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Results tagged with ap.analysis-of-pdes
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user 147073
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
0
answers
139
views
Is there a solution of a first order nonlinear PDE?
Let $a^{ij}, b_{i}, c$ and $f>0$ are smooth function. Suppose $\Lambda I\geq (a^{ij})\geq \lambda I$, where $I$ is identity matrix, $\lambda, \Lambda$ is positive constant. Is there solution of the fo …
- 153
2
votes
2
answers
234
views
Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on a Kähl...
Let M be a 2-dimensional (complex dimension) Kähler manifold and $\phi$ be a real $(1,1)$-form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$?
- 153
2
votes
1
answer
160
views
The Monge- Ampère equation with a non positive right hand side
Let $\Omega$ be a domain, $u$ and $f$ are real valued functions on $\Omega$, $(u_{ij})$ is the Hessian matrix of $u$. The function $f$ may change sign: that said, do there exist solutions for the foll …
- 153
1
vote
0
answers
157
views
Is $(e^{u}+1)\Delta u+u=0$ the Euler-Lagrange equation of a functional energy?
Does there exist a functional energy $I$ such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional $I$?
- 153
1
vote
1
answer
123
views
Are there $f, g$ such that $\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$ where...
Let $f,g$ are the functions of $S^{1}$. Are there $f, g$ such that
$$\int_{S^{1}} |f'|^{2}+|g'|^{2}d\theta-2\int_{S^{1}}f'g<0,$$
where $f'=\frac{\partial f}{\partial \theta}$?
- 153