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Corollary 1.2. Let \( S \) be a denumerable set and \( D \) an infinite subset of \( S \). Then \( D \) is denumerable. | Proof. Given an enumeration of \( S \), the subset \( D \) corresponds to a subset of \( {\mathbf{Z}}^{ + } \) in this enumeration. Using Proposition 1.1, we conclude that we can enumerate \( D \) . | No |
Proposition 1.3. Every infinite set contains a denumerable subset. | Proof. Let \( S \) be an infinite set. For every non-empty subset \( T \) of \( S \), we select a definite element \( {a}_{T} \) in \( T \) . We then proceed by induction. We let \( {x}_{1} \) be the chosen element \( {a}_{S} \) . Suppose that we have chosen \( {x}_{1},\ldots ,{x}_{n} \) having the property that for ea... | Yes |
Proposition 1.4. Let \( D \) be a denumerable set, and \( f : D \rightarrow S \) a surjective mapping. Then \( S \) is denumerable or finite. | Proof. For each \( y \in S \), there exists an element \( {x}_{y} \in D \) such that \( f\left( {x}_{y}\right) = y \) because \( f \) is surjective. The association \( y \mapsto {x}_{y} \) is an injective mapping of \( S \) into \( D \), because if\n\n\[ y, z \in S\text{ and }{x}_{y} = {x}_{z} \]\n\nthen\n\n\[ y = f\le... | Yes |
Proposition 1.5. Let \( D \) be a denumerable set. Then \( D \times D \) (the set of all pairs \( \left( {x, y}\right) \) with \( x, y \in D \) ) is denumerable. | Proof. There is a bijection between \( D \times D \) and \( {\mathbf{Z}}^{ + } \times {\mathbf{Z}}^{ + } \), so it will suffice to prove that \( {\mathbf{Z}}^{ + } \times {\mathbf{Z}}^{ + } \) is denumerable. Consider the mapping of \( {\mathbf{Z}}^{ + } \times {\mathbf{Z}}^{ + } \rightarrow {\mathbf{Z}}^{ + } \) given... | Yes |
Proposition 1.6. Let \( \\left\\{ {{D}_{1},{D}_{2},\\ldots }\\right\\} \) be a sequence of denumerable sets. Let \( S \) be the union of all sets \( {D}_{i}\\left( {i = 1,2,\\ldots }\\right) \) . Then \( S \) is denumerable. | Proof. For each \( i = 1,2,\\ldots \) we enumerate the elements of \( {D}_{i} \), as indicated in the following notation:\n\n\[ \n{D}_{1} : \\left\\{ {{x}_{11},{x}_{12},{x}_{13},\\ldots }\\right\\} \n\]\n\n\[ \n{D}_{2} : \\;\\left\\{ {{x}_{21},{x}_{22},{x}_{23},\\ldots }\\right\\} \n\]\n\n\[ \n{D}_{i} : \\;\\left\\{ {{... | Yes |
Corollary 1.7. Let \( F \) be a non-empty finite set and \( D \) a denumerable set. Then \( F \times D \) is denumerable. If \( {S}_{1},{S}_{2},\ldots \) are a sequence of sets, each of which is finite or denumerable, then the union \( {S}_{1} \cup {S}_{2} \cup \cdots \) is denumerable or finite. | Proof. There is an injection of \( F \) into \( {\mathbf{Z}}^{ + } \) and a bijection of \( D \) with \( {\mathbf{Z}}^{ + } \) . Hence there is an injection of \( F \times {\mathbf{Z}}^{ + } \) into \( {\mathbf{Z}}^{ + } \times {\mathbf{Z}}^{ + } \) and we can apply Corollary 1.2 and Proposition 1.6 to prove the first ... | No |
Let \( G \) be a group. Let \( S \) be the set of subgroups. If \( H,{H}^{\prime } \) are subgroups of \( G \), we define\n\n\[ H \leqq {H}^{\prime } \]\n\nif \( H \) is a subgroup of \( {H}^{\prime } \) . One verifies immediately that this relation defines an ordering on \( S \) . Given two subgroups \( H,{H}^{\prime ... | To prove this, let us take Example 1. Let \( T \) be a non-empty totally ordered subset of the set of subgroups of \( G \) . This means that if \( H,{H}^{\prime } \in T \), then \( H \subset {H}^{\prime } \) or \( {H}^{\prime } \subset H \) . Let \( U \) be the union of all sets in \( T \) . Then:\n\n1. \( U \) is a su... | Yes |
Theorem 2.1. (Bourbaki). Let \( A \) be a non-empty partially ordered and strictly inductively ordered set. Let \( f : A \rightarrow A \) be an increasing mapping. Then there exists an element \( {x}_{0} \in A \) such that \( f\left( {x}_{0}\right) = {x}_{0} \) . | Proof. Suppose that \( A \) were totally ordered. By assumption, it would have a least upper bound \( b \in A \), and then\n\n\[ b \leqq f\left( b\right) \leqq b \]\n\nso that in this case, our theorem is clear. The whole problem is to reduce the theorem to that case. In other words, what we need to find is a totally o... | Yes |
Lemma 2.2. We have \( {M}_{c} = M \) for every extreme point \( c \) of \( M \) . | Proof. It will suffice to prove that \( {M}_{c} \) is an admissible subset. Let \( x \in {M}_{c} \) . If \( x < c \) then \( f\left( x\right) \leqq c \) so \( f\left( x\right) \in {M}_{c} \) . If \( x = c \) then \( f\left( x\right) = f\left( c\right) \) is again in \( {M}_{c} \) . If \( f\left( c\right) \leqq x \), th... | Yes |
Corollary 2.4. Let \( A \) be a non-empty strictly inductively ordered set. Then \( A \) has a maximal element. | Proof. Suppose that \( A \) does not have a maximal element. Then for each \( x \in A \) there exists an element \( {y}_{x} \in A \) such that \( x < {y}_{x} \) . Let \( f : A \rightarrow A \) be the map such that \( f\left( x\right) = {y}_{x} \) for all \( x \in A \) . Then \( A, f \) satisfy the hypotheses of Theorem... | Yes |
Corollary 2.5. (Zorn's lemma). Let \( S \) be a non-empty inductively ordered set. Then \( S \) has a maximal element. | Proof. Let \( A \) be the set of non-empty totally ordered subsets of \( S \) . Then \( A \) is not empty since any subset of \( S \) with one element belongs to \( A \) . If \( X, Y \in A \) , we define \( X \leqq Y \) to mean \( X \subset Y \) . Then \( A \) is partially ordered, and is in fact strictly inductively o... | Yes |
Theorem 3.1. (Schroeder-Bernstein). Let \( A, B \) be sets, and suppose that \( \operatorname{card}\left( A\right) \leqq \operatorname{card}\left( B\right) \), and \( \operatorname{card}\left( B\right) \leqq \operatorname{card}\left( A\right) \) . Then\n\n\[ \operatorname{card}\left( A\right) = \operatorname{card}\left... | Proof. Let\n\n\[ f : A \rightarrow B\text{ and }g : B \rightarrow A \]\n\nbe injections. We separate \( A \) into two disjoint sets \( {A}_{1} \) and \( {A}_{2} \) . We let \( {A}_{1} \) consist of all \( x \in A \) such that, when we lift back \( x \) by a succession of inverse maps,\n\n\[ x,{g}^{-1}\left( x\right) ,\... | Yes |
Lemma 3.2. Let \( A \) be an infinite set. Then there exists a disjoint covering of \( A \) by denumerable sets. | Proof. Let \( S \) be the set whose elements are pairs \( \left( {B,\Gamma }\right) \) consisting of a subset \( B \) of \( A \), and a disjoint covering of \( B \) by denumerable sets. Then \( S \) is not empty. Indeed, since \( A \) is infinite, \( A \) contains a denumerable set \( D \), and the pair \( \left( {D,\{... | Yes |
Theorem 3.3. Let \( A \) be an infinite set, and let \( D \) be a denumerable set. Then\n\n\[ \operatorname{card}\left( {A \times D}\right) = \operatorname{card}\left( A\right) \] | Proof. By the lemma, we can write\n\n\[ A = \mathop{\bigcup }\limits_{{i \in I}}{D}_{i} \]\n\nas a disjoint union of denumerable sets. Then\n\n\[ A \times D = \mathop{\bigcup }\limits_{{i \in I}}\left( {{D}_{i} \times D}\right) \]\n\nFor each \( i \in I \), there is a bijection of \( {D}_{i} \times D \) on \( {D}_{i} \... | Yes |
Corollary 3.4. If \( F \) is a finite non-empty set, then\n\n\[ \operatorname{card}\left( {A \times F}\right) = \operatorname{card}\left( A\right) \] | Proof. We have\n\n\[ \operatorname{card}\left( A\right) \leqq \operatorname{card}\left( {A \times F}\right) \leqq \operatorname{card}\left( {A \times D}\right) = \operatorname{card}\left( A\right) . \]\n\nWe can then use Theorem 3.1 to get what we want. | No |
Corollary 3.5. Let \( A, B \) be non-empty sets, \( A \) infinite, and suppose\n\n\[ \operatorname{card}\left( B\right) \leqq \operatorname{card}\left( A\right) \]\n\nThen\n\n\[ \operatorname{card}\left( {A \cup B}\right) = \operatorname{card}\left( A\right) \] | Proof. We can write \( A \cup B = A \cup C \) for some subset \( C \) of \( B \), such that \( C \) and \( A \) are disjoint. (We let \( C \) be the set of all elements of \( B \) which are not elements of \( A \) .) Then \( \operatorname{card}\left( C\right) \leqq \operatorname{card}\left( A\right) \) . We can then co... | Yes |
Corollary 3.7. If \( A \) is an infinite set, and \( {A}^{\left( n\right) } = A \times \cdots \times A \) is the product taken \( n \) times, then\n\n\[ \operatorname{card}\left( {A}^{\left( n\right) }\right) = \operatorname{card}\left( A\right) \] | Proof. Induction. | No |
Corollary 3.8. If \( {A}_{1},\ldots ,{A}_{n} \) are non-empty sets with \( {A}_{n} \) infinite, and | Proof. We have | No |
Corollary 3.9. Let \( A \) be an infinite set, and let \( \Phi \) be the set of finite subsets of \( A \) . Then\n\n\[ \operatorname{card}\left( \Phi \right) = \operatorname{card}\left( A\right) \] | Proof. Let \( {\Phi }_{n} \) be the set of subsets of \( A \) having exactly \( n \) elements, for each integer \( n = 1,2,\ldots \) . We first show that \( \operatorname{card}\left( {\Phi }_{n}\right) \leqq \operatorname{card}\left( A\right) \) . If \( F \) is an element of \( {\Phi }_{n} \), we order the elements of ... | No |
Theorem 3.10. Let \( A \) be an infinite set, and \( T \) the set consisting of two elements \( \{ 0,1\} \) . Let \( M \) be the set of all maps of \( A \) into \( T \) . Then\n\n\[ \operatorname{card}\left( A\right) \leqq \operatorname{card}\left( M\right) \;\text{ and }\;\operatorname{card}\left( A\right) \neq \opera... | Proof. For each \( x \in A \) we let\n\n\[ {f}_{x} : A \rightarrow \{ 0,1\} \]\n\nbe the map such that \( {f}_{x}\left( x\right) = 1 \) and \( {f}_{x}\left( y\right) = 0 \) if \( y \neq x \) . Then \( x \mapsto {f}_{x} \) is obviously an injection of \( A \) into \( M \), so that \( \operatorname{card}\left( A\right) \... | Yes |
Corollary 3.11. Let \( A \) be an infinite set, and let \( S \) be the set of all subsets of \( A \) . Then \( \operatorname{card}\left( A\right) \leqq \operatorname{card}\left( S\right) \) and \( \operatorname{card}\left( A\right) \neq \operatorname{card}\left( S\right) \) . | Proof. We leave it as an exercise. [Hint: If \( B \) is a non-empty subset of \( A \) , use the characteristic function \( {\varphi }_{B} \) such that \[ {\varphi }_{B}\left( x\right) = 1\;\text{ if }\;x \in B, \] \[ {\varphi }_{B}\left( x\right) = 0\;\text{ if }\;x \notin B. \] What can you say about the association \... | No |
Example 2. Let \( S \) be a well-ordered set and let \( b \) be an element of some set, \( b \notin S \) . Let \( A = S \cup \{ b\} \) . We define \( x \leqq b \) for all \( x \in S \) . Then \( A \) is totally ordered, and is in fact well-ordered. | Proof. Let \( B \) be a non-empty subset of \( A \) . If \( B \) consists of \( b \) alone, then \( b \) is a least element of \( B \) . Otherwise, \( B \) contains some element \( a \in A \) . Then \( B \cap A \) is not empty, and hence has a least element, which is obviously also a least element for B. | No |
Lemma 2.1.1. Let \( u, v \in {C}^{2}\left( \bar{\Omega }\right) \) . Then we have Green’s 1st formula\n\n\[ \n{\int }_{\Omega }v\left( x\right) {\Delta u}\left( x\right) \mathrm{d}x + {\int }_{\Omega }\nabla u\left( x\right) \cdot \nabla v\left( x\right) \mathrm{d}x = {\int }_{\partial \Omega }v\left( z\right) \frac{\p... | Proof. With \( V\left( x\right) = v\left( x\right) \nabla u\left( x\right) \) ,(2.1.2) follows from (2.1.1). Interchanging \( u \) and \( v \) in (2.1.2) and subtracting the resulting formula from (2.1.2) yield (2.1.3). | No |
Lemma 2.1.2. The harmonic functions in \( \Omega \) form a vector space. | Proof. This follows because \( \Delta \) is a linear differential operator. | No |
Theorem 2.1.1 (Green representation formula). If \( u \in {C}^{2}\left( \bar{\Omega }\right) \), we have for \( y \in \Omega \), \[ u\left( y\right) = {\int }_{\partial \Omega }\left\{ {u\left( x\right) \frac{\partial \Gamma }{\partial {v}_{x}}\left( {x, y}\right) - \Gamma \left( {x, y}\right) \frac{\partial u}{\partia... | Proof. For sufficiently small \( \varepsilon > 0 \), \[ B\left( {y,\varepsilon }\right) \subset \Omega, \] since \( \Omega \) is open. We apply (2.1.3) for \( v\left( x\right) = \Gamma \left( {x, y}\right) \) and \( \Omega \smallsetminus B\left( {y,\varepsilon }\right) \) (in place of \( \Omega ) \) . Since \( \Gamma \... | Yes |
Theorem 2.1.2. (Poisson representation formula; solution of the Dirichlet problem on the ball): Let \( \varphi : \partial B\left( {0, R}\right) \rightarrow \mathbb{R} \) be continuous. Then \( u \), defined by\n\n\[ u\left( y\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} \frac{{R}^{2} - {\left| y\right| }^{2}}{d... | Proof. Since \( G \) is harmonic in \( y \), so is the kernel of the Poisson representation formula\n\n\[ K\left( {x, y}\right) \mathrel{\text{:=}} \frac{\partial G}{\partial {v}_{x}}\left( {x, y}\right) = \frac{{R}^{2} - {\left| y\right| }^{2}}{d{\omega }_{d}R}{\left| x - y\right| }^{-d}. \]\n\nThus \( u \) is harmoni... | Yes |
Corollary 2.1.1. For \( \varphi \in {C}^{0}\left( {\partial B\left( {0, R}\right) }\right) \), there exists a unique solution \( u \in {C}^{2} \) \( \left( {\overset{ \circ }{B}\left( {0, R}\right) }\right) \cap {C}^{0}\left( {B\left( {0, R}\right) }\right) \) of the Dirichlet problem\n\n\[ \n{\Delta u}\left( x\right) ... | Proof. Theorem 2.1.2 shows the existence. Uniqueness follows from (2.1.15); however, in (2.1.15) we have assumed \( u \in {C}^{2}\left( {B\left( {0, R}\right) }\right) \), while more generally, here we consider continuous boundary values. This difficulty is easily overcome: Since \( u \)\nis harmonic in \( B\left( {0, ... | Yes |
Corollary 2.1.2. Any harmonic function \( u : \Omega \rightarrow \mathbb{R} \) is real analytic in \( \Omega \) . | Proof. Let \( z \in \Omega \) and choose \( R \) such that \( B\left( {z, R}\right) \subset \Omega \) . Then by (2.1.27), for \( y \in \overset{ \circ }{B}\left( {z, R}\right) \)\n\n\[ u\left( y\right) = \frac{{R}^{2} - {\left| y - z\right| }^{2}}{d{\omega }_{d}R}{\int }_{\partial B\left( {z, R}\right) }\frac{u\left( x... | Yes |
Theorem 2.2.1 (Mean value formulae). A continuous or, more generally, a measurable and locally integrable \( u : \Omega \rightarrow \mathbb{R} \) is harmonic if and only if for any ball \( B\left( {{x}_{0}, r}\right) \subset \Omega \) , \[ u\left( {x}_{0}\right) = S\left( {u,{x}_{0}, r}\right) \mathrel{\text{:=}} \frac... | Proof. \ | Yes |
Theorem 2.2.2. A function \( v : \Omega \rightarrow \lbrack - \infty ,\infty ) \) (upper semicontinuous, \( ≢ - \infty \) ) is subharmonic if and only if for every ball \( B\left( {{x}_{0}, r}\right) \subset \Omega \) , \[ v\left( {x}_{0}\right) \leq S\left( {v,{x}_{0}, r}\right) \] or, equivalently, if for every such ... | Proof. \ | No |
Lemma 2.2.1. Suppose \( v \) satisfies the mean value inequality (2.2.8) or (2.2.9) for all \( B\left( {{x}_{0}, r}\right) \subset \Omega \) . Then \( v \) also satisfies the maximum principle, meaning that if there exists some \( {x}_{0} \in \Omega \) with\n\n\[ v\left( {x}_{0}\right) = \mathop{\sup }\limits_{{x \in \... | Proof. Assume\n\n\[ v\left( {x}_{0}\right) = \mathop{\sup }\limits_{{x \in \Omega }}v\left( x\right) = : M. \]\n\nThus,\n\n\[ {\Omega }^{M} \mathrel{\text{:=}} \{ y \in \Omega : v\left( y\right) = M\} \neq \varnothing . \]\n\nLet \( y \in {\Omega }^{M}, B\left( {y, r}\right) \subset \Omega \) . Since (2.2.8) implies (2... | Yes |
A function \( v \) of class \( {C}^{2}\left( \Omega \right) \) is subharmonic precisely if\n\n\[{\Delta v} \geq 0\;\text{ in }\;\Omega . | Proof. \ | No |
Corollary 2.2.3 (Strong maximum principle). Let \( u \) be harmonic in \( \Omega \) . If there exists \( {x}_{0} \in \Omega \) with\n\n\[ u\left( {x}_{0}\right) = \mathop{\sup }\limits_{{x \in \Omega }}u\left( x\right) \;\text{ or }\;u\left( {x}_{0}\right) = \mathop{\inf }\limits_{{x \in \Omega }}u\left( x\right) ,\]\n... | Proof. Otherwise, \( u \) would achieve its supremum or infimum in some interior point of \( \Omega \) . Then \( u \) would be constant by Corollary 2.2.3, and the claim would also hold true. | No |
Corollary 2.2.3 (Strong maximum principle). Let \( u \) be harmonic in \( \Omega \) . If there exists \( {x}_{0} \in \Omega \) with\n\n\[ u\left( {x}_{0}\right) = \mathop{\sup }\limits_{{x \in \Omega }}u\left( x\right) \;\text{ or }\;u\left( {x}_{0}\right) = \mathop{\inf }\limits_{{x \in \Omega }}u\left( x\right) ,\]\n... | Proof. Otherwise, \( u \) would achieve its supremum or infimum in some interior point of \( \Omega \) . Then \( u \) would be constant by Corollary 2.2.3, and the claim would also hold true. | No |
Corollary 2.2.4 (Weak maximum principle). Let \( \Omega \) be bounded and \( u \in {C}^{0}\left( \bar{\Omega }\right) \) harmonic. Then for all \( x \in \Omega \), \[ \mathop{\min }\limits_{{y \in \partial \Omega }}u\left( y\right) \leq u\left( x\right) \leq \mathop{\max }\limits_{{y \in \partial \Omega }}u\left( y\rig... | Proof. Otherwise, \( u \) would achieve its supremum or infimum in some interior point of \( \Omega \) . Then \( u \) would be constant by Corollary 2.2.3, and the claim would also hold true. | No |
Corollary 2.2.5 (Uniqueness of solutions of the Poisson equation). Let \( f \in \) \( {C}^{0}\left( \Omega \right) ,\Omega \) bounded, \( {u}_{1},{u}_{2} \in {C}^{0}\left( \bar{\Omega }\right) \cap {C}^{2}\left( \Omega \right) \) solutions of the Poisson equation\n\n\[ \n\Delta {u}_{i}\left( x\right) = f\left( x\right)... | Proof. We apply the maximum principle to the harmonic function \( {u}_{1} - {u}_{2} \). | No |
Corollary 2.2.7. Suppose that in \( \Omega \) , \[ {\Delta u}\left( x\right) = f\left( x\right) \] with a bounded function \( f \) . Let \( {x}_{0} \in \Omega \) and \( R \mathrel{\text{:=}} \operatorname{dist}\left( {{x}_{0},\partial \Omega }\right) \) . Then \[ \left| {{u}_{{x}^{i}}\left( {x}_{0}\right) }\right| \leq... | Proof. We consider the case \( i = 1 \) . For abbreviation, put \[ \mu \mathrel{\text{:=}} \mathop{\sup }\limits_{{\partial B\left( {{x}_{0}, R}\right) }}\left| u\right| ,\;M \mathrel{\text{:=}} \mathop{\sup }\limits_{{B\left( {{x}_{0}, R}\right) }}\left| f\right| . \] Without loss of generality, suppose again \( {x}_{... | Yes |
Corollary 2.2.8 (Liouville theorem). Let \( u : {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) be harmonic and bounded. Then \( u \) is constant. | Proof. For \( {x}_{1},{x}_{2} \in {\mathbb{R}}^{d} \), by (2.2.2) for all \( r > 0 \) ,\n\n\[ u\left( {x}_{1}\right) - u\left( {x}_{2}\right) = \frac{1}{{\omega }_{d}{r}^{d}}\left( {{\int }_{B\left( {{x}_{1}, r}\right) }u\left( x\right) \mathrm{d}x - {\int }_{B\left( {{x}_{2}, r}\right) }u\left( x\right) \mathrm{d}x}\r... | Yes |
Corollary 2.2.9 (Harnack inequality). Let \( u : \Omega \rightarrow \mathbb{R} \) be harmonic and nonnegative. Then for every subdomain \( {\Omega }^{\prime } \subset \subset \Omega \) there exists a constant \( c = \) \( c\left( {d,\Omega ,{\Omega }^{\prime }}\right) \) with\n\n\[ \mathop{\sup }\limits_{{\Omega }^{\pr... | Proof. We first consider the special case \( {\Omega }^{\prime } = B\left( {{x}_{0}, r}\right) \), assuming \( B\left( {{x}_{0},{4r}}\right) \subset \Omega \) . Let \( {y}_{1},{y}_{2} \in B\left( {{x}_{0}, r}\right) \) . By (2.2.2),\n\n\[ u\left( {y}_{1}\right) = \frac{1}{{\omega }_{d}{r}^{d}}{\int }_{B\left( {{y}_{1},... | Yes |
Corollary 2.2.10 (Harnack convergence theorem). Let \( {u}_{n} : \Omega \rightarrow \mathbb{R} \) be a monotonically increasing sequence of harmonic functions. If there exists \( y \in \Omega \) for which the sequence \( {\left( {u}_{n}\left( y\right) \right) }_{n \in \mathbb{N}} \) is bounded, then \( {u}_{n} \) conve... | Proof. The monotonicity and boundedness imply that \( {u}_{n}\left( y\right) \) converges for \( n \rightarrow \infty \) . For \( \varepsilon > 0 \), there thus exists \( N \in \mathbb{N} \) such that for \( n \geq m \geq N \) ,\n\n\[ 0 \leq {u}_{n}\left( y\right) - {u}_{m}\left( y\right) < \varepsilon . \]\n\nThen \( ... | Yes |
Lemma 3.1.1. Let \( u \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) ,{\Delta u} \geq 0 \) in \( \Omega \) . Then \[ \mathop{\sup }\limits_{\Omega }u = \mathop{\max }\limits_{{\partial \Omega }}u \] | Proof. We first consider the case where we even have \[ {\Delta u} > 0\;\text{ in }\Omega . \] Then \( u \) cannot assume an interior maximum at some \( {x}_{0} \in \Omega \), since at such a maximum, we would have \[ {u}_{{x}^{i}{x}^{i}}\left( {x}_{0}\right) \leq 0\;\text{ for }i = 1,\ldots, d, \] and thus also \[ {\D... | Yes |
Theorem 3.1.1. Assume \( c\left( x\right) \equiv 0 \), and let \( u \) satisfy in \( \Omega \)\n\n\[ \n{Lu} \geq 0, \n\]\n\ni.e.,\n\n\[ \n\mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) {u}_{{x}^{i}{x}^{j}} + \mathop{\sum }\limits_{{i = 1}}^{d}{b}^{i}\left( x\right) {u}_{{x}^{i}} \geq 0. \n\]\n\n(3.1.2)\n... | Proof. As in the proof of Lemma 3.1.1, we first consider the case\n\n\[ \n{Lu} > 0\text{.} \n\]\n\nSince at an interior maximum \( {x}_{0} \) of \( u \), we must have\n\n\[ \n{u}_{{x}^{i}}\left( {x}_{0}\right) = 0\;\text{ for }i = 1,\ldots, d, \n\]\n\nand\n\n\[ \n{\left( {u}_{{x}^{i}{x}^{j}}\left( {x}_{0}\right) \right... | Yes |
Corollary 3.1.1. Let \( L \) be as in Theorem 3.1.1, and let \( f \in {C}^{0}\left( \Omega \right) ,\varphi \in {C}^{0}\left( {\partial \Omega }\right) \) be given. Then the Dirichlet problem\n\n\[ \n{Lu}\left( x\right) = f\left( x\right) \;\text{ for }x \in \Omega , \n\]\n\n\[ \nu\left( x\right) = \varphi \left( x\rig... | Proof. The difference \( v\left( x\right) = {u}_{1}\left( x\right) - {u}_{2}\left( x\right) \) of two solutions satisfies\n\n\[ \n{Lv}\left( x\right) = 0\;\text{ in }\Omega \n\]\n\n\[ \nv\left( x\right) = 0\;\text{ on }\partial \Omega \n\]\n\nand by Theorem 3.1.1 it then has to vanish identically on \( \Omega \) . | Yes |
Corollary 3.1.2. Suppose \( c\left( x\right) \leq 0 \) in \( \Omega \) . Let \( u \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) \) satisfy\n\n\[ \n{Lu} \geq 0\;\text{ in }\Omega \n\]\n\nWith \( {u}^{ + }\left( x\right) \mathrel{\text{:=}} \max \left( {u\left( x\right) ,0}\right) \), we then h... | Proof. Let \( {\Omega }^{ + } \mathrel{\text{:=}} \{ x \in \Omega : u\left( x\right) > 0\} \) . Because of \( c \leq 0 \), we have in \( {\Omega }^{ + } \),\n\n\[ \n\mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) {u}_{{x}^{i}{x}^{j}} + \mathop{\sum }\limits_{{i = 1}}^{d}{b}^{i}\left( x\right) {u}_{{x}^{i}... | Yes |
Lemma 3.1.2. Suppose \( c\left( x\right) \leq 0 \) and\n\n\[ \n{Lu} \geq 0\;\text{ in }{\Omega }^{\prime } \subset {\mathbb{R}}^{d}, \n\]\n\nand let \( {x}_{0} \in \partial {\Omega }^{\prime } \) . Moreover, assume\n\n(i) \( u \) is continuous at \( {x}_{0} \) .\n\n(ii) \( u\left( {x}_{0}\right) \geq 0 \) if \( c\left(... | Proof. We may assume\n\n\[ \n\partial B\left( {y, R}\right) \cap \partial {\Omega }^{\prime } = \left\{ {x}_{0}\right\} . \n\]\n\nFor \( 0 < \rho < R \), on the annular region \( \overset{ \circ }{B}\left( {y, R}\right) \smallsetminus B\left( {y,\rho }\right) \), we consider the auxiliary\n\nfunction\n\[ \nv\left( x\ri... | Yes |
Lemma 3.2.1. For \( v \in {C}^{2}\left( \Omega \right) \), the Hessian\n\n\[{\left( {v}_{{x}^{i}{x}^{j}}\right) }_{i, j = 1,\ldots, d}\]\n\nis negative semidefinite on \( {T}^{ + }\left( v\right) \) . | Proof. For \( y \in {T}^{ + }\left( v\right) \), we consider the function\n\n\[w\left( x\right) \mathrel{\text{:=}} v\left( x\right) - v\left( y\right) - p\left( y\right) \cdot \left( {x - y}\right) .\n\]\n\nThen \( w\left( x\right) \leq 0 \) on \( \Omega \), since \( y \in {T}^{ + }\left( v\right) \) and \( w\left( y\... | Yes |
Lemma 3.2.2. Let \( v \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) \) . Then | Proof. First of all,\n\n\[{\tau }_{v}\left( \Omega \right) = {\tau }_{v}\left( {{T}^{ + }\left( v\right) }\right) = {Dv}\left( {{T}^{ + }\left( v\right) }\right) ,\]\n\n(3.2.7)\n\nsince \( v \) is differentiable. By Lemma 3.2.1, the Jacobian matrix of \( {Dv} : \Omega \rightarrow {\mathbb{R}}^{d} \) , namely, \( \left(... | Yes |
Lemma 3.2.2. Let \( v \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) \) . Then\n\n\[ \n{\mathcal{L}}_{d}\left( {{\tau }_{v}\left( \Omega \right) }\right) \leq {\int }_{{T}^{ + }\left( v\right) }\left| {\det \left( {{v}_{{x}^{i}{x}^{j}}\left( x\right) }\right) }\right| \mathrm{d}x.\n\]\n\n(3.2.... | Proof. First of all,\n\n\[ \n{\tau }_{v}\left( \Omega \right) = {\tau }_{v}\left( {{T}^{ + }\left( v\right) }\right) = {Dv}\left( {{T}^{ + }\left( v\right) }\right) ,\n\]\n\n(3.2.7)\n\nsince \( v \) is differentiable. By Lemma 3.2.1, the Jacobian matrix of \( {Dv} : \Omega \rightarrow {\mathbb{R}}^{d} \) , namely, \( \... | Yes |
Lemma 3.2.4. On \( {T}^{ + }\left( u\right) \) , \[ {\left( -1\right) }^{d}\det \left( {{u}_{{x}^{i}{x}^{j}}\left( x\right) }\right) \leq 1\det \left( {{a}^{ij}\left( x\right) }\right) {\left( -\frac{1}{d}\mathop{\sum }\limits_{{i, j = 1}}^{d}{a}^{ij}\left( x\right) {u}_{{x}^{i}{x}^{j}}\left( x\right) \right) }^{d}. \] | Proof. It is well known that for symmetric, positive definite matrices \( A \) and \( B \) , \[ \det A\det B \leq {\left( \frac{1}{d}\operatorname{trace}AB\right) }^{d} \] which is readily verified by diagonalizing one of the matrices, which is possible if that matrix is symmetric. Inserting \( A = \left( {-{u}_{{x}^{i... | Yes |
Corollary 3.2.1. Under the assumptions (3.2) and (3.2), a solution \( u \) of the Monge-Ampère equation (3.2.15) satisfies | The crucial point here is that the nonlinear Monge-Ampère equation for a solution \( u \) can be formally written as a linear differential equation. Namely, with | No |
Theorem 3.3.1. Let \( {u}_{0},{u}_{1} \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) \), and suppose\n\n(i) \( F \in {C}^{1}\left( S\right) \).\n\n(ii) \( F \) is elliptic at all functions \( t{u}_{1} + \left( {1 - t}\right) {u}_{0},0 \leq t \leq 1 \).\n\n(iii) For each fixed \( \left( {x, p, ... | Proof. We put\n\n\[ \nv \mathrel{\text{:=}} {u}_{1} - {u}_{0} \n\]\n\n\[ \n{u}_{t} \mathrel{\text{:=}} t{u}_{1} + \left( {1 - t}\right) {u}_{0}\;\text{ for }0 \leq t \leq 1, \n\]\n\n\[ \n{a}^{ij}\left( x\right) \mathrel{\text{:=}} {\int }_{0}^{1}\frac{\partial F}{\partial {r}_{ij}}\left( {x,{u}_{t}\left( x\right), D{u}... | Yes |
Theorem 3.3.2. Let \( u \in {C}^{2}\left( \Omega \right) \cap {C}^{0}\left( \bar{\Omega }\right) \) and let \( F \in {C}^{2}\left( S\right) \) . Suppose that for some \( \lambda > 0 \), the ellipticity condition\n\n\[ \n\lambda {\left| \xi \right| }^{2} \leq \mathop{\sum }\limits_{{i, j = 1}}^{d}\frac{\partial F}{\part... | Proof. We shall follow a similar strategy as in the proof of Theorem 3.3.1 and shall reduce the result to the maximum principle from Sect. 3.1 for linear equations. Here \( v \) is an auxiliary function to be determined, and \( w \mathrel{\text{:=}} u - v \) . We consider the operator\n\n\[ \n{Lw} \mathrel{\text{:=}} \... | Yes |
Theorem 4.1.1. Suppose\n\n\\[ \n{\\Delta }_{h}{u}^{h} \\geq 0\\;\\text{ in }{\\Omega }_{h} \n\\]\n\nwhere \\( {\\Omega }_{h} \\), as always, is supposed to be discretely connected. Then\n\n\\[ \n\\mathop{\\max }\\limits_{{\\bar{\\Omega }}_{h}}{u}^{h} = \\mathop{\\max }\\limits_{{\\Gamma }_{h}}{u}^{h} \n\\]\n\n(4.1.12)\... | Proof. Let \\( {x}_{0} \\) be an interior vertex, and let \\( {x}_{1},\\ldots ,{x}_{2d} \\) be its neighbors. Then\n\n\\[ \n{\\Delta }_{h}{u}^{h}\\left( x\\right) = \\frac{1}{{h}^{2}}\\left( {\\mathop{\\sum }\\limits_{{\\alpha = 1}}^{{2d}}{u}^{h}\\left( {x}_{\\alpha }\\right) - {2d}{u}^{h}\\left( {x}_{0}\\right) }\\rig... | Yes |
Corollary 4.1.1. The discrete Dirichlet problem\n\n\\[ \n{\\Delta }_{h}{u}^{h} = 0\\;\\text{ in }{\\Omega }_{h} \n\\]\n\n\\[ \n{u}^{h} = {g}^{h}\\;\\text{ on }{\\Gamma }^{h}, \n\\]\n\nfor given \\( {g}^{h} \\) has at most one solution. | Proof. This follows in the usual manner by applying the maximum principle to the difference of two solutions. | No |
Corollary 4.1.2. The discrete Dirichlet problem\n\n\\[ \n{\\Delta }_{h}{u}^{h} = 0\\;\\text{ in }{\\Omega }_{h} \n\\]\n\n\\[ \n{u}^{h} = {g}^{h}\\;\\text{ on }{\\Gamma }^{h}, \n\\]\n\nadmits a unique solution for each \\( {g}^{h} : {\\Gamma }_{h} \\rightarrow \\mathbb{R} \\) . | Proof. As already observed, the discrete problem constitutes a finite system of linear equations with the same number of equations and unknowns. Since by Corollary 4.1.1, for homogeneous boundary data \\( {g}^{h} = 0 \\), the homogeneous solution \\( {u}^{h} = 0 \\) is the unique solution, the fundamental theorem of li... | Yes |
Lemma 4.1.1. Suppose that in \( {\Omega }_{h} \), \[ {\Delta }_{h}{u}^{h}\left( x\right) = {f}^{h}\left( x\right) \] Let \( {x}_{0} \in {\Omega }_{h} \), and suppose that \( {x}_{0} \) and all its neighbors have distance greater than or equal to \( R \) from \( {\Gamma }_{h} \). Then \[ \left| {{u}_{\widetilde{\imath }... | Proof. Without loss of generality \( i = 1,{x}_{0} = 0 \) . We put \[ \mu \mathrel{\text{:=}} \mathop{\max }\limits_{{\Omega }_{h}}\left| {u}^{h}\right|, M \mathrel{\text{:=}} \mathop{\max }\limits_{{\Omega }_{h}}\left| {f}^{h}\right| . \] We consider once more the auxiliary function \[ {v}^{h}\left( x\right) \mathrel{... | Yes |
Theorem 4.1.2. Ifall solutions \( {u}^{h} \) of\n\n\[ \n{\Delta }_{h}{u}^{h} = 0\;\text{ in }{\Omega }_{h}\n\]\n\nare bounded independently of \( h \) (i.e., \( \mathop{\max }\limits_{{\Gamma }_{h}}\left| {u}^{h}\right| \leq \mu \) ), then in any subdomain \( \widetilde{\Omega } \subset \subset \Omega \), some subseque... | Convergence here first means convergence with respect to the supremum norm, i.e.,\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow 0}}\mathop{\max }\limits_{{x \in {\Omega }_{n}}}\left| {{u}_{n}\left( x\right) - u\left( x\right) }\right| = 0\n\]\n\nwith harmonic \( u \) . By the preceding considerations, however, the diff... | Yes |
Theorem 4.1.3. Let \( u \in {C}^{2}\left( \bar{\Omega }\right) \) be a solution of \[ {\Delta u} = f\;\text{ in }\Omega \] \[ u = \varphi \;\text{ on }\partial \Omega . \] Let \( {u}^{h} \) be the solution \[ {\Delta }_{h}{u}^{h} = {f}^{h}\;\text{ in }{\Omega }_{h} \] \[ {u}^{h} = {\varphi }^{h}\;\text{ on }{\Gamma }_{... | Proof. Taylor's formula implies that the second-order difference quotients (which depend on the mesh size \( h \) ) satisfy \[ {u}_{i\bar{\imath }}\left( x\right) = \frac{{\partial }^{2}u}{{\left( \partial {x}^{i}\right) }^{2}}\left( {{x}^{1},\ldots ,{x}^{i - 1},{x}^{i} + {\delta }^{i},{x}^{i + 1},\ldots ,{x}^{d}}\righ... | Yes |
Lemma 4.2.1. (i) Strong maximum principle: Let \( v \) be subharmonic in \( \Omega \) . If there exists \( {x}_{0} \in \Omega \) with \( v\left( {x}_{0}\right) = \mathop{\sup }\limits_{\Omega }v\left( x\right) \), then \( v \) is constant. In particular, if \( v \in {C}^{0}\left( \bar{\Omega }\right) \), then \( v\left... | Proof. (i) This is the strong maximum principle for subharmonic functions. Although we have not written it down explicitly, it is a direct consequence of Theorem 2.2.2 and Lemma 2.2.1. | No |
Lemma 4.2.2. Suppose \( u\left( x\right) \mathrel{\text{:=}} \mathop{\sup }\limits_{{v \in {S}_{\omega }}}v\left( x\right) \) in \( \Omega \) . If \( \xi \) is a regular point of \( \partial \Omega \) and \( \varphi \) is continuous at \( \xi \), we have\n\n\[ \mathop{\lim }\limits_{{x \rightarrow \xi }}u\left( x\right... | Proof. Let \( M \mathrel{\text{:=}} \mathop{\sup }\limits_{{\partial \Omega }}\left| \varphi \right| \) . Since \( \xi \) is regular, there exists a barrier \( \beta \), and the continuity of \( \varphi \) at \( \xi \) implies that for every \( \varepsilon > 0 \) there exists \( \delta > 0 \) and a constant \( c = c\le... | Yes |
Theorem 4.2.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be bounded. The Dirichlet problem\n\n\[ \n{\Delta u} = 0\;\text{ in }\Omega ,\n\]\n\n\[ \nu = \varphi \;\text{ on }\partial \Omega \n\]\n\nis solvable for all continuous boundary values \( \varphi \) if and only if all points \( \xi \in \partial \Omega \) are reg... | Proof. If \( \varphi \) is continuous and \( \partial \Omega \) is regular, then \( u \mathrel{\text{:=}} \mathop{\sup }\limits_{{v \in {S}_{\varphi }}}v \) solves the Dirichlet problem by Theorem 4.2.1. Conversely, if the Dirichlet problem is solvable for all continuous boundary values, we consider \( \xi \in \partial... | Yes |
Lemma 4.4.1. If \( \Omega \) satisfies an exterior sphere condition at \( y \), then \( \partial \Omega \) is regular at \( y \) . | Proof.\n\n\[ \beta \left( x\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} \frac{1}{{\rho }^{d - 2}} - \frac{1}{{\left| x - {x}_{0}\right| }^{d - 2}} & \text{ for }d \geq 3, \\ \ln \frac{\left| x - {x}_{0}\right| }{\rho } & \text{ for }d = 2, \end{array}\right. \]\n\nyields a barrier at \( y \) . Namely, \( \beta... | Yes |
Theorem 5.1.1. Let \( u \) be as in the assumptions of (5.1.1). Let \( \Omega \subset {\mathbb{R}}^{d} \) be open and bounded and\n\n\[ \n{\Delta u} - {u}_{t} \geq 0\;\text{ in }{\Omega }_{T} \n\]\n\n(5.1.17)\n\nWe then have\n\n\[ \n\mathop{\sup }\limits_{{\bar{\Omega }}_{T}}u = \mathop{\sup }\limits_{{{\partial }^{ * ... | Proof. Without loss of generality \( T < \infty \) .\n\n(i) Suppose first\n\n\[ \n{\Delta u} - {u}_{t} > 0\;\text{ in }{\Omega }_{T} \n\]\n\n(5.1.19)\n\nFor \( 0 < \varepsilon < T \), by continuity of \( u \) and compactness of \( {\bar{\Omega }}_{T - \varepsilon } \), there exists \( \left( {{x}_{0},{t}_{0}}\right) \i... | Yes |
Corollary 5.1.1. Let \( u, v \) be solutions of (5.1.1) with \( u = v \) on \( {\partial }^{ * }{\Omega }_{T} \), where \( \Omega \subset \) \( {\mathbb{R}}^{d} \) is bounded. Then \( u = v \) on \( {\bar{\Omega }}_{T} \) . | Proof. We apply Theorem 5.1.1 to \( u - v \) and \( v - u \) . | No |
Theorem 5.1.3. Let \( \Omega \subset {\mathbb{R}}^{d} \) be open and bounded and \[ {\Delta u} - {u}_{t} = 0\;\text{ in }{\Omega }_{T}, \] with the regularity properties specified at the beginning of this section. Then if there exists some \( \left( {{x}_{0},{t}_{0}}\right) \in \Omega \times (0, T\rbrack \) with \[ u\l... | Proof. The proof is the same as that of Lemma 2.2.1, using the representation formula (5.1.12). (Note that by applying (5.1.15) to the function \( u \equiv 1 \), we obtain \[ \mu {\int }_{\partial M\left( {y, T;\mu }\right) }\frac{\left| x - y\right| }{2\left( {T - t}\right) }{do}\left( {x, t}\right) = 1 \] and so a ge... | Yes |
Corollary 5.1.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be open and bounded and\n\n\[ \n{\Delta u}\left( {x, t}\right) + c\left( {x, t}\right) u\left( {x, t}\right) - {u}_{t}\left( {x, t}\right) \geq 0\;\text{ in }{\Omega }_{T}, \n\] \n\nwith some bounded function\n\n\[ \n c\left( {x, t}\right) \leq 0\;\text{ in }{\... | Proof. Our scheme of proof still applies because, since \( c \) is nonpositive, at a nonnegative maximum point \( \left( {{x}_{0},{t}_{0}}\right) \) of \( u, c\left( {{x}_{0},{t}_{0}}\right) u\left( {{x}_{0},{t}_{0}}\right) \leq 0 \) which strengthens the inequality used in the proof.\n\nAgain, we obtain a minimum prin... | Yes |
Lemma 5.1.1. Suppose the function \( c \) is bounded and satisfies \( c\left( {x, t}\right) \leq 0 \) in \( {\Omega }_{T} \) . Let \( u \) solve the differential inequality\n\n\[ \n{\Delta u}\left( {x, t}\right) + c\left( {x, t}\right) u\left( {x, t}\right) - {u}_{t}\left( {x, t}\right) \geq 0\;\text{ in }{\Omega }_{T}... | Proof. With the auxiliary function\n\n\[ \nv\left( x\right) \mathrel{\text{:=}} {\mathrm{e}}^{-\gamma \left( {{\left| x - y\right| }^{2} + {\left( t - {t}_{1}}\right) }^{2}}\right) } - {\mathrm{e}}^{-\gamma {R}^{2}}, \n\]\n\nthe proof proceeds as the one of Lemma 3.1.2, employing this time the maximum principle Theorem... | No |
Lemma 5.2.1. Let \( f : {\mathbb{R}}^{d} \rightarrow \mathbb{R} \) be bounded and continuous. Then\n\n\[ u\left( {x, t}\right) = {\int }_{{\mathbb{R}}^{d}}K\left( {x, y, t}\right) f\left( y\right) \mathrm{d}y \]\n\nis of class \( {C}^{\infty } \) on \( {\mathbb{R}}^{d} \times \left( {0,\infty }\right) \), and it solves... | Proof. That \( u \) is of class \( {C}^{\infty } \) follows, by differentiating under the integral (which is permitted by standard theorems), from the \( {C}^{\infty } \) property of \( K\left( {x, y, t}\right) \) . Consequently, we also obtain\n\n\[ \frac{\partial }{\partial t}u\left( {x, t}\right) = {\int }_{{\mathbb... | Yes |
Lemma 5.2.2. Under the assumptions of Lemma 5.2.1, we have for every \( x \in {\mathbb{R}}^{d} \)\n\n\[ \mathop{\lim }\limits_{{t \rightarrow 0}}u\left( {x, t}\right) = f\left( x\right) \] | Proof.\n\n\[ \left| {f\left( x\right) - u\left( {x, t}\right) }\right| = \left| {f\left( x\right) - {\int }_{{\mathbb{R}}^{d}}K\left( {x, y, t}\right) f\left( y\right) \mathrm{d}y}\right| \]\n\n\[ = \left| {{\int }_{{\mathbb{R}}^{d}}K\left( {x, y, t}\right) \left( {f\left( x\right) - f\left( y\right) }\right) \mathrm{d... | Yes |
Theorem 5.2.1. Let \( \Omega \) be a bounded domain in \( {\mathbb{R}}^{d} \), and let \( g\left( {x, t}\right) \) be continuous on \( \partial \Omega \times \left( {0,\infty }\right) \), and suppose\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}g\left( {x, t}\right) = g\left( x\right) \;\text{ uniformly in }x \... | Proof. We consider the difference\n\n\[ w\left( {x, t}\right) = u\left( {x, t}\right) - v\left( x\right) . \]\n\n(5.2.23)\n\nThen\n\n\[ {\Delta w}\left( {x, t}\right) - \frac{\partial }{\partial t}w\left( {x, t}\right) = F\left( {x, t}\right) - F\left( x\right) \;\text{ in }\Omega \times \left( {0,\infty }\right) , \]\... | Yes |
Lemma 5.2.3. Let \( \Omega \) be a bounded domain in \( {\mathbb{R}}^{d} \), let \( \phi \left( {x, t}\right) \) be continuous on \( \Omega \times \left( {0,\infty }\right) \), and suppose\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}\phi \left( {x, t}\right) = 0\;\text{ uniformly in }x \in \Omega . \]\n\n(5.2.... | Proof. We choose \( R > 0 \) such that\n\n\[ 2{x}^{1} < R\;\text{ for all }x = \left( {{x}^{1},\ldots ,{x}^{d}}\right) \in \Omega ,\]\n\n(5.2.29)\n\nand consider\n\n\[ k\left( x\right) \mathrel{\text{:=}} {\mathrm{e}}^{R} - {\mathrm{e}}^{{x}^{1}}. \]\n\n(5.2.30)\n\nThen\n\n\[ {\Delta k} = - {\mathrm{e}}^{{x}^{1}}. \]\n... | Yes |
Lemma 5.3.1. Let \( \Omega \) be a bounded domain of class \( {C}^{2} \) in \( {\mathbb{R}}^{d} \). Then for every \( \alpha < \frac{d}{2} + 1, T > 0 \), there exists a constant \( c = c\left( {\alpha, d,\Omega }\right) \) such that for all \( {x}_{0}, x \in \partial \Omega \), \( 0 < t \leq T \), letting \( v \) denot... | Proof. \[ \frac{\partial }{\partial {v}_{x}}K\left( {x,{x}_{0}, t}\right) = \frac{1}{{\left( 4\pi t\right) }^{\frac{d}{2}}}\frac{\partial }{\partial {v}_{x}}{\mathrm{e}}^{-\frac{{\left| x - {x}_{0}\right| }^{2}}{4t}} = - \frac{1}{{\left( 4\pi t\right) }^{\frac{d}{2}}}\frac{\left( {x - {x}_{0}}\right) \cdot {v}_{x}}{2t}... | Yes |
Lemma 5.3.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be a bounded domain of class \( {C}^{2} \) with exterior normal \( v \), and let \( \gamma \in {C}^{0}\left( {\partial \Omega \times \left\lbrack {0, T}\right\rbrack }\right) \left( {T > 0}\right) \) . We put\n\n\[ v\left( {x, t}\right) \mathrel{\text{:=}} - {\int ... | Proof. First of all, Lemma 5.3.1, with \( \alpha = \frac{3}{4} \), implies that the integral in (5.3.5) indeed exists. The \( {C}^{\infty } \) -regularity of \( v \) with respect to \( x \) then follows from the corresponding regularity of the kernel \( K \) by the change of variables \( \sigma = t - \tau \) . Equation... | Yes |
Theorem 5.3.1. The initial boundary value problem for the heat equation on a bounded domain \( \Omega \subset {\mathbb{R}}^{d} \) of class \( {C}^{2} \), namely,\n\n\[ \n{\Delta u}\left( {x, t}\right) - \frac{\partial }{\partial t}u\left( {x, t}\right) = 0\;\text{ in }\Omega \times \left( {0,\infty }\right) ,\n\]\n\n\[... | Proof. Since the series \( \mathop{\sum }\limits_{{v = 1}}^{\infty }{S}_{v} \) converges,\n\n\[ \gamma \left( {{x}_{0}, t}\right) = {2g}\left( {{x}_{0}, t}\right) + 2{\int }_{0}^{t}{\int }_{\partial \Omega }\mathop{\sum }\limits_{{v = 1}}^{\infty }{S}_{v}\left( {{x}_{0}, y, t - \tau }\right) g\left( {y,\tau }\right) {d... | Yes |
Corollary 5.3.1. Any bounded domain \( \Omega \subset {\mathbb{R}}^{d} \) of class \( {C}^{2} \) has a heat kernel, and this heat kernel is of class \( {C}^{1} \) on \( \bar{\Omega } \) with respect to the spatial variables \( y \) . The heat kernel is positive in \( \Omega \), for all \( t > 0 \) . | Proof. For each \( y \in \Omega \), by Theorem 5.3.1, we solve the boundary value problem for the heat equation with initial values 0 and\n\n\[ g\left( {x, t}\right) = - K\left( {x, y, t}\right) . \]\n\nThe solution is called \( \mu \left( {x, y, t}\right) \), and we put\n\n\[ q\left( {x, y, t}\right) \mathrel{\text{:=... | Yes |
Lemma 5.3.4 (Duhamel principle). For all functions \( u, v \) on \( \bar{\Omega } \times \left\lbrack {0, T}\right\rbrack \) with the appropriate regularity conditions, we have\n\n\[ \n{\int }_{0}^{T}{\int }_{\Omega }\left\{ {v\left( {x, t}\right) \left( {{\Delta u}\left( {x, T - t}\right) + {u}_{t}\left( {x, T - t}\ri... | Proof. Same as the proof of (5.1.12) | No |
Corollary 5.3.2. If the heat kernel \( q\left( {z, w, T}\right) \) of \( \Omega \) is of class \( {C}^{1} \) on \( \bar{\Omega } \) with respect to the spatial variables, then it is symmetric with respect to \( z \) and \( w \), i.e., \[ q\left( {z, w, T}\right) = q\left( {w, z, T}\right) \;\text{ for all }z, w \in \Om... | Proof. In (5.3.25), we put \( u\left( {x, t}\right) = q\left( {x, z, t}\right), v\left( {x, t}\right) = q\left( {x, w, t}\right) \) . The double integrals vanish by properties (i) and (ii) of Definition 5.3.1. Property (iii) of Definition 5.3.1 then yields \( v\left( {z, T}\right) = u\left( {w, T}\right) \), which is t... | Yes |
Theorem 5.3.2. Let \( \Omega \subset {\mathbb{R}}^{d} \) be a bounded domain of class \( {C}^{2} \) with heat kernel \( q\left( {x, y, t}\right) \) according to Corollary 5.3.1, and let\n\n\[ \varphi \in {C}^{0}\left( {\bar{\Omega }\times \lbrack 0,\infty }\right) ),\;g \in {C}^{0}\left( {\partial \Omega \times \left( ... | Proof. Uniqueness follows from the maximum principle. We split the existence problem into two subproblems. We solve\n\n\[ {v}_{t}\left( {x, t}\right) - {\Delta v}\left( {x, t}\right) = 0\;\text{ for }x \in \Omega, t > 0, \]\n\n\[ v\left( {x, t}\right) = g\left( {x, t}\right) \;\text{ for }x \in \partial \Omega, t > 0, ... | Yes |
Theorem 5.3.4. Any solution \( u\left( {x, t}\right) \) of the heat equation in a domain \( \Omega \) is of class \( {C}^{\infty } \) with respect to \( x \in \Omega, t > 0 \) . | Proof. Since we do not know whether the normal derivative \( \frac{\partial u}{\partial v} \) exists on \( \partial \Omega \) and is continuous there, we cannot apply (5.3.48) directly. Instead, for given \( x \in \Omega \) , we consider some ball \( B\left( {x, r}\right) \) contained in \( \Omega \) . We then apply (5... | No |
Theorem 6.1.1. Let \( \Omega \subset {\mathbb{R}}^{d} \) be a bounded domain of class \( {C}^{2} \), and let\n\n\[ g \in {C}^{0}\left( {\partial \Omega \times \left\lbrack {0,{t}_{0}}\right\rbrack }\right) ,\;f \in {C}^{0}\left( \bar{\Omega }\right) ,\]\n\n\[ \text{with}g\left( {x,0}\right) = f\left( x\right) \;\text{f... | Proof. Let \( q\left( {x, y, t}\right) \) be the heat kernel of \( \Omega \) of Corollary 5.3.1. According to (5.3.28), a solution then needs to satisfy\n\n\[ u\left( {x, t}\right) = {\int }_{0}^{t}{\int }_{\Omega }q\left( {x, y, t - \tau }\right) F\left( {y,\tau, u\left( {y,\tau }\right) }\right) \mathrm{d}y\mathrm{\;... | Yes |
Lemma 6.1.1. Let \( {u}_{1}\left( {x, t}\right) ,{u}_{2}\left( {x, t}\right) \in {C}^{0}\left( {\bar{\Omega } \times \left\lbrack {0, T}\right\rbrack }\right) \) be solutions of (6.1.8) with \( {u}_{i}\left( {x, t}\right) = g\left( {x, t}\right) \) for \( x \in \partial \Omega ,0 \leq t \leq T,\left| {{u}_{i}\left( {x,... | Proof. By the representation formula (5.3.28),\n\n\[ {u}_{1}\left( {x, t}\right) - {u}_{2}\left( {x, t}\right) = {\int }_{\Omega }q\left( {x, y, t}\right) \left( {{u}_{1}\left( {y,0}\right) - {u}_{2}\left( {y,0}\right) }\right) \mathrm{d}y \]\n\n\[ + {\int }_{0}^{t}{\int }_{\Omega }q\left( {x, y, t - \tau }\right) \lef... | Yes |
Lemma 6.1.2. Let the integrable function \( \phi : \left\lbrack {0, T}\right\rbrack \rightarrow {\mathbb{R}}^{ + } \) satisfy\n\n\[ \phi \left( t\right) \leq \phi \left( 0\right) + c{\int }_{0}^{t}\phi \left( \tau \right) \mathrm{d}\tau \]\n\n(6.1.21)\n\nfor all \( 0 \leq t \leq T \) and some constant \( c \) . Then fo... | Proof. From (6.1.21)\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}\left( {{\mathrm{e}}^{-{ct}}{\int }_{0}^{t}\phi \left( \tau \right) \mathrm{d}\tau }\right) \leq {\mathrm{e}}^{-{ct}}\phi \left( 0\right) \]\n\nhence\n\n\[ {\mathrm{e}}^{-{ct}}{\int }_{0}^{t}\phi \left( \tau \right) \mathrm{d}\tau \leq \frac{1 - {\mathrm{e}}^{-{c... | Yes |
Corollary 6.1.1. Under the assumptions of Theorem 6.1.1, suppose that the solution \( u\left( {x, t}\right) \) of (6.1.8) satisfies the a priori bound\n\n\[ \mathop{\sup }\limits_{{x \in \bar{\Omega },0 \leq \tau \leq t}}\left| {u\left( {x,\tau }\right) }\right| \leq K \]\n\n(6.1.23)\n\nfor all times \( t \) for which ... | Proof. Suppose the solution exists for \( 0 \leq t \leq T \) . Then we apply Theorem 6.1.1 at time \( T \) instead of 0, with initial values \( u\left( {x, T}\right) \) in place of the original initial values \( u\left( {x,0}\right) \) and conclude that the solution continues to exist on some interval \( \left\lbrack {... | Yes |
Lemma 6.1.3. Let \( u, v \) be of class \( {C}^{2} \) w.r.t. \( x \in \Omega \), of class \( {C}^{1} \) w.r.t. \( t \in \left\lbrack {0, T}\right\rbrack \), and satisfy\n\n\[ \n{u}_{t}\left( {x, t}\right) - {\Delta u}\left( {x, t}\right) - F\left( {x, t, u}\right) \geq {v}_{t}\left( {x, t}\right) - {\Delta v}\left( {x,... | Proof. \( w\left( {x, t}\right) \mathrel{\text{:=}} u\left( {x, t}\right) - v\left( {x, t}\right) \) satisfies \( w\left( {x,0}\right) \geq 0 \) in \( \Omega \) and \( \frac{\partial w}{\partial v} \geq 0 \) on \( \partial \Omega \times \left\lbrack {0, T}\right\rbrack \), as well as\n\n\[ \n{w}_{t}\left( {x, t}\right)... | Yes |
Theorem 6.2.1. Assume that \( u\left( {x, t}\right) \) is a bounded solution of (6.2.1) with homogeneous Neumann boundary conditions (6.2.4). Assume that\n\n\[ \delta \mathrel{\text{:=}} {d}_{0}{\lambda }_{1} - L > 0. \]\n\nThen\n\n\[ {\int }_{\Omega }\mathop{\sum }\limits_{{i = 1}}^{d}{\left| {u}_{{x}^{i}}\left( x, t\... | Proof. We put, similarly to Sect. 5.2,\n\n\[ E\left( {u\left( {\cdot, t}\right) }\right) = \frac{1}{2}{\int }_{\Omega }\mathop{\sum }\limits_{{i = 1}}^{d}\mathop{\sum }\limits_{{\alpha = 1}}^{n}\frac{1}{{d}_{\alpha }}{\left( {u}_{{x}^{i}}^{\alpha }\right) }^{2}\mathrm{\;d}x \]\n\nand compute\n\n\[ \frac{\mathrm{d}}{\ma... | Yes |
Theorem 7.1.1. The solution of the initial value problem\n\n\[ \n{u}_{tt}\left( {x, t}\right) - {u}_{xx}\left( {x, t}\right) = 0\;\text{ for }x \in \mathbb{R}, t > 0, \n\]\n\n\[ \nu\left( {x,0}\right) = f\left( x\right) \]\n\n\[ {u}_{t}\left( {x,0}\right) = g\left( x\right) \]\n\nis given by\n\n\[ u\left( {x, t}\right)... | \[ u\left( {x, t}\right) = \varphi \left( {x + t}\right) + \psi \left( {x - t}\right) \]\n\n\[ = \frac{1}{2}\{ f\left( {x + t}\right) + f\left( {x - t}\right) \} + \frac{1}{2}{\int }_{x - t}^{x + t}g\left( y\right) \mathrm{d}y. \] | Yes |
Theorem 7.3.1. Let \( u \) be a solution of (7.3.12) with\n\n\[ u\left( {x,0}\right) = f\left( x\right) ,\;{u}_{t}\left( {x,0}\right) = 0 \]\n\n(7.3.15)\n\nand let \( K \mathrel{\text{:=}} \operatorname{supp}f\left( { \mathrel{\text{:=}} \overline{\left\{ x \in {\mathbb{R}}^{d} : f\left( x\right) \neq 0\right\} }}\righ... | Proof. We show that \( f\left( y\right) = 0 \) for all \( y \in B\left( {x, T}\right) \) implies \( u\left( {x, T}\right) \geq 0 \), which is equivalent to our assertion. We put\n\n\[ \bar{E}\left( t\right) \mathrel{\text{:=}} \frac{1}{2}{\int }_{B\left( {x, T - t}\right) }\left\{ {{u}_{t}^{2} + \mathop{\sum }\limits_{... | Yes |
Theorem 7.4.1 (Darboux equation). For \( v \in {C}^{2}\left( {\mathbb{R}}^{d}\right) \) , \n\n\[ \n\left( {\frac{\partial }{\partial {r}^{2}} + \frac{d - 1}{r}\frac{\partial }{\partial r}}\right) S\left( {v, x, r}\right) = {\Delta }_{x}S\left( {v, x, r}\right) . \n\] | Proof. We have \n\n\[ \nS\left( {v, x, r}\right) = \frac{1}{d{\omega }_{d}}{\int }_{\left| \xi \right| = 1}v\left( {x + {r\xi }}\right) {do}\left( \xi \right) , \n\] \n\nand hence \n\n\[ \n\frac{\partial }{\partial r}S\left( {v, x, r}\right) = \frac{1}{d{\omega }_{d}}{\int }_{\left| \xi \right| = 1}\mathop{\sum }\limit... | Yes |
Corollary 7.4.1. Let \( u\left( {x, t}\right) \) be a solution of the initial value problem for the wave equation\n\n\[ \n{u}_{tt}\left( {x, t}\right) - \Delta \left( {x, t}\right) = 0\;\text{ for }x \in {\mathbb{R}}^{d}, t > 0, \n\]\n\n\[ \nu\left( {x,0}\right) = f\left( x\right) \n\]\n\n\[ \n{u}_{t}\left( {x,0}\right... | Proof. By the first line of (7.4.4),\n\n\[ \n\left( {\frac{{\partial }^{2}}{\partial {r}^{2}} + \frac{d - 1}{r}\frac{\partial }{\partial r}}\right) M\left( {u, x, r, t}\right) = \frac{1}{d{\omega }_{d}{r}^{d - 1}}{\int }_{\partial B\left( {x, r}\right) }{\Delta }_{y}u\left( {y, t}\right) {do}\left( y\right) \n\]\n\n\[ ... | Yes |
The unique solution of the initial value problem for the wave equation in 3 space dimensions,\n\n\[ \n{u}_{tt}\left( {x, t}\right) - {\Delta u}\left( {x, t}\right) = 0\;\text{ for }x \in {\mathbb{R}}^{3}, t > 0, \n\]\n\n\[ \nu\left( {x,0}\right) = f\left( x\right) \n\]\n\n(7.4.17)\n\n\[ \n{u}_{t}\left( {x,0}\right) = g... | Proof. First of all, (7.4.16) yields\n\n\[ \nu\left( {x, t}\right) = \frac{1}{4\pi t}{\int }_{\partial B\left( {x, t}\right) }g\left( y\right) {do}\left( y\right) + \frac{\partial }{\partial t}\left( {\frac{1}{4\pi t}{\int }_{\partial B\left( {x, t}\right) }f\left( y\right) {do}\left( y\right) }\right) .\n\]\n\n(7.4.19... | Yes |
For all \( v \in D\left( A\right) \) and all \( t \geq 0 \), we have\n\n\[ \n{T}_{t}{Av} = A{T}_{t}v \n\]\n\nThus \( A \) commutes with all the \( {T}_{t} \) . | Proof. For \( v \in D\left( A\right) \), we have\n\n\[ \n{T}_{t}{Av} = {T}_{t}\mathop{\lim }\limits_{{\tau \searrow 0}}\frac{1}{\tau }\left( {{T}_{\tau } - \mathrm{{Id}}}\right) v \n\]\n\n\[ \n= \mathop{\lim }\limits_{{\tau \searrow 0}}\frac{1}{\tau }\left( {{T}_{t}{T}_{\tau } - {T}_{t}}\right) v\text{(since}{T}_{t}\te... | Yes |
Lemma 8.2.2. For all \( v \in B \), we have\n\n\[ \mathop{\lim }\limits_{{\lambda \rightarrow \infty }}{J}_{\lambda }v = v \] | Proof. By (8.2.8),\n\n\[ {J}_{\lambda }v - v = {\int }_{0}^{\infty }\lambda {\mathrm{e}}^{-{\lambda s}}\left( {{T}_{s}v - v}\right) \mathrm{d}s. \]\n\nFor \( \delta > 0 \), let\n\n\[ {I}_{\lambda }^{1} \mathrel{\text{:=}} \begin{Vmatrix}{{\int }_{0}^{\delta }\lambda {\mathrm{e}}^{-{\lambda s}}\left( {{T}_{s}v - v}\righ... | Yes |
Theorem 8.2.1. Let \( {\left\{ {T}_{t}\right\} }_{t \geq 0} \) be a contracting semigroup with infinitesimal generator \( A \) . Then \( D\left( A\right) \) is dense in \( B \) . | Proof. We shall show that for all \( \lambda > 0 \) and all \( v \in B \) ,\n\n\[ \n{J}_{\lambda }v \in D\left( A\right) \text{.} \n\]\n\n(8.2.11)\n\nSince by Lemma 8.2.2,\n\n\[ \n\left\{ {{J}_{\lambda }v : \lambda > 0, v \in B}\right\} \n\]\n\nis dense in \( B \), this will imply the assertion. We have\n\n\[ \n\frac{1... | Yes |
Lemma 8.2.3. \( v \in D\left( A\right) \) implies \( v \in D\left( {{D}_{t}{T}_{t}}\right) \), and we have\n\n\[ \n{D}_{t}{T}_{t}v = A{T}_{t}v = {T}_{t}{Av}\;\text{ for }t \geq 0. \n\] | Proof. The second equation has already been established as shown in Lemma 8.2.1. We thus have for \( v \in D\left( A\right) \) ,\n\n\[ \n\mathop{\lim }\limits_{{h \searrow 0}}\frac{1}{h}\left( {{T}_{t + h} - {T}_{t}}\right) v = A{T}_{t}v = {T}_{t}{Av}. \n\]\n\n(8.2.15)\n\nEquation (8.2.15) means that the right derivati... | Yes |
Theorem 8.2.2. For \( \lambda > 0 \), the operator \( \left( {\lambda \operatorname{Id} - A}\right) : D\left( A\right) \rightarrow B \) is invertible \( (A \) being the infinitesimal generator of a contracting semigroup), and we have\n\n\[ \n{\left( \lambda \operatorname{Id} - A\right) }^{-1} = R\left( {\lambda, A}\rig... | Proof. In order that \( \left( {\lambda \operatorname{Id} - A}\right) \) be invertible, we need to show first that \( \left( {\lambda \operatorname{Id} - A}\right) \) is injective. So, we need to exclude that there exists \( {v}_{0} \in D\left( A\right) ,{v}_{0} \neq 0 \), with\n\n\[ \n\lambda {v}_{0} = A{v}_{0}\n\]\n\... | Yes |
Lemma 8.2.4 (Resolvent equation). Under the assumptions of Theorem 8.2.2, we have for \( \lambda ,\mu > 0 \) ,\n\n\[ R\left( {\lambda, A}\right) - R\left( {\mu, A}\right) = \left( {\mu - \lambda }\right) R\left( {\lambda, A}\right) R\left( {\mu, A}\right) . \] | Proof.\n\n\[ R\left( {\lambda, A}\right) = R\left( {\lambda, A}\right) \left( {\mu \operatorname{Id} - A}\right) R\left( {\mu, A}\right) \]\n\n\[ = R\left( {\lambda, A}\right) \left( {\left( {\mu - \lambda }\right) \operatorname{Id} + \left( {\lambda \operatorname{Id} - A}\right) }\right) R\left( {\mu, A}\right) \]\n\n... | Yes |
Lemma 8.2.6. Let \( B \) be a Banach space, \( L : B \rightarrow B \) a continuous linear operator with \( \parallel L\parallel \leq 1 \) . Then for every \( t \geq 0 \) and each \( x \in B \), the series\n\n\[ \exp \left( {tL}\right) x \mathrel{\text{:=}} \mathop{\sum }\limits_{{v = 0}}^{\infty }\frac{1}{v!}{\left( tL... | Proof. Because of \( \parallel L\parallel \leq 1 \), we also have \n\n\[ \begin{Vmatrix}{L}^{n}\end{Vmatrix} \leq 1\;\text{ for all }n \in \mathbb{N}. \] \n\n(8.2.63) \n\nThus \n\n\[ \begin{Vmatrix}{\mathop{\sum }\limits_{{v = m}}^{n}\frac{1}{v!}{\left( tL\right) }^{v}x}\end{Vmatrix} \leq \mathop{\sum }\limits_{{v = m}... | Yes |
Lemma 10.1.2. Dirichlet's integral is convex, i.e., | \[ D\left( {{tu} + \left( {1 - t}\right) v}\right) = {\int }_{\Omega }{\left| t\nabla u + \left( 1 - t\right) \nabla v\right| }^{2} \] \[ \leq {\int }_{\Omega }\left\{ {t{\left| \nabla u\right| }^{2} + \left( {1 - t}\right) {\left| \nabla v\right| }^{2}}\right\} \] because of the convexity of \( w \mapsto {\left| w\rig... | Yes |
Lemma 10.2.1. Let \( u \in {L}_{\mathrm{{loc}}}^{1}\left( \Omega \right) \), and suppose \( v = {D}_{i}u \) exists. If \( \operatorname{dist}\left( {x,\partial \Omega }\right) > h \) , we have\n\n\[ \n{D}_{i}\left( {{u}_{h}\left( x\right) }\right) = {\left( {D}_{i}u\right) }_{h}\left( x\right) .\n\] | Proof. By differentiating under the integral, we obtain\n\n\[ \n{D}_{i}\left( {{u}_{h}\left( x\right) }\right) = \frac{1}{{h}^{d}}\int \frac{\partial }{\partial {x}^{i}}\varrho \left( \frac{x - y}{h}\right) u\left( y\right) \mathrm{d}y \n\]\n\n\[ \n= \frac{-1}{{h}^{d}}\int \frac{\partial }{\partial {y}^{i}}\varrho \lef... | Yes |
Corollary 10.2.1. \( {W}^{1,2}\left( \Omega \right) \) is complete with respect to \( \parallel \cdot {\parallel }_{{W}^{1,2}} \), and is hence a Hilbert space. \( {W}^{1,2}\left( \Omega \right) = {H}^{1,2}\left( \Omega \right) \) . | Proof. Let \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) be a Cauchy sequence in \( {W}^{1,2}\left( \Omega \right) \) . Then \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \), \( {\left( {D}_{i}{u}_{n}\right) }_{n \in \mathbb{N}}\left( {i = 1,\ldots, d}\right) \) are Cauchy sequences in \( {L}^{2}\left( \Omega \rig... | Yes |
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