共查询到20条相似文献,搜索用时 199 毫秒
1.
N. A. Shananin 《Mathematical Notes》1976,19(5):447-451
In this note we show that the equation $$ - \left\{ {\left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }}} \right)^3 + \left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }}} \right)^2 + 6i\left( {\frac{1}{i}\frac{\partial }{{\partial x_1 }}} \right)\left( {\frac{1}{i}\frac{\partial }{{\partial x_2 }}} \right)x_1 } \right\}u = f$$ is locally unsolvable at the origin of the coordinate system. This equation belongs to the class that generalizes the principal type to the case of weighted derivatives. The example is interesting because the weighted principal symbol is real (in this situation, equations of principal type are solvable) but the unsolvability depends on the behavior of the lower-order terms in a neighborhood of the zeros of the weighted principal symbol. 相似文献
2.
Li Mingzhong 《数学年刊A辑(中文版)》1980,1(2):299-308
In this paper, we consider the generalized Riemann-Hilberij problem for second
order quasi-linear elliptic complex equation
\[\begin{array}{l}
\frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + {q_1}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial {z^2}}} + {q_2}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {q_3}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + {q_4}(z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}})\frac{{{\partial ^2}\bar w}}{{\partial z\partial \bar z}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1)\{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \gamma (z,w,\frac{{\partial w}}{{\partial \bar z}},\frac{{\partial w}}{{\partial z}}),z \in G
\end{array}\]
satifying the boundary condition
\[{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_1}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _1}(z),{\mathop{\rm Re}\nolimits} \left[ {{{\bar \lambda }_2}(z)\frac{{\partial w}}{{\partial \bar z}}} \right] = {\gamma _2}(z),z \in \gamma {\kern 1pt} {\kern 1pt} {\kern 1pt} (2)\]
Many authors (see that papers 1, 4-6) have studied the Diriohlet problem and Riemann-Hilbert problem for linear elliptic complex equation. In our papers 2, 3 we also
considered the generalized Riemann-Hilbert problem of the general second order linear elliptic complex equation. We obtained the existence theorem, the explicit form of
generalized solution and the sufficient and necessary conditions for the solvability of the above mentioned boundary value problem.
Based on these results and applying the property of the introduced integral operators
and Schauder's fixed-point principle, it can be proved that the analogous deductions in 3 also hold for the generalized Riemann-Hilber problem (1), (2) of the quasi-linear complex equation, i, e., we have the following theorem:
Theorem, If the coefficients of second order quasi-linear elliptic complex equation
(1) satifies some conditions then
i) When index \({n_1} \ge 0,{n_2} \ge 0\), the boundary value problem (1), (2)
is always solvable and the solution depends on 2 \(2({n_1} + {n_2} + 1)\) arbitrary real constants.
ii) When index \({n_1} \ge 0,{n_2} < 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (or{\kern 1pt} {\kern 1pt} {\kern 1pt} {n_1} < 0,{n_2} \ge 0{\kern 1pt} )\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1),(2) consists of \( - 2{n_2} - 1{\kern 1pt} {\kern 1pt} {\kern 1pt} ( - 2n, - 1)\) real equalities, if and only if the equalities are
satisfied, the boundary value problem is solvable and the solution depends on
\(2{n_1} + 1{\kern 1pt} {\kern 1pt} (2{n_2} + 1)\) arbitrary real constants.
iii)When index \({n_1} < 0,{n_2} < 0\), the sufficient and necessary condition for the solvability of the above mentioned boundary value problem (1) , (2) consists of
\( - 2({n_1} + {n_2} + 1)\) real equalities, if and only if the equalitieis are satisfied, the boundary-value problem is solvable.
Finally, in the similar way, we may farther extend the result to the case of the
nonlinear uniform elliptic complex equation. 相似文献
3.
M. A. Raupp R. A. Feijóo C. A. de Moura 《Bulletin of the Brazilian Mathematical Society》1978,9(2):39-61
In this paper we present the analysis of an algorithm of Uzawa type to compute solutions of the quasi variational inequality $$\begin{gathered} (QVI)\left( {\frac{{\partial ^2 u}}{{\partial t^2 }},\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + \left( {\frac{{\partial u}}{{\partial x}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \left( {\frac{{\partial ^2 u}}{{\partial x\partial t}},\frac{{\partial \upsilon }}{{\partial x}} - \frac{{\partial ^2 u}}{{\partial x\partial t}}} \right) + \hfill \\ + \left[ {u(1,t) + \frac{{\partial u}}{{\partial t}}(1,t)} \right]\left[ {\upsilon (1) - \frac{{\partial u}}{{\partial t}}(1,t)} \right] + J(u;\upsilon ) - J\left( {u;\frac{{\partial u}}{{\partial t}}} \right) \geqslant \hfill \\ \geqslant \left( {f,\upsilon - \frac{{\partial u}}{{\partial t}}} \right) + F(t)\left[ {\upsilon (0) - \frac{{\partial u}}{{\partial t}}(0,t)} \right],t > 0,\forall \upsilon \in H^1 (0,1), \hfill \\ \end{gathered} $$ which is a model for the dynamics of a pile driven into the ground under the action of a pile hammer. In (QVI) (...) is the scalar product inL 2(0, 1) andJ(u;.) is a convex functional onH 1(0, 1), for eachu, describing the soil-pile friction effect. 相似文献
4.
Li XUNJING 《数学年刊B辑(英文版)》1980,1(34):453-458
In this paper we consider the systems governed, by parabolioc equations
\[\frac{{\partial y}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}} ({a_{ij}}(x,t)\frac{{\partial y}}{{\partial {x_j}}}) - ay + f(x,t)\]
subject to the boundary control \[\frac{{\partial y}}{{\partial {\nu _A}}}{|_\sum } = u(x,t)\] with the initial condition \[y(x,0) = {y_0}(x)\]
We suppose that U is a compact set but may not be convex in \[{H^{ - \frac{1}{2}}}(\Gamma )\], Given \[{y_1}( \cdot ) \in {L^2}(\Omega )\] and d>0, the time optimal control problem requiers to find the control
\[u( \cdot ,t) \in U\] for steering the initial state {y_0}( \cdot )\] the final state \[\left\| {{y_1}( \cdot ) - y( \cdot ,t)} \right\| \le d\] in a minimum, time.
The following maximum principle is proved:
Theorem. If \[{u^*}(x,t)\] is the optimal control and \[{t^*}\] the optimal time, then there is a
solution to the equation
\[\left\{ {\begin{array}{*{20}{c}}
{ - \frac{{\partial p}}{{\partial t}} = \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ji}}(x,t)\frac{{\partial p}}{{\partial {x_j}}}) - \alpha p,} }\{\frac{{\partial p}}{{\partial {\nu _{{A^'}}}}}{|_\sum } = 0}
\end{array}} \right.\]
with the final condition \[p(x,{t^*}) = {y^*}(x,{t^*}) - {y_1}(x)\], such that
\[\int_\Gamma {p(x,t){u^*}} (x,t)d\Gamma = \mathop {\max }\limits_{u( \cdot ) \in U} \int_\Gamma {p(x,t)u(x)d\Gamma } \] 相似文献
5.
对构成广义Greiner算子的向量场$X_j = \frac{\partial }{\partial x_j} + 2ky_j \vert z\vert ^{2k - 2}\frac{\partial }{\partialt}$, $Y_j = \frac{\partial }{\partial y_j } - 2kx_j \vert z\vert^{2k - 2}\frac{\partial }{\partial t}$, j = 1,... ,n, x,y∈ Rn, $z = x + \sqrt { - 1} \,y$, t ∈ R, k ≥1, 得到了拟球域内和拟球域外的Hardy型不等式;建立了广义Picone型恒等式,并由此导出比文献[3]更一般的全空间上的Hardy型不等式;并在$p = 2$时建立了具最佳常数的Hardy型不等式. 相似文献
6.
We study the Γ-convergence of the following functional (p > 2)
$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega}
|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}}
\int\limits_{\Omega}
W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}
\int\limits_{\partial\Omega}
V(Tu)d\mathcal{H}^2,$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega}
|Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}}
\int\limits_{\Omega}
W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}}
\int\limits_{\partial\Omega}
V(Tu)d\mathcal{H}^2, 相似文献
7.
G. P. Kukhta 《Journal of Mathematical Sciences》1992,60(2):1396-1398
By the Fourier method a solution of the equation
|