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Proposition 7.19. \( \beta \geq \alpha \) . | Proof. Let \( K \) be a constant such that \( \left| {f}^{\prime }\right| \leq K \) . By the mean value theorem, we have\n\n\[ \left| {\mathbb{E}f\left( {X}_{t}^{\delta t}\right) - \mathbb{E}f\left( {X}_{t}\right) }\right| \leq \mathbb{E}\left| {f\left( {X}_{t}^{\delta t}\right) - f\left( {X}_{t}\right) }\right| \leq K... | Yes |
Theorem 7.20 (Convergence order). Define the length of the multi-index \( \mathbf{i} = \left( {{i}_{1},{i}_{2},\ldots ,{i}_{k}}\right) \) as\n\n\[ l\left( \mathbf{i}\right) \mathrel{\text{:=}} k,\;n\left( \mathbf{i}\right) \mathrel{\text{:=}} \{ \text{ the number of zeros in }\mathbf{i}\} \]\n\nand the set of indices\n... | The proof and detailed requirements about the smoothness conditions on \( b,\sigma \), and \( f \) may be found in [KP92, Theorems 10.6.3 and 14.5.1]. | Yes |
Lemma 7.21. Let \( {\delta t} = \mathop{\max }\limits_{n}\delta {t}_{n} \) . We have the following bounds for \( {X}_{t} \) :\n\n\[ \mathop{\sup }\limits_{{t \leq T}}\mathbb{E}{\left| {X}_{t}\right| }^{2} \leq {K}_{1}\left( T\right) ,\;\mathop{\sup }\limits_{{t \in \left\lbrack {{t}_{n},{t}_{n + 1}}\right) }}\mathbb{E}... | Proof. Applying Itô’s formula to \( {\left| {X}_{t}\right| }^{2} \), we have\n\n\[ d{\left| {X}_{t}\right| }^{2} = 2{X}_{t} \cdot \left( {b\left( {X}_{t}\right) + d{W}_{t}}\right) + {dt}. \]\n\nIntegrating from 0 to \( t \) and taking the expectation, we have\n\n\[ \mathbb{E}{\left| {X}_{t}\right| }^{2} = \mathbb{E}{\l... | Yes |
Proposition 7.22. The Euler-Maruyama scheme is of strong order 1/2. | Proof. From (7.62) we have\n\n\[ \n{X}_{{t}_{n + 1}} = {X}_{{t}_{n}} + {\int }_{{t}_{n}}^{{t}_{n + 1}}b\left( {X}_{t}\right) {dt} + \delta {W}_{n} \n\] \n\nNote that (7.63) can be rewritten as\n\n\[ \n{X}_{n + 1} = {X}_{n} + {\int }_{{t}_{n}}^{{t}_{n + 1}}b\left( {X}_{n}\right) {dt} + \delta {W}_{n} \n\] \n\nDefine \( ... | Yes |
Example 8.1 (Brownian motion). The SDE reads\n\n\[ \nd{\mathbf{X}}_{t} = d{\mathbf{W}}_{t},\;{\mathbf{X}}_{0} = 0.\n\]\n\nThus the Fokker-Planck equation is\n\n(8.8)\n\n\[ \n{\partial }_{t}p = \frac{1}{2}{\Delta p},\;p\left( {\mathbf{x},0}\right) = \delta \left( \mathbf{x}\right) .\n\] | It is well known that\n\n\[ \np\left( {\mathbf{x}, t}\right) = \frac{1}{\sqrt{2\pi t}}\exp \left( {-\frac{{\mathbf{x}}^{2}}{2t}}\right)\n\]\n\nwhich is the probability distribution density of \( N\left( {0, t\mathbf{I}}\right) \) . | Yes |
Consider the over-damped dynamics of a stochastic particle described by\n\n\[ d{\mathbf{X}}_{t} = - \frac{1}{\gamma }\nabla U\left( {\mathbf{X}}_{t}\right) {dt} + \sqrt{\frac{2{k}_{B}T}{\gamma }}d{\mathbf{W}}_{t}. \] | The Fokker-Planck equation is\n\n\[ {\partial }_{t}p - \nabla \cdot \left( {\frac{1}{\gamma }\nabla U\left( \mathbf{x}\right) p}\right) = \frac{{k}_{B}T}{\gamma }{\Delta p} = {D\Delta p} \]\n\nwhere \( D = {k}_{B}T/\gamma \) is the diffusion coefficient.\n\nEquation (8.10) is called the Smoluchowski equation in physics... | Yes |
Example 8.3 (Langevin dynamics). We have already defined the Langevin dynamics (7.41) for an inertial particle in a force field subject to friction and noise. It is straightforward to write down its Fokker-Planck equation | \[ {\partial }_{t}p + \mathbf{v} \cdot {\nabla }_{\mathbf{x}}p - {\nabla }_{\mathbf{v}} \cdot \left( {\left( {\gamma \mathbf{v} + \nabla U\left( \mathbf{x}\right) }\right) p}\right) = \frac{{k}_{B}T}{\gamma }{\Delta }_{\mathbf{v}}p. \] | Yes |
The SDE for the Brownian motion on the unit sphere \( {\mathbb{S}}^{n - 1} \) has the Stratonovich form\n\n\[ d{\mathbf{X}}_{t} = \left( {\mathbf{I} - {\mathbf{X}}_{t} \otimes {\mathbf{X}}_{t}}\right) \circ d{\mathbf{W}}_{t},{\left. \;{\mathbf{X}}_{t}\right| }_{t = 0} = {\mathbf{X}}_{0} \in {\mathbb{S}}^{n - 1}. \] | The Itô form of this SDE is given by\n\n\[ d{\mathbf{X}}_{t} = - \frac{n - 1}{2}{\mathbf{X}}_{t}{dt} + \left( {\mathbf{I} - {\mathbf{X}}_{t} \otimes {\mathbf{X}}_{t}}\right) d{\mathbf{W}}_{t}. \] | Yes |
Theorem 8.5. The function \( u\left( {\mathbf{x}, t}\right) = {\mathbb{E}}^{\mathbf{x}}f\left( {\mathbf{X}}_{t}\right) \) satisfies the Kolmogorov backward equation for \( f \in {C}_{c}^{2}\left( {\mathbb{R}}^{d}\right) \) ; i.e., \n\n\[ \n{\partial }_{t}u = \mathcal{A}u\left( \mathbf{x}\right) ,{\left. \;u\right| }_{t... | Proof. Observe that \( u\left( {\mathbf{x}, t}\right) \) is differentiable with respect to \( t \), as a result of Ito’s formula and the condition \( f \in {C}_{c}^{2}\left( {\mathbb{R}}^{d}\right) \) . For any fixed \( t > 0 \), define \( g\left( \mathbf{x}\right) = u\left( {\mathbf{x}, t}\right) \) . Then we have \n\... | Yes |
Theorem 8.6 (Feynman-Kac formula). Let \( f \in {C}_{c}^{2}\left( {\mathbb{R}}^{d}\right) \) and \( q \in C\left( {\mathbb{R}}^{d}\right) \) . Assume that \( q \) is continuous and bounded. Then the solution of (8.37) is given by \[ v\left( {\mathbf{x}, t}\right) = {\mathbb{E}}^{x}\left( {\exp \left( {{\int }_{0}^{t}q\... | The proof goes as follows. Let \( {Y}_{t} = f\left( {\mathbf{X}}_{t}\right) ,{Z}_{t} = \exp \left( {{\int }_{0}^{t}q\left( {\mathbf{X}}_{s}\right) {ds}}\right) \) and define \( v\left( {\mathbf{x}, t}\right) = {\mathbb{E}}^{\mathbf{x}}\left( {{Y}_{t}{Z}_{t}}\right) \) . Here \( v\left( {\mathbf{x}, t}\right) \) is diff... | Yes |
Theorem 8.8. Assume that \( U \subset {\mathbb{R}}^{d} \) is a bounded domain and the boundary \( \partial U \) is \( {C}^{2} \) . Assume also that \( \mathbf{b} \) and \( \mathbf{\sigma } \) satisfy the Lipschitz condition on \( \bar{U} \) . Then for \( g \in C\left( \bar{U}\right), f \in C\left( {\partial U}\right) \... | Proof. The proof of the fact that \( {\tau }_{U} \) is a stopping time and that \( {\mathbb{E}}^{\mathbf{x}}{\tau }_{U} < \infty \) is quite technical and can be found in [KS91, Fri75a].\n\nStandard PDE theory tells us that \( u \in {C}^{2}\left( U\right) \cap C\left( \bar{U}\right) \) . Applying Dynkin’s formula, we g... | No |
Lemma 8.9. Suppose that \( f \in {C}_{P}^{2\beta } \) for some \( \beta \in \{ 2,3,\ldots \} ,{\mathbf{X}}_{t} \) is time-homogeneous, and \( b \in {C}_{P}^{2\beta } \) with uniformly bounded derivatives. Then \( \partial u/\partial t \) is continuous and | \[ u\left( {\cdot, t}\right) \in {C}_{P}^{2\beta },\;t \leq T \] for any fixed \( T < \infty \) . | No |
For the SDE\n\n\\[ \nd{X}_{t} = - \\frac{1}{2}{X}_{t}{dt} + d{W}_{t},\\;{X}_{0} = 0, \n\\]\n\nwe will compute \\( u = {\\left. \\mathbb{E}{X}_{t}^{2}\\right| }_{t = 1} \\) using the Euler-Maruyama scheme. | The exact solution of \\( u \\) is\n\n\\[ \nu = {\\left. \\mathbb{E}{X}_{t}^{2}\\right| }_{t = 1} = 1 - {e}^{-1} \\approx {0.632}. \\]\n\nIn order to compute the expectation numerically, we take the Euler-Maruyama scheme with stepsize \\( {\\Delta t} = 1/M \\) ; i.e.,\n\n\\[ \n{X}_{n + 1, k} = \\left( {1 - \\frac{\\Del... | Yes |
Compute the expectation\n\n\\[ \n\\mathbb{E}\\exp \\left( {-\\frac{1}{2}{\\int }_{0}^{1}{W}_{t}^{2}{dt}}\\right) \n\\] | We have seen in Chapter [6] that the answer is \\( \\sqrt{{2e}/\\left( {1 + {e}^{2}}\\right) } \\) using the Karhunen-Loève expansion. Here we will use the path integral approach.\n\nStep 1. Finite-dimensional approximation.\n\nAssume that, as before, \\( \\left\{ {t}_{j}\\right\} \\) defines a subdivision of \\( \\lef... | Yes |
Theorem 9.2 (Girsanov Theorem I). Consider the Itô process\n\n\[ d{\\widetilde{\\mathbf{W}}}_{t} = \\mathbf{\\phi }\\left( {t,\\omega }\\right) {dt} + d{\\mathbf{W}}_{t},;{\\widetilde{\\mathbf{W}}}_{0} = 0, \]\n\nwhere \( \\mathbf{W} \\in {\\mathbb{R}}^{d} \) is a d-dimensional standard Wiener process on \( \\left( {\\... | To see how (9.15) and (9.12) are related to each other, we note that for any functional \( F \)\n\n\[ {\\left\\langle F\\left\\lbrack {\\widetilde{\\mathbf{W}}}_{t}\\right\\rbrack \\right\\rangle }_{\\widetilde{\\mathbb{P}}} = {\\left\\langle F\\left\\lbrack {\\widetilde{\\mathbf{W}}}_{t}\\right\\rbrack {Z}_{T}\\right\... | Yes |
Theorem 9.3 (Girsanov Theorem II). Let \( {\mathbf{X}}_{t},{\mathbf{Y}}_{t} \), and \( \phi \) be defined as above. Assume that \( \mathbf{b} \) and \( \mathbf{\sigma } \) satisfy the same conditions as in Theorem 7.14, \( \mathbf{\sigma } \) is nonsingular, \( \mathbf{\gamma } \) is an \( {\mathcal{F}}_{t} \) -adapted... | Details of the proof can be found in Fri75a, Oks98. | No |
Example 10.1 (The percolation model). Imagine that we have an \( N \times N \) square lattice. Let \( p \in \left\lbrack {0,1}\right\rbrack \) . For each edge on the lattice, we keep the edge with probability \( p \) and delete it with probability \( 1 - p \) . This operation is done independently for all the edges. We... | A typical question of interest is the size of the connected clusters in such a structure as \( N \rightarrow \infty \) . It was proven by \( \mathrm{H} \) . Kesten that if \( p < 1/2 \), then the probability of having infinite clusters is 0 ; if \( p > 1/2 \), then the probability of having infinite clusters is \( 1 \)... | Yes |
Example 10.4 (Gaussian white noise). Roughly speaking, Gaussian white noise is the Gaussian random field with mean 0 and covariance operator being defined by the \( \delta \) -function. In rigorous terms, the Gaussian white noise random field is a probability measure on the space of tempered distributions \( \Omega = {... | The latter equation means that the covariance operator \( \mathcal{K} = \mathcal{I} \) . This fact implies that the restrictions of the field to disjoint domains in \( {\mathbb{R}}^{d} \) are independent.\n\nThe definition (10.2) obviously implies that the random variable \( \left( {\psi ,\omega }\right) \) is Gaussian... | Yes |
Example 10.5 (The Ornstein-Uhlenbeck measure). This is the Gaussian random field with mean 0 and covariance operator\n\n(10.5)\n\n\[ \n\mathcal{K} = {\left( -\Delta + \mathcal{I}\right) }^{-1} \n\]\n\nIn one dimension, it corresponds to the stationary Ornstein-Uhlenbeck process\n\n\[ \nd{X}_{t} = - {X}_{t}{dt} + d{W}_{... | To see this, note that\n\n\[ \n\left( {\psi ,\mathcal{K}\phi }\right) = \mathbb{E}\left( {\psi, X}\right) \left( {\phi, X}\right) = {\int }_{\mathbb{R}}\psi \left( t\right) {\int }_{\mathbb{R}}\frac{1}{2}{e}^{-\left| {t - s}\right| }\phi \left( s\right) {dsdt} \n\]\n\nfor arbitrary \( \psi \) and \( \phi \), which impl... | Yes |
Example 10.6 (The Brownian sheet). This is the analog of Brownian motion in higher dimension. In two dimension, \( B\left( {s, t}\right) \) satisfies\n\n(10.6)\n\n\[ \mathbb{E}B\left( {s, t}\right) = 0,\;\mathbb{E}B\left( {s, t}\right) B\left( {\widetilde{s},\widetilde{t}}\right) = \left( {s \land \widetilde{s}}\right)... | \( B\left( {s, t}\right) \) can be constructed by the invariance principle as was done for the Wiener process. Let \( {\xi }_{i, j}\left( {i, j = 1,2,\ldots }\right) \) be a two-parameter sequence of\n\ni.i.d. random variables with mean 0 and variance 1 . Let\n\n(10.7)\n\n\[ {S}_{m, n} = \mathop{\sum }\limits_{{1 \leq... | Yes |
Theorem 10.8 (Hammersley-Clifford). The existence of the Gibbs distribution and the Markovianity of a random field are equivalent; i.e.,\n\n(1) If \( P \) is a Gibbs distribution, then it defines a Markov random field; i.e., it satisfies (10.9).\n\n(2) If \( P \) is the distribution of a Markov random field such that \... | For the proof of this result, see MD10. | No |
Example 11.1 (The entropy of mixing). On a one-dimensional lattice of size \( N \), put down \( k \) particles based on a single occupance rule. The number of ways of putting down these particles is equal to\n\n(11.36)\n\n\[ g\left( {N, k}\right) = \frac{N!}{\left( {N - k}\right) !k!}. \]\n\nFor \( N \gg 1 \), using th... | where \( x = k/N \) is the number density of the particles. Equation (11.37) gives the entropy of mixing. This result is reminiscent of the large deviation rate function considered in Example 2.10 and the related equation (2.12). | Yes |
Example 11.2 (Discrete ideal gas). Let \( D \) be a finite interval on \( \mathbb{R} \) with nonempty interior. Consider the dynamics of \( N \) noninteracting unit-mass particles in \( D \) with velocity uniformly picked from a finite set \( L \), with the property that if \( v \in L \), then \( - v \in L \) . The vel... | For this problem, one can simply integrate out the degrees of freedom for the position and consider the reduced canonical ensemble for the velocities \( \omega = \left( {{v}_{1},\ldots ,{v}_{N}}\right) \) . The probability distribution is then defined on \( {\Omega }_{N} = \) \( \left\{ {\omega = \left( {{v}_{1},\ldots... | Yes |
Example 11.3 (Langevin equation). Consider the dynamics of an inertial particle moving in a potential \( U \), subject to friction and random noise: | \[ \left\{ \begin{array}{l} {\dot{\mathbf{X}}}_{t} = {\mathbf{V}}_{t}, \\ m{\dot{\mathbf{V}}}_{t} = - \gamma {\mathbf{V}}_{t} - \nabla U\left( {\mathbf{X}}_{t}\right) + \sqrt{2\sigma }{\dot{\mathbf{W}}}_{t}. \end{array}\right. \] | No |
Lemma 12.2. The minimum action path \( \varphi \) of the Brownian dynamics is comprised of two parts defined through functions \( {\varphi }_{1} \) and \( {\varphi }_{2} \) by\n\n(12.19)\n\n\[ \n{\dot{\mathbf{\varphi }}}_{1}\left( s\right) = \nabla U\left( {{\mathbf{\varphi }}_{1}\left( s\right) }\right) ,\;{\mathbf{\v... | Proof. It is easy to see that the minimum in \( T \) in (12.18) is attained when \( T = \infty \) since \( {\mathbf{x}}_{ - },{\mathbf{x}}_{ + } \), and \( {\mathbf{x}}_{s} \) are all critical points (see Exercise 12.2). To see why the minimization problem in (12.18) is solved by the path defined above, we first note t... | Yes |
Theorem 12.3. Assume that \( \left( {\mathbf{b}\left( \mathbf{x}\right) ,\widehat{\mathbf{n}}\left( \mathbf{x}\right) }\right) < 0 \) for \( \mathbf{x} \in \partial D \), where \( \widehat{\mathbf{n}} \) is the outward normal of \( \partial D \). Then\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}\varepsilon... | The proof can be found in [FW98]. | Yes |
Lemma 12.4. We have\n\n\[ \nV\left( {\mathbf{x};\mathbf{y}}\right) = \mathop{\inf }\limits_{{t > 0}}\mathop{\inf }\limits_{{\mathbf{\varphi }\left( 0\right) = \mathbf{y},\mathbf{\varphi }\left( t\right) = \mathbf{x}}}{I}_{t}\left\lbrack \mathbf{\varphi }\right\rbrack = \mathop{\inf }\limits_{{\mathbf{\psi }\left( 0\rig... | Proof. We have for any fixed \( t > 0 \)\n\n\[ \n{I}_{t}\left\lbrack \varphi \right\rbrack = \frac{1}{2}{\int }_{0}^{t}{\left| \dot{\varphi } - \mathbf{b}\left( \varphi \right) \right| }^{2}{ds} = \frac{1}{2}{\int }_{0}^{t}\left( {{\left| \dot{\varphi }\right| }^{2} + {\left| \mathbf{b}\left( \varphi \right) \right| }^... | Yes |
Lemma 2. Assume that \( H\left( {\mathbf{x},\mathbf{p}}\right) \rightarrow \infty \) as \( \left| \mathbf{p}\right| \rightarrow \infty \) uniformly in \( \mathbf{x} \) . Then for any \( \mathbf{p} \in {\mathbb{R}}^{d} \), there exists a unique constant \( \lambda \), such that the cell problem\n\n\[ H\left( {\mathbf{y}... | This unique value of \( \lambda \) will be denoted as \( \bar{H}\left( \mathbf{p}\right) \) . \( \bar{H} \) is the effective Hamiltonian.\n\nGoing back to (2.4.48), we obtain the homogenized equation by letting \( \mathbf{p} = {\nabla }_{\mathbf{x}}{u}_{0} \):\n\n\[ \bar{H}\left( {{\nabla }_{\mathbf{x}}{u}_{0}}\right) ... | Yes |
Lemma 5. Let \( \mathbf{y} = \mathbf{y}\left( \mathbf{x}\right) \) be a nondegenerate map on \( {\mathbb{R}}^{3} \), i.e. \( J\left( \mathbf{x}\right) = \det \left( {{\nabla }_{\mathbf{x}}\mathbf{y}\left( \mathbf{x}\right) }\right) \neq 0 \) . Denote by \( d{S}_{0} \) an infinitesimal surface element at \( \mathbf{x} \... | \[ \mathbf{n}{dS} = J\left( \mathbf{x}\right) {\left( {\nabla }_{\mathbf{x}}\mathbf{y}\left( \mathbf{x}\right) \right) }^{-T}{\mathbf{n}}_{0}d{S}_{0} \] | Yes |
Lemma 6. Assume that \( \sigma \left( \omega \right) > 0 \), for all \( \omega \in {\mathbb{S}}^{2} \) . Then for any equilibrium states \( {f}_{0} \) , the following holds:\n\n\[ \n{f}_{0}\left( {\mathbf{v}}^{\prime }\right) {f}_{0}\left( {\mathbf{w}}^{\prime }\right) = {f}_{0}\left( \mathbf{v}\right) {f}_{0}\left( \m... | Proof. Let\n\n\[ \nH\left( f\right) = \iint f\ln {fd}\mathbf{x}d\mathbf{v}\n\]\n\nthen\n\n\[ \n\frac{dH}{dt} = \iint \left( {\ln f + 1}\right) \frac{\partial f}{\partial t}d\mathbf{x}d\mathbf{v}\n\]\n\n\[ \n= \iint \left( {\ln f + 1}\right) B\left( {f, f}\right) d\mathbf{x}d\mathbf{v}\n\]\n\n\[ \n= \iiint \sigma \left(... | Yes |
Under these assumptions, the numerical solutions of the coupled scheme have the following form:\n\n\[ \n{u}_{n}\left( y\right) = \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{k}^{n - i}{\xi }_{i}}\right) g\left( y\right) \n\]\n\n(7.4.1) \n\nwhere \( {u}_{n}\left( \cdot \right) \) is the velocity at the \( n \) -th iterat... | This lemma can be proved by induction. It is enough to consider the upper half channel due to the symmetry assumption. Consider the VV coupling scheme for example. In the first iteration, one starts with an equilibrium MD in the \( p \) -region. The average velocity at \( y = a \) is denoted by \( {\xi }_{1} \), and is... | Yes |
Theorem 1. Assume that (8.0.1) and (8.3.1) are uniformly elliptic. Denote by \( {U}_{0} \) and \( {U}_{\mathrm{{HMM}}} \) the solution of (8.3.1) and the HMM solution, respectively. If \( {U}_{0} \) is sufficiently smooth, then there exists a constant \( C \) independent of \( \varepsilon ,\delta \) and \( H \), such t... | \[ \n{\begin{Vmatrix}{U}_{0} - {U}_{\mathrm{{HMM}}}\end{Vmatrix}}_{1} \leq C\left( {H + e\left( \mathrm{{HMM}}\right) }\right) \]\n\n\( \left( {8.3.20}\right) \)\n\n\[ \n{\begin{Vmatrix}{U}_{0} - {U}_{\mathrm{{HMM}}}\end{Vmatrix}}_{0} \leq C\left( {{H}^{2} + e\left( \mathrm{{HMM}}\right) }\right) .\n\]\n\n(8.3.21) | Yes |
In this case, \( x \) is a slow variable. Fix \( x \), the quasi-equilibrium distribution for the fast variable \( y \) is given by a delta function \( {\mu }_{x}\left( {dy}\right) = \delta \left( {y - \phi \left( x\right) }\right) {dy} \) . The effective equation for the slow variable \( x \) is given by: | \[ \frac{dx}{dt} = \int f\left( {x, y}\right) {\mu }_{x}\left( {dy}\right) = f\left( {x,\phi \left( x\right) }\right) \] | Yes |
Theorem 1.1 (Lax-Milgram theorem). Let \( a\left( {u, v}\right) \) be a bounded, coercive bilinear form on \( H \) . Then for any \( f \in {H}^{\prime } \), there exists a unique \( u \in H \) such that\n\n\[ a\left( {u, v}\right) = \langle f, v\rangle ,\;\forall v \in H, \]\n\nand\n\n\[ \parallel u{\parallel }_{H} \le... | Proof. For each fixed \( u \in H, a\left( {u, \cdot }\right) \) is a bounded linear functional on \( H \), and therefore there exists a unique \( {Au} \in {H}^{\prime } \) such that\n\n\[ a\left( {u, v}\right) = \langle {Au}, v\rangle ,\;\forall v \in H, \]\n\nand\n\n\[ \parallel {Au}{\parallel }_{{H}^{\prime }} \leq M... | Yes |
Lemma 2.1. Let the assumptions (2.2) and (2.3) be in force, and let \( \Omega \) be an open bounded domain in \( {\mathbb{R}}^{n} \) . Then \( a\left( {u, v}\right) \) is a bounded bilinear form on \( {H}_{0}^{1}\left( \Omega \right) \) . | Proof. Using Hölder's inequality and (2.2), we obtain\n\n\[ \left| {{\int }_{\Omega }{a}^{ij}{D}_{i}u{D}_{j}{vdx}}\right| \leq \Lambda \parallel u{\parallel }_{{H}_{0}^{1}\left( \Omega \right) }\parallel v{\parallel }_{{H}_{0}^{1}\left( \Omega \right) }.\]\n\nApplying Hölder's inequality, (2.3) and the embedding theore... | Yes |
Lemma 2.2. Let the assumptions (2.2) and (2.3) be in force, and let \( \Omega \) be an open bounded domain in \( {\mathbb{R}}^{n} \). Then there exists \( \bar{\mu } > 0 \) such that \( a\left( {u, v}\right) + \mu {\left( u, v\right) }_{{L}^{2}\left( \Omega \right) } \) is coercive on \( {H}_{0}^{1}\left( \Omega \right... | Proof of Lemma 2.2. For any \( \varepsilon > 0 \), there are decompositions\n\n\[ {b}^{i} = {b}_{1}^{i} + {b}_{2}^{i},\;{d}^{i} = {d}_{1}^{i} + {d}_{2}^{i},\;c = {c}_{1} + {c}_{2}, \]\n\n such that\n\n\[ \sum {\begin{Vmatrix}{b}_{2}^{i}\end{Vmatrix}}_{{L}^{n}\left( \Omega \right) } + \sum {\begin{Vmatrix}{d}_{2}^{i}\en... | Yes |
Theorem 2.3. Let the assumptions (2.2) and (2.3) be in force, and let \( \Omega \subset {\mathbb{R}}^{n} \) be an open bounded domain on which the Sobolev embedding theorem is valid. Let \( T \in {H}^{-1}\left( \Omega \right) \) and \( g \in {H}^{1}\left( \Omega \right) \) . Then there exists \( \bar{\mu } > 0 \) such ... | Proof. By the definition of a weak solution, the bilinear form corresponding to (2.9) is given by \( a\left( {u, v}\right) + \mu {\left( u, v\right) }_{{L}^{2}\left( \Omega \right) } \) . A weak solution satisfies\n\n\( \left( {2.10}\right) \)\n\n\[ \left\{ \begin{array}{l} a\left( {u, v}\right) + \mu {\left( u, v\righ... | Yes |
Theorem 3.1. Let \( V \) be a normed linear vector space and \( A : V \rightarrow V \) be a linear compact operator. Then there are only two possibilities:\n\n(1) there exists \( x \in V, x \neq 0 \), such that \( x - {Ax} = 0 \) ; or\n\n(2) for any \( y \in V \), there exists a unique \( x \in V \) such that\n\n\[ x -... | The proof of this theorem can be found in textbooks on functional analysis. | No |
Theorem 3.2. Suppose that \( L \) and \( \Omega \) satisfy the assumptions in Theorem 2.3. Then there are only two possibilities for the problem (2.9):\n\n(1) for any \( T \in {H}^{-1}\left( \Omega \right), g \in {H}^{1}\left( \Omega \right) \), there exists a unique weak solution to \( \left( {2.9}\right) \) ; or\n\n(... | Proof. Without loss of generality we may assume that \( g \equiv 0 \) (cf. the proof of Theorem 2.3). For each fixed \( u \in {L}^{2}\left( \Omega \right) ,{\left( u, \cdot \right) }_{0} \) is a bounded linear functional on \( {H}_{0}^{1}\left( \Omega \right) \) . Therefore there exists a bounded operator \( P : {L}^{2... | Yes |
Lemma 4.1. Suppose that \( \varphi \left( t\right) \) is a nonnegative nondecreasing function defined on \( \left\lbrack {{k}_{0},\infty }\right) \) satisfying\n\n(4.1)\n\n\[ \varphi \left( h\right) \leq \frac{C}{{\left( h - k\right) }^{\alpha }}{\left\lbrack \varphi \left( k\right) \right\rbrack }^{\beta }\;\text{ for... | Proof. Set\n\n\[ {k}_{s} = {k}_{0} + d - \frac{d}{{2}^{s}}\;\left( {s = 0,1,2,\cdots }\right) . \]\n\nFrom (4.1) we deduce\n\n(4.4)\n\n\[ \varphi \left( {k}_{s + 1}\right) \leq \frac{C{2}^{\left( {s + 1}\right) \alpha }}{{d}^{\alpha }}{\left\lbrack \varphi \left( {k}_{s}\right) \right\rbrack }^{\beta }\;\left( {s = 0,1... | Yes |
Theorem 4.2 (Weak maximum principle for weak solutions). Let the assumptions (2.2), (2.3) be in force, and (4.8) \[ c - {D}_{i}{d}^{i} \geq 0\;\left( {\text{ in the sense of }{\mathcal{D}}^{\prime }\left( \Omega \right) }\right) . \] If \( u \in {H}^{1}\left( \Omega \right) \) is a weak subsolution of (2.1), then for a... | Proof. Let \( l = \mathop{\sup }\limits_{\infty }{u}^{ + } \) . Suppose that \( \mathop{\sup }\limits_{\infty }{u}^{ + } > l \) . For any \( k > l \), we choose the test function \( \varphi = {\left( u - k\right) }^{ + } \) in (4.6). Then (4.10) \[ a\left( {u,\varphi }\right) = {\int }_{\Omega }\left\{ {\left( {{a}^{ij... | Yes |
Theorem 4.3. Let the assumptions (2.1), (2.2) be in force, and let\n\n\[ c - {D}_{i}{d}^{i} \geq 0\;\left( {\text{ in the sense of }{\mathcal{D}}^{\prime }\left( \Omega \right) }\right) ,\]\n\nwhere \( \Omega \) is an open bounded domain on which the Sobolev embedding theorem is valid. Then there exists a unique weak s... | Proof. Without loss of generality, we may assume that \( g = 0 \) (cf. the proof of Theorem 2.3). If \( T = 0 \), the zero solution is the only solution for (2.4), by the weak maximum principle (Theorem 4.1). By Theorem 3.2, there exists a unique weak solution \( u \in {H}_{0}^{1}\left( \Omega \right) \) for (2.4), for... | Yes |
Theorem 5.1. Let the assumption (2.2) be in force, and let \( {a}^{ij} \in {W}^{1,\infty }\left( \Omega \right) \) , \( {b}^{i}, c \in {L}^{\infty }\left( \Omega \right), f \in {L}^{2}\left( \Omega \right) \) . If \( u \in {H}^{1}\left( \Omega \right) \) is a weak solution of (5.1), then for any \( {\Omega }^{\prime } ... | Proof. Set \( q = f - {b}^{i}{D}_{i}u - {cu} \) . Since \( u \) is a weak solution of (5.1), it satisfies\n\n(5.3)\n\n\[ {\int }_{\Omega }{a}^{ij}{D}_{i}u{D}_{j}{\varphi dx} = {\int }_{\Omega }{q\varphi dx},\;\forall \varphi \in {H}_{0}^{1}\left( \Omega \right) . \]\n\nLet \( {\tau }_{h} \) be the translation operator ... | Yes |
Theorem 5.4. In addition to the assumptions of Theorem 5.3, assume that \( \partial \Omega \in {C}^{k + 2}, g \in {W}^{k + 2,2}\left( \Omega \right) \) . If \( u \in {H}^{1}\left( \Omega \right) \) is a weak solution of (5.1) satisfying \( u - g \in {H}_{0}^{1}\left( \Omega \right) \), then \( u \in {W}^{k + 2,2}\left(... | If \( {a}^{ij},{b}^{i}, c \) and \( f \) are infinitely many times differentiable, then for any \( k \) we have \( u \in {W}_{loc}^{k + 2,2}\left( \Omega \right) \), and therefore, by the Sobolev embedding theorem, \( u \in {C}^{\infty }\left( \Omega \right) \) . | No |
Lemma 1.1. For \( u, v \in {C}^{\alpha }\left( \bar{\Omega }\right) \left( {0 < \alpha \leq 1}\right) \) | The proof is left to the reader. | No |
Theorem 1.2. Let \( \Omega \) be a bounded domain, and \( u \in {C}^{2,\alpha }\left( \bar{\Omega }\right) \left( {0 < \alpha \leq 1}\right) \) . Then for any \( \varepsilon > 0 \) ,\n\n(1.9)\n\n\[{\left\lbrack u\right\rbrack }_{2} \leq \varepsilon {\left\lbrack u\right\rbrack }_{2,\alpha } + {C}_{\varepsilon }{\left| ... | Proof. We prove only (1.9); the proof for (1.10) is similar. If there is no constant \( {C}_{\varepsilon } \) such that (1.9) is valid for all functions in \( {C}^{2,\alpha }\left( \bar{\Omega }\right) \), then for any \( N > 0 \) , there exists \( {u}_{N} \) such that\n\n(1.11)\n\n\[{\left\lbrack {u}_{N}\right\rbrack ... | Yes |
Theorem 1.3. Suppose that \( \Omega \) satisfies a cone property with \( h \) the height of the cone. Then for any \( 0 < \varepsilon \leq h \), we have\n\n\[ \n{\left\lbrack u\right\rbrack }_{2} \leq {\varepsilon }^{\alpha }{\left\lbrack u\right\rbrack }_{2,\alpha } + \frac{C}{{\varepsilon }^{2}}{\left| u\right| }_{0}... | Proof. Let \( {V}_{1} \) be a cone with the same solid angle of opening and height 1 . Then Theorem 1.2 implies that\n\n\[ \n{\left\lbrack u\right\rbrack }_{2;{V}_{1}} \leq {\left\lbrack u\right\rbrack }_{2,\alpha ;{V}_{1}} + C{\left| u\right| }_{0;{V}_{1}}\;\text{ for }u \in {C}^{2,\alpha }\left( {\bar{V}}_{1}\right) ... | Yes |
Lemma 2.1. Let \( u \in C\left( {\mathbb{R}}^{n}\right) \) . Then \( \widetilde{u}\left( {x,\tau }\right) \) converges uniformly on any compact set to \( u\left( x\right) \) as \( \tau \rightarrow 0 \), and\n\n(2.3)\n\n\[ \sup \left| \widetilde{u}\right| \leq \sup \left| u\right| \]\n\n(2.4)\n\n\[ \left| {{D}^{k}\widet... | Proof. By (2.1),\n\n\[ \widetilde{u}\left( {x,\tau }\right) - u\left( x\right) = {\tau }^{-n}{\int }_{{\mathbb{R}}^{n}}\rho \left( \frac{x - y}{\tau }\right) \left( {u\left( y\right) - u\left( x\right) }\right) {dy} \]\n\n(2.5)\n\n\[ = {\int }_{{B}_{1}\left( 0\right) }\rho \left( \eta \right) \left( {u\left( {x - {\tau... | Yes |
Lemma 2.2. Let \( u \in {C}_{loc}^{\alpha }\left( {\mathbb{R}}^{n}\right) \left( {0 < \alpha \leq 1}\right) \). Then\n\n\[ \left| {\widetilde{u}\left( {x,\tau }\right) - u\left( x\right) }\right| \leq {\tau }^{\alpha }{H}_{x}^{\alpha }\left\lbrack {u;{B}_{\tau }\left( x\right) }\right\rbrack \]\n\n\[ \left| {{D}^{k}\wi... | Proof. From (2.5) we easily derive (2.6). To prove (2.7), we denote by \( \bar{\beta } = \left( {{\beta }_{0},\beta }\right) \) the \( n + 1 \) dimensional multi-index with \( \left| \bar{\beta }\right| = k,{D}^{\bar{\beta }} = {D}_{\tau }^{{\beta }_{0}}{D}_{x}^{\beta } \). If \( \beta = 0 \), then we differentiate (2.... | Yes |
Lemma 2.3. Let \( u \in C\left( {\mathbb{R}}^{n}\right) \) . If\n\n\[ \mathop{\sup }\limits_{{y \in {B}_{R}\left( x\right) ,0 < \tau \leq R}}{\tau }^{1 - \alpha }\left| {D\widetilde{u}\left( {y,\tau }\right) }\right| < \infty \]\n\nfor some \( 0 < \alpha \leq 1, R > 0 \), then \( u \) is Hölder continuous at \( x \) wi... | Proof. For \( \left| {x - y}\right| < R,0 < \tau \leq R \), we have\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq \left| {\widetilde{u}\left( {x,\tau }\right) - u\left( x\right) }\right| + \left| {\widetilde{u}\left( {x,\tau }\right) - \widetilde{u}\left( {y,\tau }\right) }\right| \]\n\n\[ + \left| {\... | Yes |
Corollary 2.4. (1) There exists a constant \( C \), depending only on \( n,\alpha \) and the mollifier \( \rho \), such that\n\n\[ \frac{1}{C}{\left\lbrack u\right\rbrack }_{\alpha } \leq \mathop{\sup }\limits_{{\tau > 0, x \in {\mathbb{R}}^{n}}}{\tau }^{1 - \alpha }\left| {D\widetilde{u}\left( {x,\tau }\right) }\right... | Proof. (2.9) is a direct corollary of Lemmas 2.2 and 2.3. | No |
Lemma 3.1. Let \( u \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) satisfy\n\n\[ - {\Delta u} = f \]\n\nwhere \( \Delta \) is the Laplacian operator. Then for any \( R > 0 \), we have\n\n(3.1)\n\n\[ \left| {{D}_{i}u\left( x\right) }\right| \leq \frac{n}{R}\mathop{\operatorname{osc}}\limits_{{{B}_{R}\left( x\right) ... | Proof. Assume without loss of generality that \( x \) is the origin. Set \( {F}_{0} = \) sup \( \left| f\right| \) . Then by the divergence formula \( {B}_{R}\left( x\right) \)\n\n\[ {\int }_{{B}_{\rho }}\Delta \left( {{D}_{i}u}\right) {dx} = {\int }_{\partial {B}_{\rho }}\frac{\partial {D}_{i}u}{\partial r}{dS} \]\n\n... | Yes |
Theorem 3.2. Let \( u \in {C}_{0}^{2,\alpha }\left( {\mathbb{R}}^{n}\right) \left( {0 < \alpha < 1}\right) \) satisfy\n\n(3.2)\n\n\[ \n- {\Delta u} = f.\n\]\n\nThen\n\n(3.3)\n\n\[ \n{\left\lbrack {D}^{2}u\right\rbrack }_{\alpha } \leq C{\left\lbrack f\right\rbrack }_{\alpha }\n\]\n\nwhere \( C \) depends only on \( n,\... | Proof. Let \( g\left( x\right) = f\left( x\right) - f\left( {x}_{0}\right) \) in \( {B}_{R}\left( {x}_{0}\right) \) . We can rewrite the equation (3.2)\n\nas\n\n(3.4)\n\n\[ \n- {\Delta u}\left( x\right) - f\left( {x}_{0}\right) = g\left( x\right)\n\]\n\nwhere \( g \) satisfies\n\n(3.5)\n\n\[ \n\mathop{\sup }\limits_{{{... | Yes |
Theorem 3.3. Let \( u \in {C}_{0}^{2,\alpha }\left( {\mathbb{R}}^{n}\right) \left( {0 < \alpha < 1}\right) \) satisfy (3.6). Assume that the constant coefficient matrix satisfies (3.7). Then\n\n(3.8)\n\n\[ \n{\left\lbrack {D}^{2}u\right\rbrack }_{\alpha } \leq C{\lambda }^{-1}{\left\lbrack f\right\rbrack }_{\alpha }\n\... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Introduce a change of variables \( y = {Bx} \) and \( \bar{u}\left( y\right) = u\left( x\right) ,\bar{f}\left( y\right) = f\left( x\right) \) . Then a simple calculation shows that \( \bar{u}\left( y\right) \) satisfies the equation\n\n\[ \n- {\bar{a}}^{... | No |
Lemma 4.1. Let \( \varphi \left( t\right) \) be a bounded nonnegative function defined on the interval \( \left\lbrack {{T}_{0},{T}_{1}}\right\rbrack \), where \( {T}_{1} > {T}_{0} \geq 0 \) . Suppose that for any \( {T}_{0} \leq t < s \leq {T}_{1},\varphi \) satisfies\n\n(4.1)\n\n\[ \varphi \left( t\right) \leq {\thet... | Proof. Let \( {t}_{0} = \rho ,{t}_{i + 1} = {t}_{i} + \left( {1 - \tau }\right) {\tau }^{i}\left( {R - \rho }\right) \left( {i = 0,1,2,\cdots }\right) \), where \( 0 < \tau < 1 \) is to be determined. From (4.1),\n\n\[ \varphi \left( {t}_{i}\right) \leq {\theta \varphi }\left( {t}_{i + 1}\right) + \frac{A}{{\left\lbrac... | Yes |
Lemma 4.2. Suppose that the coefficients in the equation (4.3) satisfy the assumptions (4.4) and (4.5). Then there exists \( {R}_{0} \leq 1 \), depending only on \( n,\alpha ,\Lambda /\lambda \) and \( {\Lambda }_{\alpha } \), such that for any \( 0 < R \leq {R}_{0} \) with \( {B}_{R} \subset \Omega \) and any solution... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Let \( {B}_{R} \) be a ball centered at \( {x}_{0} \) . We shall use the method of freezing the coefficients by rewriting (4.3) as\n\n\[ \n- {a}^{ij}\left( {x}_{0}\right) {D}_{ij}u = \widetilde{f} \]\n\nwhere\n\n\[ \n\widetilde{f} = f + \left( {{a}^{ij}\... | Yes |
Lemma 5.1. Suppose that \( \Omega \) has a flat part of the boundary \( S,\Omega \subset {\mathbb{R}}^{n} \cap \left\{ {{x}_{n} > }\right. \) \( 0\} \), and \( S \subset \partial {\mathbb{R}}_{ + }^{n} \) . Let the assumptions (4.4) and (4.5) be in force. Then there exist \( {R}_{0} \) and \( C \), depending only on \(... | The proof is exactly the same as that of Lemma 4.2. | No |
Lemma 5.2. Let \( \Omega \) be as in Lemma 5.1. Let the assumptions (4.4) and (4.5) be in force. Suppose that \( u \in {C}^{2,\alpha }\left( {\Omega \cup S}\right) \) is a solution of the equation (4.3) satisfying \( u = 0 \) on \( S \) . Then for any \( {\Omega }^{\prime } \subset \subset \Omega \cup S \), we have\n\n... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Let \( {R}_{0} \) be the constant in Lemma 5.1, and set \( {\bar{R}}_{0} = \min \left\{ {{R}_{0},\frac{1}{2}\operatorname{dist}\left\{ {{\Omega }^{\prime },\partial \Omega \smallsetminus S}\right\} }\right\} ,{\Omega }^{\prime \prime } = {\Omega }^{\prim... | Yes |
Theorem 5.3 (Global Schauder estimates). Let the assumptions (4.4), (4.5) be in force, and \( \partial \Omega \in {C}^{2,\alpha }\left( {0 < \alpha < 1}\right) \) . Suppose that \( u \in {C}^{2,\alpha }\left( \bar{\Omega }\right) \) is a solution of the equation (4.3) satisfying the boundary condition \( {\left. u\righ... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Let \( {x}_{0} \in \partial \Omega \), and suppose that \( \phi \) is the smooth function defined in Definition 5.1 of Chapter 1 such that\n\n\[{\left| \phi \right| }_{2,\alpha ;V \cap \Omega } \leq K,\;{\left| {\phi }^{-1}\right| }_{2,\alpha ;{B}_{1}^{ ... | No |
Theorem 6.1 (Weak maximum principle for classical solutions). Suppose that \( L \) is defined in (4.3), with the coefficient matrix \( \left( {a}^{ij}\right) \) satisfying (4.4), \( {b}^{i} \) being bounded, and \( c \geq 0 \) . If \( u \in {C}^{2}\left( \Omega \right) \cap C\left( \bar{\Omega }\right) \) satisfies \( ... | Proof. The proof is divided into two steps.\n\n(1) First, we assume that \( c\left( x\right) \geq {c}_{0} > 0 \) . Let \( v = u - \mathop{\sup }\limits_{{\partial \Omega }}{u}^{ + } \) . Then \( v \) satisfies\n\n\[ {Lv} \leq f - c\mathop{\sup }\limits_{{\partial \Omega }}{u}^{ + } \leq f\;\text{ in }\Omega \]\n\n\[ v ... | Yes |
Theorem 6.2. Under the assumptions of Theorem 6.1, if \( u \in {C}^{2}\left( \Omega \right) \cap C\left( \overline{\Omega }\right) \) is a solution of the equation (4.3), then\n\n(6.5)\n\n\[{\left| u\right| }_{0;\Omega } \leq \mathop{\sup }\limits_{{\partial \Omega }}\left| u\right| + C{\left| f\right| }_{0;\Omega }\]\... | Proof. We can apply Theorem 6.1 to both \( u \) and \( - u \) . | No |
Lemma 7.1. Let \( \partial \Omega \in {C}^{\left\lbrack {n/2}\right\rbrack + 4} \) . Suppose that the coefficients in the equation (7.1) satisfy (4.4) and (4.5), \( c \geq 0, f \in {C}^{\alpha }\left( \overline{\Omega }\right) \), and \( \varphi \in {C}^{2,\alpha }\left( \overline{\Omega }\right) \), where \( 0 < \alph... | Proof. Assume without loss of generality that \( \varphi \equiv 0 \) . Let \( {a}_{N}^{ij},{b}_{N}^{i},{c}_{N},{f}_{N} \in \) \( {C}^{\infty }\left( \bar{\Omega }\right) \left( {N = 1,2,\cdots }\right) \) be sequences of functions which converge to \( {a}^{ij},{b}^{i}, c, f(i, j = \) \( 1,2,\cdots, n) \) uniformly on \... | Yes |
Lemma 1.1. Let \( f \in {L}^{p}\left( \Omega \right) \left( {1 \leq p < \infty }\right) \). Then\n\n\[{\int }_{\Omega }{\left| f\right| }^{p}{dx} = p{\int }_{0}^{\infty }{t}^{p - 1}\left| {{A}_{t}\left( f\right) }\right| {dt}.\] | Proof. By Fubini's theorem,\n\n\[{\int }_{\Omega }{\left| f\right| }^{p}{dx} = {\int }_{\Omega }{dx}{\int }_{0}^{\left| f\left( x\right) \right| }p{t}^{p - 1}{dt}\]\n\n\[= {\int }_{\Omega }{dx}{\int }_{0}^{\infty }p{t}^{p - 1}{\chi }_{{A}_{t}}{dt} = p{\int }_{0}^{\infty }{t}^{p - 1}{\int }_{\Omega }{\chi }_{{A}_{t}}{dx... | Yes |
Theorem 1.2 (Marcinkiewicz interpolation theorem). Let \( 1 \leq p < q \leq \infty \) . Suppose that a quasilinear map \( T \) is of both weak type \( \left( {p, p}\right) \) and weak type \( \left( {q, q}\right) \) , i.e., (1.7) \[ \parallel {Tf}{\parallel }_{{L}_{w}^{p}} \leq {B}_{p}\parallel f{\parallel }_{{L}^{p}},... | Proof. Let \( f \in {L}^{r}\left( \Omega \right) \) . Decompose \( f \) as \( f = {f}_{1} + {f}_{2} \), where \[ {f}_{2}\left( x\right) = \left\{ \begin{array}{l} 0\;\text{ for }\left| {f\left( x\right) }\right| > {\gamma s} \\ f\left( x\right) \;\text{ for }\left| {f\left( x\right) }\right| \leq {\gamma s} \end{array}... | Yes |
Lemma 2.1. For \( f \in {L}^{1}\left( {\mathbb{R}}^{n}\right), f \geq 0 \) and fixed \( \alpha > 0 \), there exist two sets \( F \) and \( \Omega \) such that\n\n(i) \( {\mathbb{R}}^{n} = F \cup \Omega, F \cap \Omega = \varnothing \) ,\n\n(ii) \( f\left( x\right) \leq \alpha \), a.e. \( x \in F \) ,\n\n(iii) \( \Omega ... | Proof. Since \( {\int }_{{\mathbb{R}}^{n}}f\left( x\right) {dx} \) is finite, we can decompose \( {\mathbb{R}}^{n} \) into congruent cubes, with the side so large that for any such cube \( {Q}^{\prime } \), \n\n\[ {\int }_{{Q}^{\prime }}{fdx} \leq \alpha \] \n\nWe divide each \( {Q}^{\prime } \) into \( {2}^{n} \) equa... | Yes |
If \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \), then the Newtonian potential \( w \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) and satisfies the equation\n\n\[ - {\Delta w} = f,\;\forall x \in {\mathbb{R}}^{n}. \] | Proof. We rewrite (3.2) as\n\n\[ w\left( x\right) = {\int }_{{\mathbb{R}}^{n}}\Gamma \left( \xi \right) f\left( {x - \xi }\right) {d\xi } \]\n\nThen it is obvious that \( w \in {C}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . Using integration by parts, we obtain\n\n\[ {\Delta w}\left( x\right) = {\int }_{{\mathbb{R}}^... | Yes |
Lemma 3.2. \( T \) is a bounded linear operator from \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \) to \( {L}^{2}\left( {\mathbb{R}}^{n}\right) \), and\n\n\[ \parallel T{\parallel }_{\left( 2,2\right) } \leq 1 \] | Proof. First we assume that \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . By (3.3), for any \( {B}_{R} = {B}_{R}\left( 0\right) \),\n\n\[ {\int }_{{B}_{R}}{f}^{2}{dx} = {\int }_{{B}_{R}}{\left( \Delta w\right) }^{2}{dx} = \mathop{\sum }\limits_{{ij}}{\int }_{{B}_{R}}{D}_{ii}w{D}_{jj}{wdx}. \]\n\nUsing i... | Yes |
Lemma 3.3. For the fundamental solution \( \Gamma \left( x\right) \), the following estimate holds:\n\n(3.8)\n\n\[ J \triangleq \mathop{\sup }\limits_{{\xi \neq 0,1 \leq i, j \leq n}}{\int }_{\left| x\right| \geq 2\left| \xi \right| }\left| {{D}_{ij}\Gamma \left( {x - \xi }\right) - {D}_{ij}\Gamma \left( x\right) }\rig... | Proof. By the mean value theorem,\n\n\[ {\int }_{\left| x\right| \geq 2\left| \xi \right| }\left| {{D}_{ij}\Gamma \left( {x - \xi }\right) - {D}_{ij}\Gamma \left( x\right) }\right| {dx} \leq {\int }_{\left| x\right| \geq 2\left| \xi \right| }\mathop{\sum }\limits_{{k = 1}}^{n}\left| {{D}_{ijk}\Gamma \left( {x - {\lambd... | Yes |
Theorem 3.5. For \( 1 < p < \infty, T \) is of strong type \( \left( {p, p}\right) \) . | Proof. By Lemmas 3.2 and \( {3.4}, T \) is both of strong type \( \left( {2,2}\right) \) and of weak type \( \left( {1,1}\right) \) . If follows from the Marcinkiewicz interpolation theorem that \( T \) is of strong type \( \left( {p, p}\right) \), for any \( 1 < p \leq 2 \) .\n\nFor \( 2 < p < \infty \), we set \( {p}... | Yes |
Theorem 3.6. Suppose that \( u \in {W}_{0}^{2, p}\left( {B}_{R}\right) \) satisfies\n\n\[ - {\Delta u} = f.\]\n\nThen for \( 1 < p < \infty \), there exists a constant \( C \), depending only on \( n, p \), such that\n\n\[ {\begin{Vmatrix}{D}^{2}u\end{Vmatrix}}_{{L}^{p}\left( {B}_{R}\right) } \leq C\parallel f{\paralle... | Proof. Assume without loss of generality that \( u \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{n}\right) \) . By Lemma 3.1,\n\n\[ u\left( x\right) = {\int }_{{\mathbb{R}}^{n}}\Gamma \left( {x - \xi }\right) \left( {-{\Delta u}\left( \xi \right) }\right) {d\xi }\n\n= {\int }_{{\mathbb{R}}^{n}}\Gamma \left( {x - \xi }\righ... | No |
Lemma 4.1. Let the assumptions (4.3)-(4.5) be in force. Then there exists \( {R}_{0} > 0 \), depending only on \( n, p,\Lambda /\lambda \) and the modulus of continuity of \( {a}^{ij} \), such that \( \textit{for any}\;0 < R \leq {R}_{0}\textit{ and any}\;u \in {W}_{0}^{2, p}\left( {B}_{R}\right) \;\left( {1 < p < \inf... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Let \( {B}_{R} \) be a ball centered at \( {x}_{0} \) . We freeze the coefficients at \( {x}_{0} \) and rewrite (4.1) as\n\n(4.7)\n\n\[ \n- {a}^{ij}\left( {x}_{0}\right) {D}_{ij}u = \widetilde{f} \]\n\nwhere\n\n\[ \n\widetilde{f} = f + \left\lbrack {{a}^... | Yes |
Theorem 4.2. Let the assumptions (4.3)-(4.5) be in force. Suppose that \( u \in \) \( {W}_{loc}^{2, p}\left( \Omega \right) \) satisfies (4.1) almost everywhere. Then for any \( {\Omega }^{\prime } \subset \subset \Omega \) , \[ \parallel u{\parallel }_{{W}^{2, p}\left( {\Omega }^{\prime }\right) } \leq C\left\{ {\frac... | Proof. Assume without loss of generality that \( \lambda = 1 \) . Let \( {R}_{0} \) be the constant in Lemma 4.1 and \( {\bar{R}}_{0} = \min \left\{ {{R}_{0},\frac{1}{2}\operatorname{dist}\left\{ {{\Omega }^{\prime },\partial \Omega }\right\} }\right\} \) . For any \( {x}_{0} \in {\Omega }^{\prime } \) and \( {\bar{R}}... | Yes |
Lemma 5.1. Let \( u \in {W}^{2, p}\left( {B}_{R}^{ + }\right) \cap {W}_{0}^{1, p}\left( {B}_{R}^{ + }\right) \), where \( {B}_{R}^{ + } = {B}_{R}\left( 0\right) \cap \left\{ {{x}_{n} > 0}\right\} \) . Suppose that \( u \) vanishes in a neighborhood of \( \partial {B}_{R}^{ + } \cap \left\{ {{x}_{n} > 0}\right\} \), and... | Proof. Let\n\n\[ \n\widetilde{u} = \left\{ \begin{array}{l} u\left( {{x}^{\prime },{x}_{n}}\right) \;\text{ for }{x}_{n} \geq 0 \\ - u\left( {{x}^{\prime }, - {x}_{n}}\right) \;\text{ for }{x}_{n} < 0 \end{array}\right.\n\]\n\nThen \( \widetilde{u} \in {W}_{0}^{2, p}\left( {B}_{R}\right) \) and satisfies the equation\n... | Yes |
Theorem 6.3. Let \( \Omega \) be a bounded domain with \( \partial \Omega \in {C}^{1,1} \) . Suppose that for any \( L \in \mathcal{L} \), the Dirichlet problem (4.1),(4.2) admits at most one solution in \( {W}^{2, p}\left( \Omega \right) ,1 < p < \infty \) . Then for any \( L \in \mathcal{L} \), the solution \( u \in ... | Proof. Assume without loss of generality that \( \lambda = 1 \) . If (6.7) does not hold, then for any \( N \), there exist \( {L}_{N} = - {a}_{N}^{ij}{D}_{ij} + {b}_{N}^{i}{D}_{i} + {c}_{N} \in \mathcal{L},{f}_{N} \in {L}^{p}\left( \Omega \right) \) , \( {u}_{N} \in {W}^{2, p}\left( \Omega \right) \cap {W}_{0}^{1, p}\... | Yes |
Lemma 1.1. Suppose \( \Phi \left( s\right) \in {C}_{loc}^{0,1}\left( \mathbb{R}\right) \) is a convex function \( \left( {\Phi \left( {\sigma {s}_{1} + \left( {1 - \sigma }\right) {s}_{2}}\right) }\right. \) \( \leq {\sigma \Phi }\left( {s}_{1}\right) + \left( {1 - \sigma }\right) \Phi \left( {s}_{2}\right) \) for \( \... | Proof. We prove only (1). First assume that \( \Phi \in {C}_{loc}^{2}\left( \mathbb{R}\right) \) . Let \( v = \Phi \left( u\right) \) . By assumption, \( {\Phi }^{\prime }\left( s\right) \geq 0 \), and \( u \) is a weak subsolution of \( {\left( {1.1}\right) }^{\prime } \) . Thus for any \( \varphi \in {C}_{0}^{\infty ... | Yes |
Theorem 1.4 (Harnack inequality). Let the assumption (1.2) be in force on \( {B}_{R} \) . Suppose that \( u \) is a nonnegative, bounded weak solution of \( {\left( {1.1}\right) }^{\prime } \) on \( {B}_{R} \) . Then for any \( 0 < \theta < 1 \) , (1.18) \[ \underset{{B}_{\theta R}}{\operatorname{ess}\sup }u \leq C\und... | Proof. Let \( {R}_{1} = \frac{\left( 1 + \theta \right) }{2}R \) . By Lemma 1.2, \[ \underset{{B}_{\theta R}}{\operatorname{ess}\sup }u \leq C{\left\lbrack {f}_{{B}_{{R}_{1}}}{\left| u\right| }^{{p}_{0}}dx\right\rbrack }^{1/{p}_{0}} \] where \( {p}_{0} \) is the constant in Lemma 1.3. By Lemma 1.3, \[ \underset{{B}_{\t... | Yes |
Lemma 2.1. Let \( \omega \left( P\right) \) be a nonnegative, nondecreasing function on \( \left\lbrack {0,{R}_{0}}\right\rbrack \) . Suppose that for some \( 0 < \theta ,\eta < 1,0 < \alpha \leq 1 \) and \( K \geq 0,\omega \) satisfies\n\n(2.1)\n\n\[ \omega \left( {\theta R}\right) \leq {\eta \omega }\left( R\right) +... | Proof. Let \( {\widetilde{R}}_{0} \in \left( {\theta {R}_{0},{R}_{0}}\right\rbrack \) and \( {R}_{s} = {\theta }^{s}{\widetilde{R}}_{0}\left( {s = 1,2,\cdots }\right) \) . By (2.1),\n\n(2.3)\n\n\[ \omega \left( {R}_{s + 1}\right) \leq {\eta \omega }\left( {R}_{s}\right) + K{R}_{s}^{\alpha }\;\left( {s = 1,2,\cdots }\ri... | Yes |
Theorem 2.2. Let the assumption (1.2) be in force. Suppose that \( u \) is a bounded weak solution of \( {\left( {1.1}\right) }^{\prime } \) . Then there exist \( C \geq 0 \) and \( 0 < \gamma < 1 \) such that for any \( {B}_{R}\left( x\right) \subset \Omega \) ,\n\n(2.4)\n\n\[ \underset{{B}_{R}\left( x\right) }{\opera... | Proof. Let \( M\left( R\right) = \underset{{B}_{R}\left( {x}_{0}\right) }{\operatorname{ess}\sup }u, m\left( R\right) = \underset{{B}_{R}\left( {x}_{0}\right) }{\operatorname{ess}\inf }u \) and \( \omega \left( R\right) = M\left( R\right) - m\left( R\right) \) , where \( 0 < R \leq {d}_{{x}_{0}} \) . Then the function ... | Yes |
Lemma 3.1. Suppose that \( v \) is a bounded weak subsolution of \( {\left( {1.1}\right) }^{\prime },{x}_{0} \in \partial \Omega \) , \( M = \mathop{\sup }\limits_{{\partial \Omega \cap {B}_{R}\left( {x}_{0}\right) }}{v}^{ + } \) . Then for any \( p > 0,0 < \theta < 1 \) ,\n\n(3.3)\n\n\[ \mathop{\sup }\limits_{{{B}_{\t... | Proof. We choose the test function \( \varphi = {\zeta }^{2}{\left\lbrack {v}^{p - 1} - {M}^{p - 1}\right\rbrack }^{ + } \), where \( \zeta \) is a cutoff function on \( {B}_{R}\left( {x}_{0}\right) \) . Clearly, \( \varphi \in {W}_{0}^{1,2}\left( \Omega \right) \) and \( \varphi \geq 0 \) . The remaining proof is simi... | No |
Lemma 3.2. Suppose that \( v \) is a bounded, nonnegative weak supersolution of \( {\left( {1.1}\right) }^{\prime },\Omega \) satisfies a uniform exterior cone condition, and \( \left( {a}^{ij}\right) \) satisfies (1.2). For \( {x}_{0} \in \partial \Omega \), we set \( m = \mathop{\inf }\limits_{{\partial \Omega \cap {... | Proof. From Lemma 1.1, we derive that \( {v}_{m}^{ - } \) is a weak supersolution of \( {\left( {1.1}\right) }^{\prime } \) and \( {\left( {v}_{m}^{ - }\right) }^{-p} \) is a weak subsolution of \( {\left( {1.1}\right) }^{\prime } \) . Using translation and scaling if necessary, we may assume without loss of generality... | Yes |
Theorem 3.3. Suppose that \( \Omega \) satisfies a uniform exterior cone condition, and \( \left( {a}^{ij}\right) \) satisfies (1.2). Let \( u \) be a weak solution of \( {\left( {1.1}\right) }^{\prime } \) with \( {\left\lbrack u\right\rbrack }_{{\varepsilon }_{1},\partial \Omega } < \infty \), where \( {\varepsilon }... | Proof. Set \( {\Omega }_{R} = \Omega \cap {B}_{R}\left( {x}_{0}\right) ,\partial {\Omega }_{R} = \partial \Omega \cap {B}_{R}\left( {x}_{0}\right) \) and\n\n\[ M\left( R\right) = \mathop{\sup }\limits_{{\Omega }_{R}}u,\;m\left( R\right) = \mathop{\inf }\limits_{{\Omega }_{R}}u,\;\omega \left( R\right) = M\left( R\right... | Yes |
Theorem 3.4. Suppose that \( \Omega \) satisfies a uniform exterior cone condition, the coefficients of (3.6) satisfy (1.2), and \( f \in {L}^{{q}_{ * }}\left( \Omega \right) ,{f}^{i} \in {L}^{q}\left( \Omega \right) \) for some \( q > n \), where \( {q}_{ * } = {nq}/\left( {n + q}\right) \) . If \( u \) is a weak solu... | The proof is exactly the same as that of Theorem 2.3. | No |
Theorem 3.5. Under the assumptions of Theorem 3.4, there exist \( C > 0 \) and \( 0 < \gamma < 1 \) such that\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq C{\left| x - y\right| }^{\gamma }\left( {{\left| u\right| }_{0;\Omega } + {\left\lbrack u\right\rbrack }_{{\varepsilon }_{1};\partial \Omega } + \... | Proof. Assume without loss of generality that \( {d}_{xy} = {d}_{x} \) . For simplicity we let \( \delta = {d}_{x} \) . Theorem 3.4 implies the following interior estimate (similar to the corollary of Theorem 2.2):\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq C{\left| x - y\right| }^{\gamma }\left( {... | Yes |
Theorem 1.1. Suppose that the structure conditions (1.2), (1.3), (1.4) and (1.7) hold, \( {F}_{0} \triangleq \parallel g{\parallel }_{{L}^{q}} + \parallel f{\parallel }_{{L}^{q * }} < \infty \) for some \( q > n \), where \( {q}_{ * } = {nq}/\left( {n + q}\right) \) . Then for a weak subsolution \( u \in {W}^{1,2}\left... | Proof. Assume that \( \mathop{\sup }\limits_{{\partial \Omega }}{u}^{ + } < \infty \) . For \( k \geq \mathop{\sup }\limits_{{\partial \Omega }}{u}^{ + } \), we choose the test function \( \varphi = {\left( u - k\right) }^{ + } \) in (1.6), and we set \( v = {\left( u - k\right) }^{ + }, A\left( k\right) = \{ x \in \Om... | Yes |
Theorem 2.2. Let the structure conditions (1.2), (1.3) and (1.4) be in force. Suppose that \( u \in {W}^{1,2}\left( {B}_{\sigma R}\right) \left( {\sigma > 1}\right) \) is a bounded weak supersolution of (1.1) on \( {B}_{\sigma R} \) and that (2.1) holds for some \( q > n \) . Then there exists \( {p}_{0} > 0 \) such th... | Proof. Using \( {F}_{0} + \varepsilon \) instead of \( {F}_{0} \) if necessary, we may assume without loss of generality that \( {F}_{0} > 0 \) . We also assume that \( R = 1 \) and choose a cutoff function \( \zeta \) on \( {B}_{1} \) . Choosing the test function \( \varphi = {\zeta }^{2}{\widetilde{u}}^{-\left( {{2p}... | Yes |
Theorem 2.5. Let \( \Omega \) satisfy a uniform exterior cone condition, and let the structure conditions (1.2),(1.3) and (1.4) be in force. Suppose that \( u \) is a bounded weak supersolution of (1.1) on \( \Omega ,{x}_{0} \in \partial \Omega \) and \( m = \mathop{\inf }\limits_{{{B}_{R}\left( {x}_{0}\right) \cap \pa... | \[ \mathop{\inf }\limits_{{{B}_{\theta R}\left( {x}_{0}\right) }}{u}_{m}^{ - } \geq \frac{1}{C}{\left\lbrack {\int }_{{B}_{R}\left( {x}_{0}\right) }{\left( {u}_{m}^{ - }\right) }^{p}dx\right\rbrack }^{1/p} - C{F}_{0} \] where \( C \) depends only on \( n,\Lambda, q,\Omega ,{\left( 1 - \theta \right) }^{-1} \) and \( \p... | Yes |
Lemma 1.1. Let \( u \in {W}_{loc}^{2,1}\left( \Omega \right) \cap C\left( \Omega \right) \). Then\n\n\[ \chi \left( y\right) = \{ {Du}\left( y\right) \} ,\; - {D}^{2}u\left( y\right) \geq 0,\;\text{ a.e. }y \in {\Gamma }_{u}. \] | Proof. Let \( w\left( x\right) \) be defined by (1.3). For each fixed direction \( \xi \in {\mathbb{R}}^{n},\left| \xi \right| = 1 \) , we have\n\n\[ \frac{w\left( {y + {h\xi }}\right) - w\left( y\right) }{h} \rightarrow \frac{\partial w}{\partial \xi } \]\n\n\[ \frac{w\left( {y + {h\xi }}\right) + w\left( {y - {h\xi }... | Yes |
Lemma 1.2. Let \( u \in C\left( \Omega \right) \) . Then\n\n(1) for any \( y \in {\Gamma }_{u} \) ,\n\n(1.10)\n\n\[ \left| p\right| \leq \frac{2\sup \left| u\right| }{\operatorname{dist}\{ y,\partial \Omega \} },\;\forall p \in \chi \left( y\right) \]\n\n\( \left( 2\right) \) the normal mapping maps any compact subset ... | Proof. For \( y \in {\Gamma }_{u} \) ,\n\n(1.11)\n\n\[ u\left( y\right) + p \cdot \left( {x - y}\right) \geq u\left( x\right) ,\;\forall x \in \Omega . \]\n\nThe ray starting at \( y \) with direction \( - p \) intersects \( \partial \Omega \) at \( {x}_{0} \), i.e.,\n\n\( \left( {1.12}\right) \)\n\n\[ {x}_{0} = y - \f... | Yes |
Lemma 1.3. Suppose that \( \Omega, A \) are open domains in \( {\mathbb{R}}^{n} \) .\n\n(2) If the diameter of \( \Omega \) is \( d \), then\n\n(1.13)\n\n\[ \left| {\Omega \left\lbrack {{x}_{0},\lambda }\right\rbrack }\right| \geq {\left( \frac{\lambda }{d}\right) }^{n}{\omega }_{n} \]\n\nwhere \( \left| \cdot \right| ... | Proof. (1) is obvious. We prove (2). Clearly, \( {B}_{d}\left( {x}_{0}\right) \supset \Omega \) . Let \( A = {B}_{d}\left( {x}_{0}\right) \) . By (1) and (1.8),\n\n\[ \left| {\Omega \left\lbrack {{x}_{0},\lambda }\right\rbrack }\right| \geq \left| {A\left\lbrack {{x}_{0},\lambda }\right\rbrack }\right| = \left| {{B}_{\... | Yes |
Lemma 1.4. Suppose that \( u \in {C}^{2}\left( \Omega \right), g \in C\left( \bar{\Omega }\right), g \geq 0 \), and \( E \) is a measurable subset of \( {\Gamma }_{u} \) . Then\n\n(1.14)\n\n\[{\int }_{{Du}\left( E\right) }g\left( {\xi \left( p\right) }\right) {dp} \leq {\int }_{E}g\left( x\right) \det \left( {-{D}^{2}u... | Proof. Let \( J\left( x\right) = \det \left( {-{D}^{2}u}\right), S = \{ x \in \Omega \mid J\left( x\right) = 0\} \) . By Sard’s theorem (cf. Appendix 2), \( \left| {{Du}\left( S\right) }\right| = 0 \) .\n\nFirst we assume that \( E \) is open. Then \( E \smallsetminus S \) is an open set. Thus there exist cubes \( {\le... | Yes |
Lemma 1.5. Suppose that \( u \in C\left( \bar{\Omega }\right), u \leq 0 \) on \( \partial \Omega ,{x}_{0} \in \Omega \), and \( u\left( {x}_{0}\right) > 0 \) . Then\n\n(1.15)\n\n\[ \Omega \left\lbrack {{x}_{0}, u\left( {x}_{0}\right) }\right\rbrack \subset \chi \left( {\Gamma }_{u}^{ + }\right) \]\n\nwhere \( {\Gamma }... | Proof. This lemma is obvious from the geometric picture. However, here we shall give a rigorous analytical proof. Let \( p \in \Omega \left\lbrack {{x}_{0}, u\left( {x}_{0}\right) }\right\rbrack \) . From Definition 1.4,\n\n(1.16)\n\n\[ u\left( {x}_{0}\right) + p \cdot \left( {x - {x}_{0}}\right) \geq 0,\;\forall x \in... | Yes |
Lemma 1.6. Suppose that \( u \in {C}^{2}\left( \bar{\Omega }\right) \) and \( u \leq 0 \) on \( \partial \Omega \) . Then\n\n(1.19)\n\n\[ \mathop{\sup }\limits_{\Omega }u \leq \frac{d}{\sqrt[n]{{\omega }_{n}}}{\left\lbrack {\int }_{{\Gamma }_{u}^{ + }}\det \left( -{D}^{2}u\right) dx\right\rbrack }^{1/n} \]\n\nwhere \( ... | Proof. For \( {x}_{0} \in \Omega, u\left( {x}_{0}\right) > 0 \), we can derive from (1.15) and (1.13) that\n\n\[ \left| {\chi \left( {\Gamma }_{u}^{ + }\right) }\right| \geq \left| {\Omega \left\lbrack {{x}_{0}, u\left( {x}_{0}\right) }\right\rbrack }\right| \geq {\omega }_{n}{\left\lbrack \frac{u\left( {x}_{0}\right) ... | Yes |
Theorem 1.8. Suppose that \( u \in C\left( \bar{\Omega }\right) \cap {W}_{loc}^{2, n}\left( \Omega \right) \) satisfies \( {L}_{0}u \leq f \) almost everywhere in \( \Omega \) and the coefficients \( {a}^{ij} \) satisfy (1.28). Then\n\n(1.32)\n\n\[ \mathop{\sup }\limits_{\Omega }u\left( x\right) \leq \mathop{\sup }\lim... | Proof. We assume that the right-hand side of (1.32) is finite. Set \( A = \left( {a}^{ij}\right) \) , \( U = \left( {-{D}^{2}u}\right) \) . By Lemma 1.1, \( U \geq 0 \) almost everywhere in \( {\Gamma }_{v} \) . Using the inequality that the arithmetic mean is greater than the geometric mean, we get\n\n\[ - {a}^{ij}{D}... | Yes |
Theorem 2.4 (Harnack inequality). Let the coefficients of \( L \) satisfy the uniform ellipticity assumption (2.2). Suppose that \( u \in {W}^{2, n}\left( \Omega \right) \) satisfies \( {Lu} = f \) almost everywhere in \( \Omega, f/\lambda \in {L}^{n}\left( \Omega \right) \), and \( u \geq 0 \) on \( {B}_{2R}\left( y\r... | Proof. For \( p \) determined in Theorem 2.3, we apply Theorems 2.1 and 2.3 to conclude that\n\n\[ \mathop{\sup }\limits_{{{B}_{R/2}\left( y\right) }}u \leq C\left\lbrack {{\left( {\int }_{{B}_{R}\left( y\right) }{u}^{p}dx\right) }^{1/p} + R\parallel f/\lambda {\parallel }_{{L}^{n}\left( {{B}_{2R}\left( y\right) }\righ... | Yes |
Theorem 2.5. Let the coefficients of \( L \) satisfy the uniform ellipticity assumption (2.2). Suppose that \( u \in {W}_{loc}^{2, n}\left( \Omega \right) \) satisfies \( {Lu} = f \) almost everywhere in \( \Omega \) and \( f/\lambda \in {L}^{n}\left( \Omega \right) \) . Then for any \( {B}_{{R}_{0}}\left( y\right) \su... | Proof. Set \[ m\left( R\right) = \mathop{\inf }\limits_{{{B}_{R}\left( y\right) }}u,\;M\left( R\right) = \mathop{\sup }\limits_{{{B}_{R}\left( y\right) }}u, \] \[ \omega \left( R\right) = M\left( R\right) - m\left( R\right) ,\;{F}_{0} = \parallel f/\lambda {\parallel }_{{L}^{n}\left( \Omega \right) }. \] It is clear th... | Yes |
Theorem 3.2. Let the assumptions of Lemma 3.1 be in force. If \( u \) is a solution of (2.1), then there exist \( C > 0 \) and \( 0 < \beta < 1 \) such that\n\n(3.4)\n\n\[ \left| {u\left( x\right) - u\left( y\right) }\right| \leq C{\left| x - y\right| }^{\beta }\left( {{\left| u\right| }_{0} + {\left\lbrack u\right\rbr... | Proof. If we apply Lemma 3.1 to \( u \) and \( - u \), then \( \forall x \in \Omega ,{x}_{0} \in \partial \Omega \) ,\n\n\[ \left| {u\left( x\right) - u\left( {x}_{0}\right) }\right| \leq C{\left| x - {x}_{0}\right| }^{\alpha /\left( {1 + \alpha }\right) } \cdot \left( {{\left| u\right| }_{0} + {\left\lbrack u\right\rb... | Yes |
Theorem 1.1. Under the structure conditions (F1) and (F5), a solution \( u \in \) \( C\left( \bar{\Omega }\right) \cap {W}_{\text{loc }}^{2, n}\left( \Omega \right) \) of the Dirichlet problem\n\n(1.1)\n\n\[ \nF\left( {x, u,{Du},{D}^{2}u}\right) = 0\;\text{ in }\Omega ,\n\]\n\n(1.2)\n\n\[ \nu = \varphi \;\text{ on }\Om... | Proof. We rewrite the equation (1.1) as\n\n\[ \n- {a}^{ij}{D}_{ij}u - F\left( {x, u,{Du},0}\right) = 0, \n\]\n\nwhere\n\n\[ \n{a}^{ij}\left( {x, z, p, r}\right) = {\int }_{0}^{1}\frac{\partial F}{\partial {r}_{ij}}\left( {x, z, p,{\tau r}}\right) {d\tau }. \n\]\n\nUsing (F5) on the subdomain \( {\Omega }^{ + } = \{ x \... | Yes |
Lemma 1.2. Suppose that \( u \in {W}_{loc}^{2, n}\left( \Omega \right) \) satisfies the inequality\n\n(1.3)\n\n\[ \n{Lu} = - {a}^{ij}\left( x\right) {D}_{ij}u \geq - \lambda \left( {{\mu }_{0}{\left| Du\right| }^{2} + g\left( x\right) }\right) \]\n\nwhere \( {\mu }_{0} > 0, g \in {L}^{n}\left( \Omega \right) \), and \(... | Proof. Set\n\n\[ \nv = 1 - {e}^{-{\mu }_{0}u},\;M = {\left| u\right| }_{0,{B}_{2R}}.\n\]\n\nFrom (1.3), we deduce that\n\n\[ \n{Lv} = - {a}^{ij}{D}_{ij}v = {\mu }_{0}{e}^{-{\mu }_{0}u}\left\lbrack {{Lu} + {\mu }_{0}{a}^{ij}{D}_{i}u{D}_{j}u}}\right\rbrack \n\]\n\n\[ \n\geq - \lambda {\mu }_{0}\left| {g\left( x\right) }\... | Yes |
Theorem 1.3. Suppose that \( u \in {W}_{loc}^{2, n}\left( \Omega \right) \) satisfies the inequality\n\n(1.5)\n\n\[ \left| {Lu}\right| \leq \lambda \left( {{\mu }_{0}{\left| Du\right| }^{2} + g\left( x\right) }\right) \;\text{ in }\Omega \]\n\nwhere we assume the same conditions on \( L,{\mu }_{0} \) and \( g \) as in ... | Proof. Set\n\n\[ M\left( R\right) = \mathop{\sup }\limits_{{B}_{R}}u,\;m\left( R\right) = \mathop{\inf }\limits_{{B}_{R}}u \]\n\n\[ \omega \left( R\right) = M\left( R\right) - m\left( R\right) ,\;{F}_{0} = \parallel g{\parallel }_{{L}^{n}\left( {B}_{R}\right) }.\]\n\nApplying the weak Harnack inequality (Lemma 1.2) to ... | Yes |
Theorem 1.4. Suppose that \( u \in {W}_{loc}^{2, n}\left( \Omega \right) \) is a solution of (1.1) satisfying \( {\left| u\right| }_{0;\Omega } \leq {M}_{0} \) and that \( F\left( {x, z, r, p}\right) \) satisfies the structure conditions (F1) and (F2). Then for any \( {B}_{R} \subset \Omega \) and \( 0 < \sigma \leq 1 ... | Proof. The structure condition (F2) implies that\n\n\[ \left| {{a}^{ij}{D}_{ij}u}\right| = \left| {F\left( {x, u,{Du},0}\right) }\right| \leq \lambda {\mu }_{2}\left( {1 + {\left| Du\right| }^{2}}\right) ,\]\n\nwhere\n\n\[ {a}^{ij} = {\int }_{0}^{1}\frac{\partial F}{\partial {r}_{ij}}\left( {x, u,{Du},\tau {D}^{2}u}\ri... | Yes |
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