# Tagged Questions

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

207 views

### Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...
150 views

### Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
272 views

### Recovering full regularity by energy method in the heat equation

Consider the heat equation $$u_t = u_{xx} + f,$$ on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
189 views

270 views

### Controlling the Second Eigenvalue of a Schrödinger Operator

Consider a bounded domain $\Omega$ (with smooth boundary) in some Riemannian $n$-manifold $M^n$. Let $L$ be the operator $$L=\Delta+V$$ where $\Delta$ is the Laplace-beltrami operator on $M$ (so is ...
166 views

### Generalizations of group algebras for arbitrary manifolds?

In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take ...
16 views

### Calculation of characteristic strip, solutions of a geometric PDE

Say I have a family of spheres of radius $1$ with centers in the $xy$-plane$$u = G(x, y, \lambda, \mu) = \sqrt{1 - (x - \lambda)^2 - (y - \mu)^2}.$$I have a few questions. What are all the ...
37 views

### Strengthening of the local smoothing estimates for the free Laplacian

The classical local-smoothing estimates for the free Laplacian asserts that: $$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$ where ...
35 views

### Interior regularity for elliptic operators with non smooth coefficients

I need a pretty standard interior regularity result for a second order elliptic operator of the form $$-\nabla^b \cdot (A(x) \nabla^b v)+c v=f, \qquad \nabla^b=\nabla+ib(x)$$ where $A(x)$ is a ...
61 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. ...
41 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) ...
42 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 ...
70 views

111 views

78 views

73 views

### Parametrices for the wave equation on manifolds with boundary

I am trying to understand parametrices for the solution operator $G_t = \sin(t\Delta)/\Delta$ to the wave equation $$(\partial_{tt} + \Delta)u_t=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$ on a ...
186 views

Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ... 0answers 207 views ### Reference request: The compactness and compact embedding in Besov Space? Let \Omega\subset \mathbb R^N be open bounded with smooth boundary. Let 0<s<1, 1\leq p<\infty, and 1\leq \theta\leq\infty. We denote by B^{s,p,\theta}(\Omega) the Besov space. For ... 0answers 82 views ### What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian? Let (M,g) be a smooth, closed Riemannian manifold and suppose that \phi_1,\dots,\phi_m are eigenfunctions of the Laplacian on M. Write f = \phi_1 + \dots + \phi_m. How big can the set ... 0answers 129 views ### Complex sum of squares of vector fields (hypoelliptic operators) Consider a compact M of dimension n. Consider real smooth vector fields X_0, X_1, X_2,..., X_n on M and consider the differential operator \mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0. Now, by ... 0answers 41 views ### Hess-Schrader-Uhlenbrock inequality for non-symmetric operators Let M be a (compact, let's say) Riemannian manifold, \mathcal{V} a vector bundle over M with covariant derivative \nabla and a fiber metric. Let L = - \mathrm{tr}(\nabla^2) + V with some ... 0answers 93 views ### Find U \in H^1(\Omega \times (0,\infty)) such that \nabla E(u-\bar u)\nabla U \geq 0? (PDE harmonic extension) Let \Omega be a bounded smooth domain. Given u \in H^{\frac 12}(\Omega) with mean value \bar u = 0, let Eu = v \in H^1(\Omega \times (0,\infty)) solve$$\int_0^\infty\int_\Omega \nabla v\nabla ...
For an compact manifold $M$ without boundary, we consider the eigenfunctions $(f_1,f_2,\ldots)$ of some elliptic operator (e.g $\Delta$) with eigenvalue $\lambda_{1},\lambda_{2},\ldots$. To define ...