# Tagged Questions

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

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### 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 $f\in L^\infty H^s$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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) ...
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### 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 ...
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### 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 ...
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Consider the following 2nd order nonlinear elliptic equation on $\mathbb{R}^n$: $$-\Delta \varphi_\varepsilon + \sum_i a_i(x, \varepsilon)\partial_i \varphi_\varepsilon + \varphi_\varepsilon = ... 0answers 204 views ### Reference request: The compactness and compact embedding in Besov Space? Let \Omega\subset \mathbb R^N be open bounded with smooth boundary. Let 0<s<1, 1\leq p<\infty, and 1\leq \theta\leq\infty. We denote by B^{s,p,\theta}(\Omega) the Besov space. For ... 0answers 82 views ### What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian? Let (M,g) be a smooth, closed Riemannian manifold and suppose that \phi_1,\dots,\phi_m are eigenfunctions of the Laplacian on M. Write f = \phi_1 + \dots + \phi_m. How big can the set ... 0answers 129 views ### Complex sum of squares of vector fields (hypoelliptic operators) Consider a compact M of dimension n. Consider real smooth vector fields X_0, X_1, X_2,..., X_n on M and consider the differential operator \mathcal{L}_1 = \sum_{i = 1}^n X_i^2 + X_0. Now, by ... 0answers 41 views ### Hess-Schrader-Uhlenbrock inequality for non-symmetric operators Let M be a (compact, let's say) Riemannian manifold, \mathcal{V} a vector bundle over M with covariant derivative \nabla and a fiber metric. Let L = - \mathrm{tr}(\nabla^2) + V with some ... 0answers 93 views ### Find U \in H^1(\Omega \times (0,\infty)) such that \nabla E(u-\bar u)\nabla U \geq 0? (PDE harmonic extension) Let \Omega be a bounded smooth domain. Given u \in H^{\frac 12}(\Omega) with mean value \bar u = 0, let Eu = v \in H^1(\Omega \times (0,\infty)) solve$$\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 ...