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Results tagged with ap.analysis-of-pdes
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user 15856
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
19
votes
4
answers
6k
views
unique continuation principle
I recently encountered a paper by Protter ("Unique continuation of elliptic equations") that starts out by saying "any solution of an elliptic equation that is defined on a domain $D$ must vanish on a …
- 1,701
7
votes
Does elliptic regularity guarantee analytic solutions?
Also: there is a classical result due to Charles Morrey, "Analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations", that says that if $F(x,u,\nabla u,\na …
- 1,701
5
votes
PDE on manifolds
The tour-de-force of elliptic pde on manifolds is the Yamabe problem. There the pde is a second-order, elliptic, and semilinear with a Sobolev critical exponent. The analysis can become incredibly dif …
- 1,701
4
votes
1
answer
627
views
Higher order Sobolev inequality
Let $(M,g)$ be a closed, Riemannian manifold of dimension $n>4$. Let $K$ be the best constant for the Sobolev inequality
$||u||^2_p \leq K \int_{{\Bbb{R}}^n} (\Delta u)^2 dx,$
where $p=\frac{2n}{n-4 …
- 1,701
4
votes
1
answer
283
views
non-negativity to positivity
Let $(M,g)$ be a closed Riemannian manifold of dimension $n>7$. In this setting I have been able to prove that the Green's function of a positive Paneitz-Branson operator is non-negative. Furthermore, …
- 1,701
3
votes
0
answers
488
views
kernel of the conformal Laplacian
Let $M$ be a smooth, closed manifold of dimension $n>2$. Let $L_g$ be the conformal Laplacian of the metric $g$. That is, $L_g=-\Delta_g + \frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature o …
- 1,701
3
votes
1
answer
225
views
meromorphic family of pseudo-differential operators
Let $(M,g)$ be a closed, Riemannian manifold. Let $S(z)$ be a holomorphic family of pseudo-differential operators, with $z \in \Bbb{C}$. Let $u$ be a smooth function. Does it follow that $\lim_{y \rig …
- 1,701
3
votes
1
answer
308
views
ellipticity and invertible differential operators
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}_+$. Suppose that the differential equation $Pu=f$ has a unique $H^r(M …
- 1,701
2
votes
1
answer
186
views
ellipticity independent of metric?
I am new to the theory of pseudo-differential operators on compact manifolds, but I need to use a result related to this theory in a proof I'm working on. The problem is as follows: Let $(M,g)$ be a c …
- 1,701
2
votes
2
answers
506
views
Positivity of Second-Order Elliptic Differential Operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $\Delta = -div\nabla$ be the Laplace-Beltrami operator. Let $h$ be a smooth function on $M$. Is there a condition on $h$ weaker than non-negati …
- 1,701
1
vote
Positivity of Second-Order Elliptic Differential Operators
I believe the answer is no if $n>2$. Let $g$ be a metric with a negative Yamabe constant. There will be a metric $h$ in the conformal class of $g$ such that $\int_M R_h dv_g> 0$. Let $L_h$ be the conf …
- 1,701
1
vote
Higher order Sobolev inequality
The desired inequality is correct. It is easily generalized from Aubin's proof (which is given in Lee&Parker's "Yamabe Problem") of the classical Sobolev inequality for Riemannian manifolds. One simpl …
- 1,701
1
vote
1
answer
264
views
radial limits of subharmonic functions
Let $u$ be a non-negative subharmonic function on the unit ball in $\Bbb{R}^n$. Does it follow that there exists a radial limit (including limits of infinity or negative infinity) along almost every l …
- 1,701
0
votes
0
answers
96
views
partial maximum principle for elliptic differential operators
Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest eigenv …
- 1,701