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Results tagged with ap.analysis-of-pdes
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user 16702
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
0
answers
68
views
Hess-Schrader-Uhlenbrock inequality for non-symmetric operators
Let $M$ be a (compact, let's say) Riemannian manifold, $\mathcal{V}$ a vector bundle over $M$ with covariant derivative $\nabla$ and a fiber metric. Let $L = - \mathrm{tr}(\nabla^2) + V$ with some pot …
- 9,853
0
votes
1
answer
233
views
What is the Newtonian Capacity of a subset of $S^n$?
In their paper "Conformally flat Manifolds, Kleinian Groups and Scalar Curvature", Schoen and Yau repeatedly use the term "Newtonian capacity" for a subset of $S^n$.
I know the following definition: …
- 9,853
10
votes
1
answer
467
views
Special Second-Order PDE
Let $\Phi$ be a given smooth function on a neighborhood of zero in $\mathbb{R}^n$ with
$$\Phi(0) = 0, ~~~~D \Phi(0) = 0, ~~~~ D^2\Phi(0) >0,$$
the latter meaning that the Hessian is positive definite. …
- 9,853
2
votes
0
answers
77
views
Well-posedness of a certain linear Cauchy-problem
I am interested in solutions to the linear Cauchy problem
$$\Bigl(\frac{\partial^2}{\partial t^2} + a(t, x)\frac{\partial}{\partial t} + \sum_{j=1}^n b_j(t, x) \frac{\partial}{\partial x_j}\Bigr)u(t, …
- 9,853
8
votes
1
answer
2k
views
Solutions to the eikonal equation
Theorem. Let V be a $C^\infty$ function on a riemannian manifold $M$ and $p$ be a nondegenerate local minimum with $V(p)=0$. Then there is a unique positive function $\varphi \in C^\infty(U)$ such tha …
- 9,853
5
votes
1
answer
232
views
Zeta-Determinant Theorem
Recently, someone asked on MO about lecture notes from Graeme Segal's "Stanford lectures" on TQFT, and the answer was to check here.
When scrolling over the notes, I stumpled of Prop. 2.8.2 in lectu …
- 9,853
2
votes
0
answers
184
views
Heat kernel on manifold with boundary
Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. …
- 9,853
2
votes
0
answers
104
views
Existence of harmonic maps between loops
Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
$$E[ …
- 9,853
2
votes
0
answers
103
views
Inhomogeneous heat kernel estimates
I am looking for existence results on inhomogeneous linear heat equations. Concretely, I have the equation
$$ \frac{\partial}{\partial t} u(t, x) = \Delta_t u(t, x), ~~~~~u(0, x) = u_0(x)$$
where $\De …
- 9,853
7
votes
2
answers
1k
views
First order Elliptic operator
Assume that there exists a first order elliptic operator $D$ acting on functions from $\mathbb{R}^n$ to some vector space $V$. What can we conclude about $V$?
For example, is the dimension of $V$ alw …
- 9,853
3
votes
1
answer
728
views
Decay of Solutions to the Heat equation
Consider the heat equation
$$ (\partial_t + \Delta + V)u = 0$$
on a complete (open) Riemannian manifold with bounded geometry, where $V$ is a smooth and bounded potential.
Consider the semigroup gene …
- 9,853
6
votes
1
answer
215
views
Boundary values of boundary value problems
Let $M$ be a manifold with smooth boundary. We can consider the Dirichlet or the Neumann problem on $M$. Let $(\phi_k)$ be an orthonormal basis of eigenfunctions to the Dirichlet problem and let $(\ps …
- 9,853
5
votes
0
answers
219
views
Parametrices for the wave equation on manifolds with boundary
I am trying to understand parametrices for the solution operator $G_t = \sin(t\sqrt{\Delta})/\sqrt{\Delta}$ to the wave equation
$$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$ …
- 9,853
-1
votes
2
answers
258
views
$L^{n/2}$ norm of scalar curvature
In the Wikipedia article on scalar curvature, it is noticed that the Hölder inequality implies
$$Y(g) \geq - \left(\int_M |R(g)|^{n/2} \mathrm{d}V_g\right)^{n/2}$$
for the Yamabe functional $Y(g)$ and …
- 9,853
11
votes
How to prove Liouville measure is invariant under geodesic flow?
A hands down proof not using the theory of Hamiltonian systems can be done by just proving that the Jacobian determinant of the transformation is zero.
We have
$$ TSM \cong \pi^* TM \oplus VSM,$$
wh …
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