# Tagged Questions

Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

1answer
277 views

### Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose $$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$ where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
0answers
75 views

1answer
189 views

### Critical case of Sobolev Embedding

I got stuck in the following lemma: Lemma: Let $B$ be the unit ball in the 4 dimensional Euclidean space. Suppose that $u\in W^{2,2}(B)$, then $e^{u}\in L^{q}$ for any $q>1$. As we know this is ...
1answer
104 views

### Is the Lopatinski-Shapiro condition invariant under diffeomorphism?

If a PDE (eg. the heat equation with Robin BCs, or the elliptic version) on a bounded smooth domain $U$ satisfies the Lopatinski-Shapiro condition (for a definition see eg. Wloka), and if $T:U \to W$ ...
0answers
22 views

2answers
116 views

### Symplectic formulation of compressible Euler equation

It has been widely known that the compressible Euler equation can be cast into the Hamiltonian form. For example, in the book "Dubrovin B A, Fomenko A T, Novikov S P. Modern geometry—methods and ...
0answers
61 views

0answers
59 views

### On the principal eigenvector of an elliptic operator

Suppose I have an open domain $U \subset \mathbb{R}^n$ and an elliptic operator $L$ acting on all square-integrable $C^2$ functions $\rho:U\to \mathbb{R}$ which converge to zero at $\partial U$: \...
2answers
471 views

### Unexpected regularity of the distance from a $C^2$ submanifold

Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...
0answers
52 views

### Is $\Delta u+f\in (H^1(\Omega))^*$ with $u\in H^1_0(\Omega)$ and $f\in L^2(\Omega)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $\partial \Omega$ being $C^2$. Suppose $u\in H^1_0(\Omega)$, $f\in L^2(\Omega)$ and $\nu>0$. It is said in the Navier-Stokes Equations ...
0answers
75 views

1answer
138 views

### Bounded solutions for Schrödinger equation at the edge of the essential spectrum

Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
2answers
68 views

0answers
68 views

### What is known for harmonic map flow in dimension > 2?

I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. ...
0answers
44 views

### Can we say translation/dilation of the $L^p-$multiplier is again a $L^{p}-$multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ such that $\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}$ (where $C$ is some constant, $f\in L^{p}$). (That is, $m$ is an $L^{p}-$ multiplier) ...
0answers
44 views

### 1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm \$\|f\|_{L^2(...