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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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, …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar
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 …
Viktor Bundle's user avatar