) ,\n\]\n\nand thus in particular,\n\n\[ \n{\int }_{\Omega }D\left( {{u}_{1} - {u}_{2}}\right) D\left( {{u}_{1} - {u}_{2}}\righ... | Yes |
Theorem 10.5.1. Let \( \left( {H,\left( {\cdot , \cdot }\right) }\right) \) be a Hilbert space with norm \( \parallel \cdot \parallel, V \subset H \) convex and closed, \( A : H \times H \rightarrow \mathbb{R} \) a continuous symmetric elliptic bilinear form, \( L \) : \( H \rightarrow \mathbb{R} \) a continuous linear... | Proof. By ellipticity of \( A, J \) is bounded from below, namely,\n\n\[ J\left( v\right) \geq \lambda \parallel v{\parallel }^{2} - \parallel L\parallel \parallel v\parallel \geq - \frac{\parallel L{\parallel }^{2}}{4\lambda }. \]\n\nWe put\n\n\[ \kappa \mathrel{\text{:=}} \mathop{\inf }\limits_{{v \in V}}J\left( v\ri... | Yes |
Corollary 10.5.1. The other assumptions of the previous theorem remaining in force, now let \( V \) be a closed linear (hence convex) subspace of \( H \) . Then there exists precisely one \( u \in V \) that solves\n\n\[ \n{2A}\left( {u,\varphi }\right) + L\left( \varphi \right) = 0\;\text{ for all }\varphi \in V.\n\]\n... | Proof. The point \( u \) is a critical point (e.g., a minimum) of the functional\n\n\[ \nJ\left( v\right) = A\left( {v, v}\right) + L\left( v\right)\n\]\n\nin \( V \) precisely if\n\n\[ \n{2A}\left( {v,\varphi }\right) + L\left( \varphi \right) = 0\;\text{ for all }\varphi \in V.\n\]\n\nNamely, that \( u \) is a critic... | Yes |
Let \( A : H \times H \rightarrow \mathbb{R} \) be a continuous, symmetric, elliptic, bilinear form in the sense of Definition 10.5.1, and let \( L : H \rightarrow \mathbb{R} \) be linear and continuous. We consider once more the problem\n\n\[ J\left( v\right) \mathrel{\text{:=}} A\left( {v, v}\right) + L\left( v\right... | Proof. By Corollary 10.5.1,\n\n\[ {2A}\left( {u,\varphi }\right) + L\left( \varphi \right) = 0\;\text{ for all }\varphi \in H, \]\n\n\[ {2A}\left( {{u}_{V},\varphi }\right) + L\left( \varphi \right) = 0\;\text{ for all }\varphi \in V, \]\n\nhence also\n\n\[ {2A}\left( {u - {u}_{V},\varphi }\right) = 0\;\text{ for all }... | Yes |
Theorem 10.5.2. Let \( A : H \times H \rightarrow \mathbb{R} \) be a continuous, symmetric, elliptic, bilinear form on the Hilbert space \( \left( {H,\left( {\cdot , \cdot }\right) }\right) \) with norm \( \parallel \cdot \parallel \), and let \( L : H \rightarrow \mathbb{R} \) be linear and continuous. We consider the... | Proof. Let\n\n\[ \kappa \mathrel{\text{:=}} \mathop{\inf }\limits_{{v \in H}}J\left( v\right) \]\n\nWe want to show that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}J\left( {u}_{n}\right) = \kappa \]\n\nIn that case, \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) will be a minimizing sequence for \( J \) in... | Yes |
Theorem 10.6.1. Let \( \Omega \subset {\mathbb{R}}^{d} \) be open, and consider a function\n\n\[ f : \Omega \times {\mathbb{R}}^{d} \rightarrow \mathbb{R} \]\n\nsatisfying:\n\n(i) \( f\left( {\cdot, v}\right) \) is measurable for all \( v \in {\mathbb{R}}^{d} \).\n\n(ii) \( f\left( {x, \cdot }\right) \) is convex for a... | To simplify our further considerations, we first observe that it suffices to consider the case \( g = 0 \). Namely, otherwise, we consider, for \( w = u - g \), \n\n\[ \widetilde{f}\left( {x, w\left( x\right) }\right) \mathrel{\text{:=}} f\left( {x, w\left( x\right) + g\left( x\right) }\right) .\n\nThe function \( \wid... | No |
Lemma 10.6.1. Suppose that \( f \) is as in Theorem 10.6.1, but with (ii) weakened to (ii’) \( f\left( {x, \cdot }\right) \) is continuous for all \( x \in \Omega \) , and supposing in (iii) only \( \kappa \in \mathbb{R} \), but not necessarily \( \kappa > 0 \) . Then \[ J\left( v\right) \mathrel{\text{:=}} {\int }_{\O... | Proof. We first observe that if \( v \) is in \( {L}^{2} \), it is measurable, and since \( f\left( {x, v}\right) \) is continuous with respect to \( v, f\left( {x, v\left( x\right) }\right) \) then is measurable by a basic result in Lebesgue integration theory. \( {}^{5} \) Now let \( {\left( {v}_{n}\right) }_{n \in \... | Yes |
Lemma 10.6.2. Let \( f \) be as in Theorem 10.6.1, without necessarily requiring \( \kappa \) in (iii) to be positive. Then\n\n\[ J\left( v\right) = {\int }_{\Omega }f\left( {x, v\left( x\right) }\right) \mathrm{d}x \]\n\nis convex on \( {L}^{2}\left( {\Omega ;{\mathbb{R}}^{d}}\right) \) . | Proof. Let \( {v}_{0},{v}_{1} \in {L}^{2}\left( {\Omega ,{\mathbb{R}}^{d}}\right) ,0 \leq t \leq 1 \) . We have\n\n\[ J\left( {t{v}_{0} + \left( {1 - t}\right) {v}_{1}}\right) = \int f\left( {x, t{v}_{0}\left( x\right) + \left( {1 - t}\right) {v}_{1}\left( x\right) }\right) \]\n\n\[ \leq \int \left( {{tf}\left( {x,{v}_... | Yes |
Lemma 10.6.3. Let \( f \) be as in Theorem 10.6.1, still not necessarily requiring \( \kappa > 0 \) . With our previous simplification \( g = 0 \) (10.6.3), the functional \[ I\left( u\right) = {\int }_{\Omega }f\left( {x,{Du}\left( x\right) }\right) \mathrm{d}x \] is a convex and lower semicontinuous functional on \( ... | Proof. We first verify the auxiliary statement about the uniqueness and existence of \( {u}_{\lambda } \) . We let \( {\left( {y}_{n}\right) }_{n \in \mathbb{N}} \) be a minimizing sequence for (10.6.4), i.e., \[ I\left( {y}_{n}\right) + \lambda {\begin{Vmatrix}u - {y}_{n}\end{Vmatrix}}^{2} \rightarrow \mathop{\inf }\l... | Yes |
Theorem 11.1.1.\n\n\[ \n{H}_{0}^{1, p}\left( \Omega \right) \subset \left\{ \begin{array}{ll} {L}^{\frac{dp}{d - p}}\left( \Omega \right) & \text{ for }p < d, \\ {C}^{0}\left( \bar{\Omega }\right) & \text{ for }p > d. \end{array}\right.\n\]\n\nMoreover, for \( u \in {H}_{0}^{1, p}\left( \Omega \right) \) ,\n\n\[ \n\par... | Proof of Theorem 11.1.1: We shall first prove the inequalities (11.1.2) and (11.1.3) for \( u \in {C}_{0}^{1}\left( \Omega \right) \) . We put \( u = 0 \) on \( {\mathbb{R}}^{d} \smallsetminus \Omega \) again. As in the proof of Theorem 10.2.2,\n\n\[ \n\left| {u\left( x\right) }\right| \leq {\int }_{-\infty }^{{x}^{i}}... | Yes |
Lemma 11.1.2. For \( \mu \in (0,1\rbrack, f \in {L}^{1}\left( \Omega \right) \) let\n\n\[ \n\left( {{V}_{\mu }f}\right) \left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }{\left| x - y\right| }^{d\left( {\mu - 1}\right) }f\left( y\right) \mathrm{d}y.\n\]\n\nLet \( 1 \leq p \leq q \leq \infty \), \n\n\[ \n0 \leq \delt... | Proof. Let\n\n\[ \n\frac{1}{r} \mathrel{\text{:=}} 1 + \frac{1}{q} - \frac{1}{p} = 1 - \delta \n\]\n\nThen\n\n\[ \n\ell \left( {x - y}\right) \mathrel{\text{:=}} {\left| x - y\right| }^{d\left( {\mu - 1}\right) } \in {L}^{r}\left( \Omega \right) , \n\]\n\nand as in the proof of Lemma 10.2.4, we choose \( R \) such that... | Yes |
Corollary 11.1.1.\n\n\[ \n{H}_{0}^{k, p}\left( \Omega \right) \subset \left\{ {\begin{array}{ll} {L}^{\frac{dp}{d - {kp}}}\left( \Omega \right) & \text{ for }{kp} < d, \\ {C}^{m}\left( \Omega \right) & \text{ for }0 \leq m < k - \frac{d}{p}. \end{array}.}\right. \n\] | Proof. The first embedding iteratively follows from Theorem 11.1.1, and the second one then from the first and the case \( p > d \) in Theorem 11.1.1. | No |
Corollary 11.1.3. Let \( 1 \leq p < d \) and \( u \in {H}^{1, p}\left( {B\left( {{x}_{0}, R}\right) }\right) \) . Then\n\n\[{\left( {\int }_{B\left( {{x}_{0}, R}\right) }{\left| u\right| }^{\frac{dp}{d - p}}\right) }^{\frac{d - p}{dp}} \leq {c}_{0}{\left( {R}^{p}{\int }_{B\left( {{x}_{0}, R}\right) }{\left| Du\right| }... | Proof. Without loss of generality, \( {x}_{0} = 0 \) . Likewise, we may assume \( R = 1 \) , since we may consider the functions \( \widetilde{u}\left( x\right) = u\left( {Rx}\right) \) and check that the expressions in (11.1.9) scale in the right way. Thus, let \( u \in {H}^{1, p}\left( {B\left( {0,1}\right) }\right) ... | Yes |
Lemma 11.1.3. Let \( \Omega \subset {\mathbb{R}}^{d} \) be convex, \( B \subset \Omega \) measurable with \( \left| B\right| > 0, u \in \) \( {W}^{1,1}\left( \Omega \right) \) . Then we have for almost all \( x \in \Omega \) ,\n\n\[ \left| {u\left( x\right) - {u}_{B}}\right| \leq \frac{{\left( \operatorname{diam}\Omega... | Proof. As before, it suffices to prove the inequality for \( u \in {C}^{1}\left( \Omega \right) \) . Since \( \Omega \) is convex, if \( x \) and \( y \) are contained in \( \Omega \), so is the straight line joining them, and we have\n\n\[ u\left( x\right) - u\left( y\right) = - {\int }_{0}^{\left| x - y\right| }\frac... | Yes |
Lemma 11.1.4. Let \( f \in {L}^{1}\left( \Omega \right) \), and suppose that for all balls \( B\left( {{x}_{0}, R}\right) \subset {\mathbb{R}}^{d} \), \[ {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }\left| f\right| \leq K{R}^{d\left( {1 - \frac{1}{p}}\right) } \] with some fixed \( K \). Moreover, let \( p > 1,1/p... | Proof. We put \( f = 0 \) in the exterior of \( \Omega \). With \( r = \left| {x - y}\right| \), then \[ \left| {{V}_{\mu }f\left( x\right) }\right| \leq {\int }_{\Omega }{r}^{d\left( {\mu - 1}\right) }\left| {f\left( y\right) }\right| \mathrm{d}y \] \[ = {\int }_{0}^{\operatorname{diam}\Omega }{r}^{d\left( {\mu - 1}\r... | Yes |
Corollary 11.1.4. Let \( \Omega \subset {\mathbb{R}}^{d} \) be convex, and \( u \in {W}^{1, p}\left( \Omega \right) \) . We then have for every measurable \( B \subset \Omega \) with \( \left| B\right| > 0 \) ,\n\n\[ \n{\left( {\int }_{\Omega }{\left| u - {u}_{B}\right| }^{p}\right) }^{\frac{1}{p}} \leq \frac{{\omega }... | Proof. By Lemma 11.1.3,\n\n\[ \n\left| {u\left( x\right) - {u}_{B}}\right| \leq \frac{{\left( \operatorname{diam}\Omega \right) }^{d}}{d\left| B\right| }{V}_{\frac{1}{d}}\left( \left| {Du}\right| \right) ,\n\]\n\nand by Lemma 11.1.2, then,\n\n\[ \n{\begin{Vmatrix}{V}_{\frac{1}{d}}\left( \left| Du\right| \right) \end{Vm... | Yes |
Theorem 11.1.3. Assume \( u \in {W}^{1,1}\left( \Omega \right) ,\Omega \subset {\mathbb{R}}^{d} \), and that there exist constants \( K < \infty ,0 < \alpha < 1 \), such that for all balls \( B\left( {{x}_{0}, R}\right) \subset {\mathbb{R}}^{d} \) ,\n\n\[ \n{\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }\left| {Du}\... | Proof. We have\n\n\[ \n\mathop{\operatorname{osc}}\limits_{{\Omega \cap B\left( {z, r}\right) }}u \leq 2\mathop{\sup }\limits_{{x \in B\left( {z, r}\right) \cap \Omega }}\left| {u\left( x\right) - {u}_{B\left( {z, r}\right) }}\right|\n\]\n\n\[ \n\leq {c}_{1}\mathop{\sup }\limits_{{x \in B\left( {z, r}\right) \cap \Omeg... | Yes |
Corollary 11.1.5. Let \( u \in {H}_{0}^{1, p}\left( \Omega \right) \) with \( p > d \) . Then\n\n\[ u \in {C}^{1 - \frac{d}{p}}\left( \bar{\Omega }\right) . \]\n\nMore precisely, for every ball \( B\left( {z, r}\right) \subset {\mathbb{R}}^{d} \),\n\n\[ \mathop{\operatorname{osc}}\limits_{{\Omega \cap B\left( {z, r}\ri... | Proof of Corollary 11.1.5: By Hölder's inequality\n\n\[ {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }\left| {Du}\right| \leq {\left| B\left( {x}_{0}, R\right) \right| }^{1 - \frac{1}{p}}{\left( {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }{\left| Du\right| }^{p}\right) }^{\frac{1}{p}} \]\n\n(11.1.34)\n\n\[ \l... | Yes |
Corollary 11.1.6. Let \( u \in {W}^{1,2}\left( \Omega \right) \), and suppose there exist constants \( {K}^{\prime } < \infty \) , \( 0 < \alpha < 1 \) such that for all balls \( B\left( {{x}_{0}, R}\right) \subset {\mathbb{R}}^{d} \), \[ {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }{\left| Du\right| }^{2} \leq {K... | Proof. By Hölder's inequality \[ {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }\left| {Du}\right| \leq {\left| B\left( {x}_{0}, R\right) \right| }^{\frac{1}{2}}{\left( {\int }_{\Omega \cap B\left( {{x}_{0}, R}\right) }{\left| Du\right| }^{2}\right) }^{\frac{1}{2}} \] \[ \leq {c}_{4}{\left( {K}^{\prime }\right) }^{\... | Yes |
Lemma 11.2.1. Assume \( u \in {W}^{1,2}\left( \Omega \right) ,{\Omega }^{\prime } \subset \subset \Omega ,\left| h\right| < \operatorname{dist}\left( {{\Omega }^{\prime },\partial \Omega }\right) \) . Then \( {\Delta }_{i}^{h}u \in {L}^{2}\left( {\Omega }^{\prime }\right) \) and\n\n\[ \n{\begin{Vmatrix}{\Delta }_{i}^{h... | Proof. By an approximation argument, it again suffices to consider the case \( u \in \) \( {C}^{1}\left( \Omega \right) \cap {W}^{1,2}\left( \Omega \right) \) . Then\n\n\[ \n{\Delta }_{i}^{h}u\left( x\right) = \frac{u\left( {x + h{e}_{i}}\right) - u\left( x\right) }{h}\n\]\n\n\[ \n= \frac{1}{h}{\int }_{0}^{h}{D}_{i}u\l... | Yes |
Lemma 11.2.2. Let \( u \in {L}^{2}\left( \Omega \right) \), and suppose there exists \( K < \infty \) with \( {\Delta }_{i}^{h}u \in \) \( {L}^{2}\left( {\Omega }^{\prime }\right) \) and\n\n\[ \n{\begin{Vmatrix}{\Delta }_{i}^{h}u\end{Vmatrix}}_{{L}^{2}\left( {\Omega }^{\prime }\right) } \leq K \n\]\n\n(11.2.2)\n\nfor a... | Proof. For \( \varphi \in {C}_{0}^{1}\left( \Omega \right) \) and \( 0 < h < \operatorname{dist}\left( {\operatorname{supp}\varphi ,\partial \Omega }\right) \) (supp \( \varphi \) is the closure of \( \{ x \in \Omega : \varphi \left( x\right) \neq 0\} ) \), we have\n\n\[ \n{\int }_{\Omega }{\Delta }_{i}^{h}{u\varphi } ... | Yes |
Lemma 11.2.3. Let \( u \) be a weak solution of \( {\Delta u} = f \) with \( f \in {L}^{2}\left( \Omega \right) \) . We then have for any \( {\Omega }^{\prime } \subset \subset \Omega \), \[ \parallel {Du}{\parallel }_{{L}^{2}\left( {\Omega }^{\prime }\right) }^{2} \leq \frac{17}{{\delta }^{2}}\parallel u{\parallel }_{... | So far, we have not used that we are temporarily assuming \( u \in {W}^{2,2}\left( {\Omega }^{\prime }\right) \) for any \( {\Omega }^{\prime } \subset \subset \Omega \). Now, however, we come to the estimate of the \( {W}^{2,2} \) -norm, so we shall need that assumption. Let \( u \in {W}^{2,2}\left( {\Omega }^{\prime ... | Yes |
Theorem 11.2.2. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak solution of \( {\Delta u} = f, f \in {W}^{k,2}\left( \Omega \right) \). For any \( {\Omega }^{\prime } \subset \subset \Omega \) then \( u \in {W}^{k + 2,2}\left( {\Omega }^{\prime }\right) \), and\n\n\[ \parallel u{\parallel }_{{W}^{k + 2,2}\left... | Proof. From Theorem 11.2.2 and Corollary 11.1.2. | No |
Theorem 11.2.2. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak solution of \( {\Delta u} = f, f \in {W}^{k,2}\left( \Omega \right) \). For any \( {\Omega }^{\prime } \subset \subset \Omega \) then \( u \in {W}^{k + 2,2}\left( {\Omega }^{\prime }\right) \), and | \[ \parallel u{\parallel }_{{W}^{k + 2,2}\left( {\Omega }^{\prime }\right) } \leq \operatorname{const}\left( {\parallel u{\parallel }_{{L}^{2}\left( \Omega \right) } + \parallel f{\parallel }_{{W}^{k,2}\left( \Omega \right) }}\right) ,\] where the constant depends on \( d, k \), and \( \operatorname{dist}\left( {{\Omeg... | Yes |
Corollary 11.2.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{\infty }\left( \Omega \right) \), then also \( u \in {C}^{\infty }\left( \Omega \right) \) . | Proof. From Theorem 11.2.2 and Corollary 11.1.2. | No |
Lemma 11.3.1. Let \( u \) be a weak solution of \( {\Delta u} = f, u - g \in {H}_{0}^{1,2}\left( \Omega \right) \) in the bounded region \( \Omega \right) . Then\n\n\[ \parallel u{\parallel }_{{W}^{1,2}\left( \Omega \right) } \leq c\left( {\parallel g{\parallel }_{{W}^{1,2}\left( \Omega \right) } + \parallel f{\paralle... | Proof. We insert the test function \( v = u - g \) into the weak differential equation\n\n\[ {\int }_{\Omega }{Du} \cdot {Dv} = - {\int }_{\Omega }{fv}\;\text{ for all }v \in {H}_{0}^{1,2}\left( \Omega \right) \]\n\nto obtain\n\n\[ {\int }_{\Omega }{\left| Du\right| }^{2} = \int {Du} \cdot {Dg} - \int {fu} + \int {fg} ... | Yes |
Theorem 11.3.1. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak solution of \( {Lu} = f \) ; i.e., let (11.3.12) hold. Let the ellipticity assumption (A11.3) hold. Moreover, let all coefficients \( {a}^{ij}\left( x\right) ,\ldots, d\left( x\right) \) as well as \( f\left( x\right) \) be of class \( {C}^{\infty... | Let us discuss the Proof of Theorem 11.3.1: We first reduce the proof to the case \( {b}^{j},{c}^{i}, d \equiv 0 \), i.e., to the regularity of weak solutions of\n\n\[ \n{Mu} \mathrel{\text{:=}} \mathop{\sum }\limits_{{i, j}}\frac{\partial }{\partial {x}^{j}}\left( {{a}^{ij}\left( x\right) \frac{\partial }{\partial {x}... | Yes |
Theorem 11.3.3. Let \( u \) be a weak solution of \( {Mu} = f \) in \( \Omega \) with \( u - g \in {H}_{0}^{1,2}\left( \Omega \right) \) . As always, suppose (A11.3). Let \( f \in {W}^{k,2}\left( \Omega \right), g \in {W}^{k + 2,2}\left( \Omega \right) \) . Let \( \Omega \) be of class \( {C}^{k + 2} \), and let the co... | Proof. As explained at the beginning of this section, we may assume that \( \partial \Omega \) is locally a hyperplane, by considering the composition \( u \circ {\phi }^{-1} \) in place of \( u \), where \( \phi \) is a diffeomorphism of the type described in Definition 11.3.1. Namely, by (10.4.12), our equation \( {M... | No |
Theorem 11.4.3. Let \( \Omega \subset {\mathbb{R}}^{d} \) be a bounded domain of class \( {C}^{\infty } \), and let \( g \in \) \( {C}^{\infty }\left( {\partial \Omega }\right), f \in {C}^{\infty }\left( \bar{\Omega }\right) \) . Then the Dirichlet problem\n\n\[ \n{\Delta u} = f\;\text{ in }\Omega , \n\]\n\n\[ \nu = g\... | Proof. As explained in the beginning of Sect. 11.3, we may restrict ourselves to the case where \( g = 0 \), by considering \( \bar{u} = u - g \) in place of \( u \), where we have extended \( g \) as a \( {C}^{\infty } \) -function to all of \( \bar{\Omega } \) . (Since \( \bar{\Omega } \) is bounded, \( {C}^{\infty }... | Yes |
Theorem 11.4.5. Let (11.4.7) be satisfied for all \( v \in {W}^{1,2}\left( \Omega \right) \), on some \( {C}^{\infty } \) - domain \( \Omega \), for some function \( f \in {C}^{\infty }\left( \bar{\Omega }\right) \) . Then also\n\n\[ u \in {C}^{\infty }\left( \bar{\Omega }\right) \text{.} \] | The Proof follows the scheme presented in Sect. 11.3. We obtain differentiability results on the boundary \( \partial \Omega \) (note that here we conclude that \( u \) is smooth even on the boundary and not only in \( \Omega \) as in Theorem 11.3.1) by applying the version stated in Theorem 11.4.1 of the Sobolev embed... | Yes |
Corollary 11.4.1. Let \( u \) be a solution of (11.4.8), for all \( v \in {W}^{1,2}\left( \Omega \right) \) . If the domain \( \Omega \) is of class \( {C}^{\infty } \), then \( u \in {C}^{\infty }\left( \bar{\Omega }\right) \) . | We return to the equation\n\n\[ \n{\int }_{\Omega }{Du} \cdot {Dv} + {\int }_{\Omega }{fv} = 0 \n\]\n\non a \( {C}^{\infty } \) -domain \( \Omega \), for \( f \in {C}^{\infty }\left( \bar{\Omega }\right) \) . Since \( u \) is smooth up to the boundary by Theorem 11.4.5, we may integrate by parts to obtain\n\n\[ \n- {\i... | Yes |
Theorem 11.5.1. Let \( \Omega \subset {\mathbb{R}}^{d} \) be connected, open, and bounded. Then the eigenvalue problem\n\n\[ \n{\Delta u} + {\lambda u} = 0,\;u \in {H}_{0}^{1,2}\left( \Omega \right)\n\]\n\nhas countably many eigenvalues\n\n\[ \n0 < {\lambda }_{1} < {\lambda }_{2} \leq \cdots \leq {\lambda }_{m} \leq \c... | The Proofs of Theorems 11.5.1 and 11.5.2 are now easy: We first check\n\n\[ \n\mathop{\lim }\limits_{{m \rightarrow \infty }}{\lambda }_{m} = \infty\n\]\n\nIndeed, otherwise,\n\n\[ \n\begin{Vmatrix}{D{u}_{m}}\end{Vmatrix} \leq c\;\text{ for all }m\text{ and some constant }\mathrm{c}.\n\]\n\nBy Rellich’s theorem again, ... | Yes |
Theorem 11.5.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be bounded, open, and of class \( {C}^{\infty } \). Then the eigenvalue problem\n\n\[ \n{\Delta u} + {\lambda u} = 0,\;u \in {W}^{1,2}\left( \Omega \right)\n\]\nhas countably many eigenvalues\n\n\[ \n0 = {\lambda }_{0} \leq {\lambda }_{1} \leq \cdots \leq {\lamb... | The Proofs of Theorems 11.5.1 and 11.5.2 are now easy: We first check\n\n\[ \n\mathop{\lim }\limits_{{m \rightarrow \infty }}{\lambda }_{m} = \infty\n\]\n\nIndeed, otherwise,\n\n\[ \n\begin{Vmatrix}{D{u}_{m}}\end{Vmatrix} \leq c\;\text{ for all }m\text{ and some constant }\mathrm{c}.\n\]\n\nBy Rellich’s theorem again, ... | Yes |
Corollary 11.5.1. For \( v \in {H}_{0}^{1,2}\left( \Omega \right) \) , \[ {\lambda }_{1}\langle v, v\rangle \leq \langle {Dv},{Dv}\rangle \] where \( {\lambda }_{1} \) is the first Dirichlet eigenvalue according to Theorem 11.5.1. | Proof. The inequalities (11.5.23) and (11.5.24) readily follow from (11.5.14), noting that in the second case, \( v - \bar{v} \) is orthogonal to the constants, the eigenfunctions for \( {\lambda }_{0} = 0 \), since \[ {\int }_{\Omega }\left( {v\left( x\right) - \bar{v}}\right) \mathrm{d}x = 0 \] (11.5.26) As an altern... | Yes |
Theorem 11.5.3. Under the above assumptions, let \( {P}^{k} \) be the collection of all \( k \) - dimensional linear subspaces of the Hilbert space \( H \) . Then the \( k \) th eigenvalue of \( \Delta \) (i.e., \( {\lambda }_{k} \) in the Dirichlet case, \( {\lambda }_{k - 1} \) in the Neumann case) is characterized a... | Proof. We have seen that\n\n\[ {\lambda }_{m} = \min \left\{ {\frac{\langle {Du},{Du}\rangle }{\langle u, u\rangle } : u \neq 0, u\text{ orthogonal to the }{u}_{i}\text{ with }i \leq m - 1}\right\} .\n\]\n\n(11.5.30)\n\nIt is also clear that\n\n\[ {\lambda }_{m} = \max \left\{ {\frac{\langle {Du},{Du}\rangle }{\langle ... | Yes |
Corollary 11.5.2. Under the above assumptions, we let \( 0 < {\lambda }_{1}^{D} \leq {\lambda }_{2}^{D} \leq \cdots \) be the Dirichlet eigenvalues, and \( 0 = {\lambda }_{0}^{N} < {\lambda }_{1}^{N} \leq {\lambda }_{2}^{N} \leq \cdots \) be the Neumann eigenvalues. Then\n\n\[ \n{\lambda }_{j - 1}^{N} \leq {\lambda }_{... | Proof. The Hilbert space for the Dirichlet case, namely, \( {H}_{0}^{1,2}\left( \Omega \right) \), is a subspace of that for the Neumann case, namely, \( {W}^{1,2}\left( \Omega \right) \), and so (11.5.33) applies. | Yes |
Corollary 11.5.3. Let \( {\Omega }_{1} \subset {\Omega }_{2} \) be bounded open subsets of \( {\mathbb{R}}^{d} \) . We denote the eigenvalues for the Dirichlet case of the domain \( \Omega \) by \( {\lambda }_{k}\left( \Omega \right) \) . Then\n\n\[ \n{\lambda }_{k}\left( {\Omega }_{2}\right) \leq {\lambda }_{k}\left( ... | Proof. Any \( v \in {H}_{0}^{1,2}\left( {\Omega }_{1}\right) \) can be extended to a function \( \widetilde{v} \in {H}_{0}^{1,2}\left( {\Omega }_{2}\right) \), simply by putting\n\n\[ \n\widetilde{v}\left( x\right) = \left\{ \begin{array}{ll} v\left( x\right) & \text{ for }x \in {\Omega }_{1}, \\ 0 & \text{ for }x \in ... | Yes |
Theorem 11.5.4. Let \( {\lambda }_{1} \) be the first eigenvalue of \( \Delta \) on the open and bounded domain \( \Omega \subset {\mathbb{R}}^{d} \) with Dirichlet boundary conditions. Then \( {\lambda }_{1} \) is a simple eigenvalue, meaning that the corresponding eigenspace is one-dimensional. Moreover, an eigenfunc... | Proof. Let\n\n\[ \Delta {u}_{1} + {\lambda }_{1}{u}_{1} = 0\;\text{ in }\Omega . \]\n\nBy Corollary 10.2.2, we know that \( \left| {u}_{1}\right| \in {W}^{1,2}\left( \Omega \right) \), and\n\n\[ \frac{\left\langle D\left| {u}_{1}\right|, D\left| {u}_{1}\right| \right\rangle }{\left\langle \left| {u}_{1}\right| ,\left| ... | Yes |
Corollary 12.1.1. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak solution of \( {\Delta u} = f \), for \( f \in \) \( {C}^{\infty }\left( \Omega \right) \) . Then \( u \in {C}^{\infty }\left( \Omega \right) \) . | Proof. Theorem 12.1.3 and Corollary 11.1.2. | No |
Theorem 12.2.1. Let \( 1 < p < \infty, f \in {L}^{p}\left( \Omega \right) \) ( \( \Omega \subset {\mathbb{R}}^{d} \) open and bounded), and let \( w \) be the Newton potential (12.1.1) of \( f \) . Then \( w \in {W}^{2, p}\left( \Omega \right) ,{\Delta w} = f \) almost everywhere in \( \Omega \), and\n\n\[{\begin{Vmatr... | In contrast to the case \( p = 2 \), i.e., Theorem 12.1.1, where \( c\left( {d,2}\right) = 1 \) for all \( d \) and the proof is elementary, the proof of the general case is relatively involved; we refer the reader to Bers-Schechter [2] or Gilbarg-Trudinger [12]. | No |
Theorem 12.2.2. Let \( u \in {W}^{1,1}\left( \Omega \right) \) be a weak solution of \( {\Delta u} = f, f \in {L}^{p}\left( \Omega \right) \) , \( 1 < p < \infty \), i.e., \[ \int {Du} \cdot {D\varphi } = - \int {f\varphi }\;\text{ for all }\varphi \in {C}_{0}^{\infty }\left( \Omega \right) . \] Then \( u \in {W}^{2, p... | We do not provide a complete proof of this result either. This time, however, we shall present at least a sketch of the proof. Apart from the fact that (12.1.8) needs to be replaced by the inequality \[ {\begin{Vmatrix}{D}^{2}v\end{Vmatrix}}_{{L}^{p}\left( {B\left( {x, R}\right) }\right) } \leq \text{ const. }\parallel... | No |
Lemma 13.1.1. If \( {f}_{1},{f}_{2} \in {C}^{\alpha }\left( G\right) \) on \( G \subset {\mathbb{R}}^{d} \), then \( {f}_{1}{f}_{2} \in {C}^{\alpha }\left( G\right) \), and\n\n\[ \n{\left| {f}_{1}{f}_{2}\right| }_{{C}^{\alpha }\left( G\right) } \leq \left( {\mathop{\sup }\limits_{G}\left| {f}_{1}\right| }\right) {\left... | Proof.\n\n\[ \n\frac{\left| {f}_{1}\left( x\right) {f}_{2}\left( x\right) - {f}_{1}\left( y\right) {f}_{2}\left( y\right) \right| }{{\left| x - y\right| }^{\alpha }} \leq \frac{\left| {f}_{1}\left( x\right) - {f}_{1}\left( y\right) \right| }{{\left| x - y\right| }^{\alpha }}\left| {{f}_{2}\left( x\right) }\right| + \fr... | Yes |
Theorem 13.1.1. As always, let \( \Omega \subset {\mathbb{R}}^{d} \) be open and bounded,\n\n\[ u\left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }\Gamma \left( {x, y}\right) f\left( y\right) \mathrm{d}y \]\n\n(13.1.3)\n\nwhere \( \Gamma \) is the fundamental solution defined in Sect. 2.1.\n\n(a) If \( f \in {L}^{\i... | Proof. (a) Up to a constant factor, the first derivatives of \( u \) are given by\n\n\[ {v}^{i}\left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }\frac{{x}^{i} - {y}^{i}}{{\left| x - y\right| }^{d}}f\left( y\right) \mathrm{d}y\;\left( {i = 1,\ldots, d}\right) . \]\n\nFrom this formula,\n\n\[ \left| {{v}^{i}\left( {x}... | Yes |
Theorem 13.1.2. As always, let \( \Omega \subset {\mathbb{R}}^{d} \) be open and bounded, and \( {\Omega }_{0} \subset \subset \Omega \) . Let \( u \) be a weak solution of \( {\Delta u} = f \) in \( \Omega \) .\n\n(a) If \( f \in {C}^{0}\left( \Omega \right) \), then \( u \in {C}^{1,\alpha }\left( \Omega \right) \), a... | Proof. We demonstrate the estimates (13.1.20) and (13.1.21) first under the assumption \( u \in {C}^{2,\alpha }\left( \Omega \right) \) . We may cover \( {\Omega }_{0} \) by finitely many balls that are contained in \( \Omega \) . Therefore, it suffices to verify the estimates for the case\n\n\[ {\Omega }_{0} = B\left(... | Yes |
(a) There exists a constant \( {c}_{a} \) such that for every \( \rho > 0 \) and any function \( v \in \) \( {C}^{1}\left( {B\left( {0,\rho }\right) }\right) \) :\n\n\[ \parallel v{\parallel }_{{C}^{0}\left( {B\left( {0,\rho }\right) }\right) } \leq \parallel {Dv}{\parallel }_{{C}^{0}\left( {B\left( {0,\rho }\right) }\... | Proof. If (a) did not hold, for every \( n \in \mathbb{N} \), we could find a radius \( {\rho }_{n} \) and a function \( {v}_{n} \in {C}^{1}\left( {B\left( {0,{\rho }_{n}}\right) }\right) \) with\n\n\[ 1 = {\begin{Vmatrix}{v}_{n}\end{Vmatrix}}_{{C}^{0}\left( {B\left( {0,{\rho }_{n}}\right) }\right) } \geq {\begin{Vmatr... | Yes |
Theorem 13.1.3. Let \( u \) be a weak solution of \( {\Delta u} = f \) in \( \Omega \) ( \( \Omega \) a bounded domain in \( {\mathbb{R}}^{d} \) ), \( f \in {L}^{p}\left( \Omega \right) \) for some \( p > d,{\Omega }_{0} \subset \subset \Omega \) . Then \( u \in {C}^{1,\alpha }\left( \Omega \right) \) for some \( \alph... | Proof. Again, we consider the Newton potential\n\n\[ w\left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }\Gamma \left( {x, y}\right) f\left( y\right) \mathrm{d}y \]\n\nand\n\n\[ {v}^{i}\left( x\right) \mathrel{\text{:=}} {\int }_{\Omega }\frac{{x}^{i} - {y}^{i}}{{\left( x - y\right) }^{d}}f\left( y\right) \mathrm{d}y... | Yes |
Corollary 13.1.1. If \( u \in {W}^{1,2}\left( \Omega \right) \) is a weak solution of \( {\Delta u} = f \) with \( f \in \) \( {C}^{k,\alpha }\left( \Omega \right), k \in \mathbb{N},0 < \alpha < 1 \), then \( u \in {C}^{k + 2,\alpha }\left( \Omega \right) \), and for \( {\Omega }_{0} \subset \subset \Omega \) , \[ \par... | Proof. Since \( u \in {C}^{2,\alpha }\left( \Omega \right) \) by Theorem 13.1.2, we know that it weakly solves \[ \Delta \frac{\partial }{\partial {x}^{i}}u = \frac{\partial }{\partial {x}^{i}}f \] Theorem 13.1.2 then implies \[ \frac{\partial }{\partial {x}^{i}}u \in {C}^{2,\alpha }\left( \Omega \right) \;\left( {i \i... | No |
Theorem 13.2.1. Let \( f \in {C}^{\alpha }\left( \Omega \right) \), and suppose \( u \in {C}^{2,\alpha }\left( \Omega \right) \) satisfies\n\n\[ \n{Lu} = f \n\]\n\nin \( \Omega \left( {0 < \alpha < 1}\right) \) . For any \( {\Omega }_{0} \subset \subset \Omega \), we then have\n\n\[ \n\parallel u{\parallel }_{{C}^{2,\a... | Proof of Theorem 13.2.1: We shall show that for every \( {x}_{0} \in {\bar{\Omega }}_{0} \) there exists some ball \( B\left( {{x}_{0}, r}\right) \) on which the desired estimate holds. The radius \( r \) of this ball will depend only on \( \operatorname{dist}\left( {{\Omega }_{0},\partial \Omega }\right) \) and the Hö... | Yes |
Lemma 13.2.1. Let the symmetric matrix \( {\left( {A}^{ij}\right) }_{i, j = 1,\ldots, d} \) satisfy\n\n\[ \lambda {\left| \xi \right| }^{2} \leq \mathop{\sum }\limits_{{i, j = 1}}^{d}{A}^{ij}{\xi }_{i}{\xi }_{j} \leq \Lambda {\left| \xi \right| }^{2}\;\text{ for all }\xi \in {\mathbb{R}}^{d} \]\n\nwith\n\n\[ 0 < \lambd... | Proof. We shall employ the following notation:\n\n\[ A \mathrel{\text{:=}} {\left( {A}^{ij}\right) }_{i, j = 1,\ldots, d},\;{D}^{2}u \mathrel{\text{:=}} {\left( \frac{{\partial }^{2}u}{\partial {x}^{i}\partial {x}^{j}}\right) }_{i, j = 1,\ldots, d}. \]\n\nIf \( B \) is a nonsingular \( d \times d \) -matrix and if \( y... | Yes |
Theorem 13.2.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be a bounded domain of class \( {C}^{2,\alpha } \) (analogously to Definition 11.3.1, we require the same properties as there, except that (iii) is replaced by the condition that \( \phi \) and \( {\phi }^{-1} \) are of class \( {C}^{2,\alpha } \) ). Let \( f \i... | The Proof essentially is a modification of that of Theorem 13.2.1, with modifications that are similar to those employed in the proof of Theorem 11.3.3. We shall therefore provide only a sketch of the proof. We start with a simplified model situation, namely, the Poisson equation in a half-ball, from which we shall der... | No |
Corollary 13.2.1. In addition to the assumptions of Theorem 13.2.2, suppose that \( c\left( x\right) \leq 0 \) in \( \Omega \) . Then\n\n\[ \parallel u{\parallel }_{{C}^{2,\alpha }\left( \Omega \right) } \leq {c}_{16}\left( {\parallel f{\parallel }_{{C}^{\alpha }\left( \Omega \right) } + \parallel g{\parallel }_{{C}^{2... | Proof. Because of \( c \leq 0 \), the maximum principle (see, e.g., Theorem 3.3.2) implies\n\n\[ \mathop{\sup }\limits_{\Omega }\left| u\right| \leq \mathop{\max }\limits_{{\partial \Omega }}\left| u\right| + {c}_{17}\mathop{\sup }\limits_{\Omega }\left| f\right| = \mathop{\max }\limits_{{\partial \Omega }}\left| g\rig... | Yes |
Theorem 13.3.1. Let \( \Omega \) be a bounded domain of class \( {C}^{\infty } \) in \( {\mathbb{R}}^{d}, f \in {C}^{\alpha }\left( \bar{\Omega }\right) \) , \( g \in {C}^{2,\alpha }\left( \bar{\Omega }\right) \) . The Dirichlet problem\n\n\[ \n{\Delta u} = f\;\text{ in }\Omega , \n\]\n\n\[ \nu = g\;\text{ on }\partial... | Proof. Uniqueness follows from the maximum principle (see Corollary 3.1.1). For the existence proof, we first assume that \( f \) and \( g \) are of class \( {C}^{\infty } \) . The variational methods of Sect. 10.3 yield a weak solution, which then is of class \( {C}^{\infty }\left( \Omega \right) \) by Theorem 11.3.1.... | Yes |
Theorem 13.3.2. Let \( \Omega \) be a bounded domain of class \( {C}^{\infty } \) in \( {\mathbb{R}}^{d} \). Let the differential operator\n\n\[ L = \mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) \frac{{\partial }^{2}}{\partial {x}^{i}\partial {x}^{j}} + \mathop{\sum }\limits_{{i = 1}}^{d}{b}^{i}\left( x... | Proof. Considering, as usual, \( \bar{u} = u - g \) in place of \( u \), we may assume \( g = 0 \), as our problem is equivalent to\n\n\[ L\bar{u} = \bar{f} \mathrel{\text{:=}} f - {Lg} \in {C}^{\alpha }\left( \Omega \right) ,\n\n\[ \bar{u} = 0\;\text{ on }\partial \Omega \]\n\nWe thus assume \( g = 0 \) (and write \( ... | Yes |
Theorem 13.3.3. Let \( {L}_{0},{L}_{1} : {B}_{1} \rightarrow {B}_{2} \) be bounded linear operators between the Banach spaces \( {B}_{1},{B}_{2} \) . We put\n\n\[ \n{L}_{t} \mathrel{\text{:=}} \left( {1 - t}\right) {L}_{0} + t{L}_{1}\;\text{ for }0 \leq t \leq 1.\n\]\n\nWe assume that there exists a constant \( c \) th... | Proof. Let \( {L}_{\tau } \) be surjective for some \( \tau \in \left\lbrack {0,1}\right\rbrack \) . By (13.3.11), \( {L}_{\tau } \) then is injective as well, and thus bijective. We therefore have an inverse operator\n\n\[ \n{L}_{\tau }^{-1} : {B}_{2} \rightarrow {B}_{1}\n\]\n\nFor \( t \in \left\lbrack {0,1}\right\rb... | Yes |
Lemma 14.1.1. (i) Let \( u \) be a subsolution, i.e. \( u \in {C}^{2}\left( \Omega \right) ,{Lu} \geq 0 \), and let \( f \in {C}^{2}\left( \mathbb{R}\right) \) be convex with \( {f}^{\prime } \geq 0 \) . Then \( f \circ u \) is a subsolution as well. | Proof.\n\n\[ \nL\left( {f \circ u}\right) = \mathop{\sum }\limits_{{i, j}}\frac{\partial }{\partial {x}^{j}}\left( {{a}^{ij}{f}^{\prime }\left( u\right) \frac{\partial u}{\partial {x}^{i}}}\right) = {f}^{\prime \prime }\left( u\right) \mathop{\sum }\limits_{{i, j}}{a}^{ij}\frac{\partial u}{\partial {x}^{i}}\frac{\parti... | Yes |
Lemma 14.1.3. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak subsolution of \( L \), and \( k \in \mathbb{R} \) . Then\n\n\[ v\left( x\right) \mathrel{\text{:=}} \max \left( {u\left( x\right), k}\right) \]\n\nis a weak subsolution as well. | Proof. We consider the function\n\n\[ f : \mathbb{R} \rightarrow \mathbb{R} \]\n\n\[ f\left( y\right) \mathrel{\text{:=}} \max \left( {y, k}\right) . \]\n\nThen\n\n\[ v = f \circ u. \]\n\nWe approximate \( f \) by a sequence \( {\left( {f}_{n}\right) }_{n \in N} \) of convex functions of class \( {C}^{2} \) with\n\n\[ ... | Yes |
Lemma 14.1.3. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak subsolution of \( L \), and \( k \in \mathbb{R} \) . Then\n\n\[ v\left( x\right) \mathrel{\text{:=}} \max \left( {u\left( x\right), k}\right) \]\n\nis a weak subsolution as well. | Proof. We consider the function\n\n\[ f : \mathbb{R} \rightarrow \mathbb{R} \]\n\n\[ f\left( y\right) \mathrel{\text{:=}} \max \left( {y, k}\right) . \]\n\nThen\n\n\[ v = f \circ u. \]\n\nWe approximate \( f \) by a sequence \( {\left( {f}_{n}\right) }_{n \in N} \) of convex functions of class \( {C}^{2} \) with\n\n\[ ... | Yes |
Theorem 14.1.1. Let \( u \) be a subsolution in the ball \( B\left( {{x}_{0},{4R}}\right) \subset {\mathbb{R}}^{d}\left( {R > 0}\right) \), and assume \( p > 1 \) . Then | \[ \mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}u \leq {c}_{1}{\left( \frac{p}{p - 1}\right) }^{\frac{2}{p}}{\left( {\int }_{B\left( {{x}_{0},{2R}}\right) }{\left( \max \left( u\left( x\right) ,0\right) \right) }^{p}\mathrm{\;d}x\right) }^{\frac{1}{p}}, \] | Yes |
Corollary 14.1.2. Let \( u \) be a positive (weak) solution of \( {Lu} = 0 \) in a domain \( \Omega \) of \( {\mathbb{R}}^{d} \), and let \( {\Omega }_{0} \subset \subset \Omega \) . Then\n\n\[ \mathop{\sup }\limits_{{\Omega }_{0}}u \leq c\mathop{\inf }\limits_{{\Omega }_{0}}u \]\n\n(14.1.7)\n\nwith \( c \) depending o... | Proof. This Harnack inequality on \( {\Omega }_{0} \) follows by the standard ball chain argument: Since \( {\bar{\Omega }}_{0} \) is compact, it can be covered by finitely many balls \( {B}_{i} \mathrel{\text{:=}} B\left( {{x}_{i}, R}\right) \) with \( B\left( {{x}_{i}, R}\right) \subset \Omega \) (we choose, e.g., \(... | Yes |
\[ \mathop{\lim }\limits_{{p \rightarrow \infty }}\phi \left( {p, R}\right) = \mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}u = : \phi \left( {\infty, R}\right) \] | Proof. By Hölder’s inequality, \( \phi \left( {p, R}\right) \) is monotonically increasing with respect to \( p \) . Namely, for \( p < {p}^{\prime } \) and \( u \in {L}^{{p}^{\prime }}\left( \Omega \right) \), \[ {\left( \frac{1}{\left| \Omega \right| }{\int }_{\Omega }{u}^{p}\right) }^{\frac{1}{p}} \leq \frac{1}{{\le... | Yes |
Corollary 14.1.3. Let \( v \) be a bounded weak subsolution on \( B\left( {{x}_{0},{4R}}\right) \) . There exists a constant \( 0 < {\delta }_{0} < 1 \), independent of \( v \) and \( R \), with\n\n\[ \mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}v \leq \left( {1 - {\delta }_{0}}\right) \mathop{\sup }\limits_{{... | Proof. We abbreviate\n\n\[ {v}_{+, R} \mathrel{\text{:=}} \mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}v \]\n\nand have\n\n\[ {v}_{+,{4R}} - {v}_{R} = {f}_{B\left( {{x}_{0}, R}\right) }\left( {{v}_{+,{4R}} - v}\right) \]\n\n\[ \leq {2}^{d}{\int }_{B\left( {{x}_{0},{2R}}\right) }\left| {{v}_{+,{4R}} - v}\right|... | Yes |
Lemma 14.2.1. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a weak subsolution of \( L \), i.e., \[ {Lu} = \mathop{\sum }\limits_{{i, j = 1}}^{d}\frac{\partial }{\partial {x}^{j}}\left( {{a}^{ij}\left( x\right) \frac{\partial }{\partial {x}^{i}}u\left( x\right) }\right) \geq 0\text{ weakly,} \] with \( L \) satisfy... | Proof. By Lemma 14.1.3, for any positive \( k \) , \[ v\left( x\right) \mathrel{\text{:=}} \max \left( {u\left( x\right), k}\right) \] is a positive subsolution (by the way, in place of \( v \), one might also employ the approximating subsolutions \( {f}_{n} \circ u \) from the proof of Lemma 14.1.3). The local bounded... | Yes |
Theorem 14.2.2. Let \( u \in {W}^{1,2}\left( \Omega \right) \) satisfy \( {Lu} \geq 0 \) weakly, the coefficients \( {a}^{ij} \) of \( L \) again satisfying\n\n\[ \n\lambda {\left| \xi \right| }^{2} \leq \mathop{\sum }\limits_{{i, j}}{a}^{ij}\left( x\right) {\xi }_{i}{\xi }_{j},\;\left| {{a}^{ij}\left( x\right) }\right... | Proof. If (14.2.9) holds, we may find some ball \( B\left( {{x}_{0},{R}_{0}}\right) \) with \( B\left( {{x}_{0},4{R}_{0}}\right) \subset \Omega \) and\n\n\[ \n\mathop{\sup }\limits_{{B\left( {{x}_{0},{R}_{0}}\right) }}u = \mathop{\sup }\limits_{\Omega }u.\n\]\n\n(14.2.10)\n\nWithout loss of generality \( \mathop{\sup }... | Yes |
Theorem 14.2.3. Any bounded (weak) solution of \( {Lu} = 0 \) that is defined on all of \( {\mathbb{R}}^{d} \), where \( L \) has measurable bounded coefficients \( {a}^{ij}\left( x\right) \) satisfying\n\n\[ \n\lambda \left| \xi \right| \leq \mathop{\sum }\limits_{{i, j}}{a}^{ij}\left( x\right) {\xi }_{i}{\xi }_{j},\;... | Proof. Since \( u \) is bounded, \( \mathop{\inf }\limits_{{\mathbb{R}}^{d}}u \) and \( \mathop{\sup }\limits_{{\mathbb{R}}^{d}}u \) are finite. Thus, for any\n\n\[ \n\mu < \mathop{\inf }\limits_{{\mathbb{R}}^{d}}u \n\]\n\n\( u - \mu \) is a positive solution of \( {Lu} = 0 \) on \( {\mathbb{R}}^{d} \) . Therefore, by ... | Yes |
Lemma 14.3.1. Let \( u : B\left( {{x}_{0},{4R}}\right) \rightarrow \mathbb{R}\left( {B\left( {{x}_{0},{4R}}\right) }\right. \) a ball in \( \left. {\mathbb{R}}^{d}\right) \) be bounded, with\n\n\[ \mathop{\sup }\limits_{{{y}_{1},{y}_{2} \in B\left( {{x}_{0},{2R}}\right) }}\left| {u\left( {y}_{1}\right) - u\left( {y}_{2... | Proof. By (14.3.7), we can find some \( {x}_{1} \in B\left( {{x}_{0},{2R}}\right) \) with\n\n\[ \operatorname{meas}\left( \left\{ {x \in B\left( {{x}_{0}, R}\right) : \left| {u\left( x\right) - u\left( {x}_{1}\right) }\right| \leq \frac{M}{4}}\right\} \right) \geq \frac{1}{4}\operatorname{meas}\left( {B\left( {{x}_{0},... | Yes |
Theorem 14.3.1. Let \( u \) be a bounded solution of \[ {\int }_{\Omega }\left( {\mathop{\sum }\limits_{i}{D}_{i}u{D}_{i}\varphi - \Gamma \left( u\right) {\left| Du\right| }^{2}\varphi }\right) \mathrm{d}x = 0\text{ for all }\varphi \in {H}_{0}^{1,2} \cap {L}^{\infty }\left( \Omega \right) \] with a smooth and bounded ... | Proof. We choose \( \eta \in {H}_{0}^{1,2}\left( {B\left( {{x}_{0}, R}\right) }\right) \) with \[ 0 \leq \eta \leq 1 \] \[ \eta \equiv 1\;\text{ on }B\left( {{x}_{0}, r}\right) ;\text{ hence }{D\eta } \equiv 0\;\text{ on }B\left( {{x}_{0}, r}\right) , \] \[ \left| {D\eta }\right| \leq \frac{2}{R - r}. \] As in Sect. 11... | No |
Lemma 14.3.2. Let \( u \in {W}^{1,2}\left( \Omega \right) \) be a bounded and continuous weak solution of (14.3.18) in \( \Omega \). Assume \( \left| {\Gamma \left( u\right) }\right| \leq a \). For all \( {x}_{0} \in \Omega \), there then exists a radius \( {R}_{0} < \operatorname{dist}\left( {{x}_{0},\partial \Omega }... | Proof. We choose \( \eta \in {H}_{0}^{1,2}\left( {B\left( {{x}_{0}, R}\right) }\right) \) with\n\n\[ 0 \leq \eta \leq 1 \]\n\n\[ \eta \equiv 1\;\text{ on }B\left( {{x}_{0}, r}\right) ;\text{ hence }{D\eta } \equiv 0\;\text{ on }B\left( {{x}_{0}, r}\right) ,\]\n\n\[ \left| {D\eta }\right| \leq \frac{2}{R - r}. \]\n\nAs ... | Yes |
Lemma 14.4.1. Suppose that the assumptions of Theorem 14.4.1 hold. We then have for all \( \varphi \in {H}_{0}^{1,2}\left( \Omega \right) \), \[ {\int }_{\Omega }\mathop{\sum }\limits_{{i = 1}}^{d}{F}_{{p}_{i}}\left( {Du}\right) {D}_{i}\varphi = 0 \] (using the abbreviation \( {F}_{{p}_{i}} = \frac{\partial F}{\partial... | Proof. By (i), \[ {\int }_{\Omega }\mathop{\sum }\limits_{{i = 1}}^{d}{F}_{{p}_{i}}\left( {Dv}\right) {D}_{i}\varphi \leq {dK}{\int }_{\Omega }\left| {Dv}\right| \left| {D\varphi }\right| \leq {dK}\parallel {Dv}{\parallel }_{{L}^{2}\left( \Omega \right) }\parallel {D\varphi }{\parallel }_{{L}^{2}\left( \Omega \right) }... | Yes |
Lemma 14.4.4. Let \( {\left( {A}^{ij}\right) }_{i, j = 1,\ldots, d} \) be a matrix with \( \left| {A}^{ij}\right| \leq \Lambda \) for all \( i, j \), and\n\n\[ \n\lambda {\left| \xi \right| }^{2} \leq \mathop{\sum }\limits_{{i, j = 1}}^{d}{A}^{ij}{\xi }_{i}{\xi }_{j}\;\text{ for all }\xi \in {\mathbb{R}}^{d} \n\]\n\nwi... | Proof. We choose \( \eta \in {H}_{0}^{1,2}\left( {B\left( {{x}_{0}, R}\right) }\right) \) with\n\n\[ \n0 \leq \eta \leq 1 \n\]\n\n\[ \n\eta \equiv 1\;\text{ on }B\left( {{x}_{0}, r}\right) ,\text{ hence }{D\eta } \equiv 0\;\text{ on }B\left( {{x}_{0}, r}\right) , \n\]\n\n\[ \n\left| {D\eta }\right| \leq \frac{2}{R - r}... | Yes |
Lemma 14.4.5. Under the assumptions of Lemma 14.4.4, we have\n\n\[ \n{\int }_{B\left( {{x}_{0}, r}\right) }{\left| u\right| }^{2} \leq {c}_{3}{\left( \frac{r}{R}\right) }^{d}{\int }_{B\left( {{x}_{0}, R}\right) }{\left| u\right| }^{2} \n\]\n\n(14.4.17)\n\nas well as\n\n\[ \n{\int }_{B\left( {{x}_{0}, r}\right) }{\left|... | Proof. Without loss of generality \( r < \frac{R}{2} \) . We choose \( k > d \) . By the Sobolev embedding theorem (Theorem 11.1.1) or an extension of this result analogous to Corollary 11.1.3,\n\n\[ \n{W}^{k,2}\left( {B\left( {{x}_{0}, R}\right) }\right) \subset {C}^{0}\left( {B\left( {{x}_{0}, R}\right) }\right) . \n... | Yes |
Lemma 14.4.6. Let \( \sigma \left( r\right) \) be a nonnegative, monotonically increasing function satisfying\n\n\[ \sigma \left( r\right) \leq \gamma \left( {{\left( \frac{r}{R}\right) }^{\mu } + \delta }\right) \sigma \left( R\right) + \kappa {R}^{v} \]\n\nfor all \( 0 < r \leq R \leq {R}_{0} \), with \( \mu > v \) a... | Proof. Let \( 0 < \tau < 1, R < {R}_{0} \) . Then by assumption\n\n\[ \sigma \left( {\tau R}\right) \leq \gamma {\tau }^{\mu }\left( {1 + \delta {\tau }^{-\mu }}\right) \sigma \left( R\right) + \kappa {R}^{v}. \]\n\nWe choose \( 0 < \tau < 1 \) such that\n\n\[ {2\gamma }{\tau }^{\mu } = {\tau }^{\lambda } \]\n\nwith \(... | Yes |
Lemma 1.1.3. Let \( R \) be a subring of a ring \( S \) and let \( x \in S \) . Then the following conditions are equivalent:\n\n1. \( x \) satisfies a monic polynomial with coefficients in \( R \) ,\n\n2. \( R\left\lbrack x\right\rbrack \) is a finitely generated \( R \) -module,\n\n3. \( x \) lies in a subring that i... | Proof. The implications \( \left( 1\right) \Rightarrow \left( 2\right) \Rightarrow \left( 3\right) \) are clear. To prove \( \left( 3\right) \Rightarrow \left( 1\right) \), let \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) be a set of \( R \) -module generators for a subring \( {S}_{0} \) containing \( x \), then the... | Yes |
Lemma 1.1.5 (Nakayama's Lemma). Let \( R \) be a local ring with maximal ideal \( P \) and let \( M \) be a nonzero finitely generated \( R \) -module. Then \( {PM} \varsubsetneq M \) . | Proof. Let \( M = R{m}_{1} + \cdots + R{m}_{n} \), where \( n \) is minimal, and put \( {M}_{0} \mathrel{\text{:=}} R{m}_{2} + \cdots + \) \( R{m}_{n} \) . Then \( {M}_{0} \) is a proper submodule. If \( M = {PM} \), we can write\n\n\[ {m}_{1} = \mathop{\sum }\limits_{{i = 1}}^{n}{a}_{i}{m}_{i} \]\n\nwith \( {a}_{i} \i... | Yes |
Theorem 1.1.6 (Valuation Extension Theorem). Let \( R \) be a subring of a field \( K \) and let \( P \) be a nonzero prime ideal of \( R \) . Then there exists a valuation ring \( \mathcal{O} \) of \( K \) with maximal ideal \( M \) such that \( R \subseteq \mathcal{O} \subseteq K \) and \( M \cap R = P \) . | Proof. Consider the set of pairs \( \left( {{R}^{\prime },{P}^{\prime }}\right) \) where \( {R}^{\prime } \) is a subring of \( K \) and \( {P}^{\prime } \) is a prime ideal of \( {R}^{\prime } \) . We say that \( \left( {{R}^{\prime \prime },{P}^{\prime \prime }}\right) \) extends \( \left( {{R}^{\prime },{P}^{\prime ... | Yes |
Corollary 1.1.7. Suppose that \( k \subseteq K \) are fields and \( x \in K \) . If \( x \) is transcendental over \( k \), there exists a \( k \) -valuation \( v \) of \( K \) with \( v\left( x\right) > 0 \) . If \( x \) is algebraic over \( k \) , \( v\left( x\right) = 0 \) for all \( k \) -valuations \( v \) . | Proof. If \( x \) is transcendental over \( k \), apply (1.1.6) with \( \mathcal{O} \mathrel{\text{:=}} k\left\lbrack x\right\rbrack \) and \( P \mathrel{\text{:=}} \left( x\right) \) to obtain a \( k \) -valuation \( v \) with \( v\left( x\right) > 0 \) . Conversely, if\n\n\[ \mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}... | Yes |
Corollary 1.1.8. Let \( R \) be a subring of a field \( K \) . Then the intersection of all valuation rings of \( K \) containing \( R \) is the integral closure of \( R \) in \( K \) . | Proof. If \( x \in K \) satisfies a monic polynomial of degree \( n \) over \( R \) and \( v \) is a valuation of \( K \) that is nonnegative on \( R \), then there are \( {r}_{i} \in R \) such that\n\n\[ \n{nv}\left( x\right) = v\left( {x}^{n}\right) = v\left( {\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{r}_{i}{x}^{i}}\... | Yes |
Lemma 1.1.9. Let \( \mathcal{O} \) be a valuation ring. Then finitely generated torsion-free O-modules are free. In particular, finitely generated ideals are principal. | Proof. Let \( P \) be a torsion-free \( \mathcal{O} \) -module with generating set \( \left\{ {{m}_{1},\ldots ,{m}_{n}}\right\} \) . Supposing there to be a relation \( \mathop{\sum }\limits_{i}{a}_{i}{m}_{i} = 0 \) where not all \( {a}_{i} \) are zero, we may choose notation so that \( v\left( {a}_{n}\right) = \min \l... | Yes |
Lemma 1.1.10. Let \( t \) be an element of a subring \( \mathcal{O} \) of a field \( K \) . Then \( \mathcal{O} \) is a discrete valuation ring of \( K \) with local parameter \( t \) if and only if every element \( x \in K \) can be written \( x = u{t}^{i} \) for some unit \( u \in \mathcal{O} \) . | Proof. If every element of \( K \) is of the form \( u{t}^{i} \), put \( {\mathcal{O}}_{0} \mathrel{\text{:=}} \left\{ {u{t}^{i} \in K \mid i \geq 0}\right\} \subseteq \mathcal{O} \) . It is obvious that \( {\mathcal{O}}_{0} \) is both a valuation ring of \( K \) and a maximal subring of \( K \), and that \( {K}^{ \tim... | Yes |
Theorem 1.1.12 (Smith Normal Form). Let \( \mathcal{O} \) be a discrete valuation ring with local parameter \( t \) and let \( A \) be a matrix with entries in \( \mathcal{O} \) . Then there exist matrices \( U, V \) with entries in \( \mathcal{O} \) and unit determinant, and nonnegative integers\n\n\[ \n{e}_{1} \leq {... | Proof. If \( A = 0 \), there is nothing to prove. Otherwise, multiplying by permutation matrices as necessary, we may assume that \( {e}_{1} \mathrel{\text{:=}} v\left( {a}_{11}\right) \leq v\left( {a}_{ij}\right) \) for all \( i, j \) . Multiplying row 1 by a unit, we may assume that \( {a}_{11} = {t}^{{e}_{1}} \) .\n... | Yes |
Corollary 1.1.13. Let \( \mathcal{O} \) be a discrete valuation ring with local parameter \( t \), let \( M \) be a free \( \mathcal{O} \) -module of finite rank, and let \( N \subseteq M \) be a nonzero submodule. Then \( N \) is free, and there exists a basis \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) for \( M \... | Proof. We first argue by induction on the rank of \( M \) that \( N \) is finitely generated. If \( M \) has rank one, this follows from (1.1.11). If \( M \) has rank \( n > 1 \), let \( {M}_{0} \) be a free submodule of rank \( n - 1 \) . Then \( N \cap {M}_{0} \) and \( N/\left( {N \cap {M}_{0}}\right) \) are finitel... | Yes |
Theorem 1.1.14. Let \( v \) be a valuation of \( K \mathrel{\text{:=}} k\left( X\right) \) . Then either \( v = {v}_{p} \) for some irreducible polynomial \( p \in k\left\lbrack X\right\rbrack \), or \( v\left( {f\left( X\right) /g\left( X\right) }\right) = \deg \left( g\right) - \deg \left( f\right) \), where \( f \) ... | Proof. If \( v\left( X\right) \geq 0 \), then \( k\left\lbrack X\right\rbrack \subseteq {\mathcal{O}}_{v} \) and \( {P}_{v} \cap k\left\lbrack X\right\rbrack \) is a prime ideal \( \left( p\right) \) for some irreducible polynomial \( p \) . This implies that the localization \( k{\left\lbrack X\right\rbrack }_{\left( ... | Yes |
Lemma 1.1.15. Let \( \\left\\{ {{v}_{1},\\ldots ,{v}_{n}}\\right\\} \) be a set of distinct discrete valuations of a field \( K \), and let \( m \) be a positive integer. Then there exists \( e \in K \) such that \( {v}_{1}\\left( {e - 1}\\right) > m \) and \( {v}_{i}\\left( e\\right) > m \) for \( i > 1 \) . | Proof. We first find an element \( x \in K \) such that \( {v}_{1}\\left( x\\right) > 0 \) and \( {v}_{i}\\left( x\\right) < 0 \) for \( i > 1 \) . Namely, if \( n = 2 \), we choose \( {x}_{i} \in {\\mathcal{O}}_{{v}_{i}} \\smallsetminus {\\mathcal{O}}_{{v}_{3 - i}} \) for \( i = 1,2 \) . This is possible since \( {\\m... | Yes |
Theorem 1.1.16 (Weak Approximation Theorem). Suppose that \( {v}_{1},\ldots ,{v}_{n} \) are distinct discrete valuations of a field \( K,{m}_{1},\ldots ,{m}_{n} \) are integers, and \( {x}_{1},\ldots ,{x}_{n} \in K \) . Then there exists \( x \in K \) such that \( {v}_{i}\left( {x - {x}_{i}}\right) = {m}_{i} \) for \( ... | Proof. Choose elements \( {a}_{i} \in K \) such that \( {v}_{i}\left( {a}_{i}\right) = {m}_{i} \) for all \( i \), and let \( {m}_{0} \mathrel{\text{:=}} \) \( \mathop{\max }\limits_{i}{m}_{i} \) . Now choose an integer \( M \) such that\n\n\[ M + \mathop{\min }\limits_{{i, j}}\left\{ {{\nu }_{i}\left( {x}_{j}\right) ,... | Yes |
Corollary 1.1.18. With the above notation, we have\n\n\[ K\left( {\mathcal{V};m}\right) + K\left( {{\mathcal{V}}^{\prime };{m}^{\prime }}\right) = K\left( {\mathcal{V} \cap {\mathcal{V}}^{\prime };\min \left\{ {m,{m}^{\prime }}\right\} }\right) \]\n\nfor \( m \) and \( {m}^{\prime } \) nonnegative. | Proof. It is obvious that \( K\left( {\mathcal{V};m}\right) + K\left( {{\mathcal{V}}^{\prime };{m}^{\prime }}\right) \subseteq K\left( {\mathcal{V} \cap {\mathcal{V}}^{\prime };\min \left\{ {m,{m}^{\prime }}\right\} }\right) \). Conversely, let \( y \in K\left( {\mathcal{V} \cap {\mathcal{V}}^{\prime };\min \left\{ {m,... | Yes |
Lemma 1.1.20. Let \( \mathcal{O} \) be a discrete valuation ring with field of fractions \( K \) , maximal ideal \( P \), and residue field \( F \) . Let \( M \) be a torsion-free \( \mathcal{O} \) -module with \( {\dim }_{K}K{ \otimes }_{\mathcal{O}}M = n \) . Then \( {\dim }_{F}M/{PM} \leq n \) with equality if and o... | Proof. If \( M \) is finitely generated, it is free by (1.1.9) and therefore free of rank \( n \) , whence \( {\dim }_{F}M/{PM} = n \) as well.\n\nSuppose that \( {x}_{1},{x}_{2},\ldots ,{x}_{m} \in M \) . If we have a nontrivial dependence relation\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{a}_{i}{x}_{i} = 0 \]\n\nwith... | Yes |
Lemma 1.1.21. Let \( \left| {{K}^{\prime } : K}\right| = n \), let \( {\mathcal{O}}_{v} \) be a discrete valuation ring of \( K \), and let \( R \) be any subring of \( {K}^{\prime } \) containing the integral closure of \( {\mathcal{O}}_{v} \) in \( {K}^{\prime } \) . Then the map\n\n\[ K{ \otimes }_{{\mathcal{O}}_{v}... | Proof. We first argue that the map \( x \otimes y \mapsto {xy} \) is an embedding. Let \( t \) be a local parameter for \( v \) . Then any element of the kernel can be written \( x = \mathop{\sum }\limits_{{i = 0}}^{n}{t}^{-{e}_{i}} \otimes {x}_{i} \) , where notation can be chosen so that \( {e}_{0} = \mathop{\max }\l... | Yes |
Theorem 1.1.22. Let \( {K}^{\prime } \) be a finite extension of \( K \) and let \( \mathcal{O} \) be a discrete valuation ring of \( K \) with maximal ideal \( P \) and residue field \( F \) . Let \( \left\{ {{\mathcal{O}}_{1},\ldots ,{\mathcal{O}}_{r}}\right\} \) be distinct valuation rings of \( {K}^{\prime } \) con... | Proof. Let \( {v}_{i} \) be the valuation afforded by \( {\mathcal{O}}_{i} \) for all \( i \), and let \( \mathcal{V} = \left\{ {{v}_{1},\ldots ,{v}_{r}}\right\} \) . Note that \( R = K\left( {\mathcal{V};0}\right) \) and that any valuation ring of \( {K}^{\prime } \) containing \( R \) also contains \( \mathcal{O} \) ... | Yes |
Theorem 1.1.23. Suppose that \( \mathcal{O} \) is a discrete valuation ring of \( K \) with maximal ideal \( P \) and residue field \( F \) . Let \( \left\{ {\left( {{\mathcal{O}}_{i},{P}_{i}}\right) \mid 1 \leq i \leq r}\right\} \) be the set of distinct extensions of \( \left( {\mathcal{O}, P}\right) \) to some finit... | Proof. Since \( g\left( X\right) \) has distinct roots \( {\;\operatorname{mod}\;P} \), there is certainly a factorization\n\n\[ \bar{g}\left( X\right) = \mathop{\prod }\limits_{{i = 1}}^{{r}^{\prime }}{g}_{i}\left( X\right) \]\n\ninto distinct irreducibles over \( F\left\lbrack X\right\rbrack \), where \( \bar{g}\left... | Yes |
Theorem 1.1.24. Suppose that \( \left| {{K}^{\prime } : K}\right| = n \) and that \( v \) is a discrete valuation of \( K \) with \( e\left( {{v}^{\prime } \mid v}\right) = n \) for some discrete valuation \( {v}^{\prime } \) of \( {K}^{\prime } \) . Let \( s \) be a local parameter at \( v \) . Then \( {K}^{\prime } =... | Proof. For any \( n \) -tuple \( \left\{ {{a}_{0},\ldots ,{a}_{n - 1}}\right\} \) of elements of \( K \), let \( I \) be the set of indices \( i \) for which \( {a}_{i} \neq 0 \) . Then for all \( i \in I \) we have \( {v}^{\prime }\left( {a}_{i}\right) \equiv 0{\;\operatorname{mod}\;n} \) and thus \( {v}^{\prime }\lef... | Yes |
Lemma 1.1.25. Suppose \( {K}_{0} \subseteq {K}_{1} \subseteq {K}_{2} \) are three fields with \( \left| {{K}_{2} : {K}_{0}}\right| < \infty \), and \( {v}_{i} \) is a discrete valuation of \( {K}_{i}\left( {0 \leq i \leq 2}\right) \) with \( {v}_{2}\left| {v}_{1}\right| {v}_{0} \) . Then\n\n\[ e\left( {{v}_{2} \mid {v}... | Proof. The first statement is immediate from the definition of \( e \) and the fact that restriction of functions is transitive. The second statement follows from the natural inclusion of residue fields \( {F}_{0} \subseteq {F}_{1} \subseteq {F}_{2} \) and (A.0.2). | Yes |
Lemma 1.2.3. A ring \( R \) is complete at the ideal \( I \) if and only if the following two conditions are satisfied:\n\n1. \( { \cap }_{n = 0}^{\infty }{I}^{n} = 0 \), and\n\n2. Given any sequence \( {r}_{n} \in R \) with \( {r}_{n} \equiv {r}_{n + 1}{\;\operatorname{mod}\;{I}^{n}} \) for all \( n \), there exists \... | Proof. As already noted, 1) is equivalent to the injectivity of the natural map \( R \rightarrow {\widehat{R}}_{I} \) and one verifies easily that 2) is equivalent to its surjectivity. If \( {I}^{n} = 0 \) for some \( n \), the sequences satisfying 2) are eventually constant and we can take \( r = {r}_{n} \) for any su... | Yes |
Lemma 1.2.5. Suppose that \( S \) is a subring of \( R, I \) is an ideal of \( R \), and \( J \) is an ideal of \( S \) contained in \( I \) . Then there is a natural map \( \phi : {\widehat{S}}_{J} \rightarrow {\widehat{R}}_{I} \) making all diagrams commutative, where \( {\phi }_{n} \) is induced by inclusion. If \( ... | Proof. Since \( {J}^{n} \subseteq {I}^{n} \) for any \( n \), there are natural maps \[ {\widehat{S}}_{J}\overset{{\pi }_{n}}{ \rightarrow }S/{J}^{n}\overset{{\phi }_{n}}{ \rightarrow }R/{I}^{n} \] that commute with \( R/{I}^{n + 1} \rightarrow R/{I}^{n} \), so \( \phi \mathrel{\text{:=}} \mathop{\lim }\limits_{n}\left... | Yes |
Lemma 1.2.6. If \( R \) is complete at \( I \) and \( u \in R \) is invertible modulo \( I \), then \( u \) is invertible. | Proof. By hypothesis there is an element \( y \in R \) with \( a = 1 - {uy} \in I \) . Put \( {s}_{n} \mathrel{\text{:=}} \) \( 1 + a + {a}^{2} + \cdots + {a}^{n} \) . Then \( \left\{ {s}_{n}\right\} \) is a strong Cauchy sequence, which therefore converges to some element \( s \in R \) . Since \( \left( {1 - a}\right)... | Yes |
Lemma 1.2.7 (Newton's Algorithm). Let \( R \) be a ring with an ideal \( l \) and suppose that for some polynomial \( f \in R\left\lbrack X\right\rbrack \) there exists \( a \in R \) such that \( f\left( a\right) \equiv 0{\;\operatorname{mod}\;I} \) and \( {f}^{\prime }\left( a\right) \) is invertible, where \( {f}^{\p... | Proof. We have \( b \equiv a{\;\operatorname{mod}\;I} \) because \( f\left( a\right) \in I \) . For any element \( a \in R \) and any \( n \geq 0 \) we have the identity\n\n\[ {X}^{n} = {\left( X - a + a\right) }^{n} = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{matrix} n \\ i \end{matrix}\right) {\left( X - a\rig... | Yes |
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