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

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3
votes
0answers
367 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D ...
3
votes
0answers
237 views

Best Poincare constants on the surface of a ball

I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first ...
3
votes
0answers
174 views

Numerical solution

Last time, I asked this question but after discussing with some friends, I have given up finding closed-form solutions. Now I have a simpler question.Let $g_i: i=1,2$ be $C^2 =C^{2}(-\infty,\infty)$ ...
3
votes
0answers
351 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 ...
3
votes
0answers
359 views

PDES - from Vector fields whose inner product with their vector Laplacian equals norm of the vector field

Let $g(x_{1},........,x_{n}) = \sum_{i=1}^{n}g_{i}(x_{1},\cdots,x_{n})e_{i}$ be a function in $\mathbb{C}^n$ ($e_{i}$ are the standard bases). Let $\nabla^{2}$ be the vector Laplacian. Let ...
3
votes
0answers
339 views

Existence of solutions to elliptic PDE in undbounded domain

Specifically, I have $Lu=f$, where $L$ is a linear elliptic pseudodifferential operator, on an unbounded domain of the form $\Omega\times [0,\infty)$, $\Omega$ has Lipschitz boundary, $u$ is 0 on all ...
3
votes
0answers
335 views

problem with non linear pde

I have the following pde which i cannot solve. any suggestions, tips on how to approach a solution? $$\left(1-x^3 \frac{\partial y}{\partial x} \partial_{y}\right) f(y(x))-1/4 \left(1-\frac{1}{x^3 ...
3
votes
0answers
243 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 ...
3
votes
0answers
163 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 ...
2
votes
0answers
76 views

Inverse Function Theorem on Zygmund Spaces, is the inverse in the same Zygmund Space?

Preliminary Definition We define the Zygmund spaces $C^r_{*}$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed to have values in $\mathbb{R}^m$, via working with ...
2
votes
0answers
92 views

Newtonian potential for continuous $f$

Suppose $f(x)$ is a continuous compactly supported function in $ R^N$ where $N \ge 3$. Consider the Newtonian potential of $f$ (at least I think this is what it is called) $$ v(x)=\int_{R^N} ...
2
votes
0answers
50 views

Approximating a superharmonic function, by smooth superharmonic functions

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$, $u\ge 0$ and $-\Delta u\ge 0$ in the sense of distributions ($u$ is superharmonic). The standard ...
2
votes
0answers
77 views

Variant form of the gronwall inequality

I know the following statement for gronwall inequality: Given $f$ non negative and absolutely continuous on $[0;T]$ and $\phi \in L^1(0;T)$if we have, $f' \leq \phi f$ and $f(0)=0$ then $f=0$ Now is ...
2
votes
0answers
54 views

Holder continuity of Poisson equation with divergence free drift

I am interested in the following PDE. Suppose $u_m$ is a smooth solution of a elliptic equation of the form $$ -\Delta u_m(x) + a_m(x) \cdot \nabla u_m(x) = f_m(x) \qquad B_1 $$ with $ u_m=0 $ on ...
2
votes
0answers
106 views

Variational inequality on Manifold

Let $(M,g)$ be a Riemannian manifold. Consider $A : W^{1,r}(M,\mathbb{R}) \rightarrow W^{-1,r'}(M,\mathbb{R}), k \mapsto Ak$, where $Ak$ is defined by $(Ak)(\varphi) = \int_{M}g(\nabla k, \nabla ...
2
votes
0answers
139 views

Showing $\langle \frac{\partial b(v)}{\partial t}, v \rangle_{H^{-1}(\Omega), H^1(\Omega)} = \frac{d}{dt}\int_{\Omega}\Psi^*(b(v))$

Let $b$ be continuous and increasing with $b(0) = 0$. Define $\Psi(t) = \int_0^t b(s)\;ds$ and $\Psi^*(s) = \sup_{r \in \mathbb{R}} (sr-\Psi(r))$. (Note $\Psi^*(b(s)) + \Psi(s) = sb(s)$). Let $v ...
2
votes
0answers
82 views

Minimizing some $H^{-1}$ functional over (a subset of) probability densities in $R^d$

Let me consider the following subset of probability measures in $R^d$ $$ \mathcal{K}_M=\left\{0\leq u(x)\in L^1(R^d):\quad \int u(x)dx =1,\,\int|x|^2u(x)dx\leq M,\,\int u(x)|\log u(x)|dx\leq M\right\} ...
2
votes
0answers
129 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
2
votes
0answers
145 views

$C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...
2
votes
0answers
57 views

Integrability of $D^2u$ for $\infty$-harmonic function $u$?

Consider infinity harmonic functions; that is, functions satisfying $\Delta_\infty u = 0$ with $$\Delta_\infty u = \langle Du, D^2 u \, Du \rangle = \sum_{i,j} \frac{\partial^2 u}{\partial x_i \, ...
2
votes
0answers
74 views

Weighted energy estimate for the heat equation of higher order

The question is originally related to Hardy's uncertainty principle, convexity and Schrodinger evolutions. In this work the authors deduce a convex property of Schrodinger equation by doing it first ...
2
votes
0answers
61 views

Changing the test function space in a weak formulation of parabolic PDE

Suppose we are interested in the existence of a $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that $$(u(T),\varphi(t))_H -\int_0^T \langle \varphi'(t), u(t) \rangle_{V^*,V} + \int_0^T ...
2
votes
0answers
152 views

Scattering for rapidly decaying solutions of NLS

Cazenave and Weissler proved in their paper "Rapidly Decaying Solutions of the Nonlinear Schrödinger Equation" the following property. Given the problem \begin{equation} \left\{ \begin{array}{rl} ...
2
votes
0answers
87 views

Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?

I would be interested to know if a certain set of periodic solutions for the two-dimensional Navier-Stokes equations is closed generically. Many similar (yet not identical) set-ups can be found in the ...
2
votes
0answers
112 views

geometric irregularities in pde's

The following question is intended for a person more acquainted with the works of Yves Laurent. see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French) ...
2
votes
0answers
107 views

How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?

I am reading this paper. Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$ On page 5 of ...
2
votes
0answers
131 views

Estimates on gradients of diffusion semigroups

Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form $$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
2
votes
0answers
93 views

Almost a Green formula

Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
2
votes
0answers
70 views

Deduce global estimate from scaling-invariant local estimate

Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
2
votes
0answers
87 views

slightly subcritical elliptic pde; the linearized equations

Let $ p_m \nearrow \frac{N+2}{N-2}$ and consider the family of elliptic problems $$-\Delta u_m(x)=u_m(x)^{p_m} \quad B \qquad \quad u_m =0 \quad \partial B,$$ where $B$ is the unit ball ...
2
votes
0answers
78 views

Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain

Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
2
votes
0answers
68 views

What's a good resource for Hormander symbols of type (1/2, 1/2)?

I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...
2
votes
0answers
135 views

W^2,p regularity for solutions of elliptic equations

I'm stucked in the following (maybe classical) issue concerning the $W^{2,p}$ regularity of solutions of a second order elliptic equations like $Lu=f$ in a bounded domain (say a ball) $\Omega$. I have ...
2
votes
0answers
97 views

Why pseudoconvexity is important in Partial differential equation theory?

I am a new researcher in mathematics and I work on convexity. Are convexity and pseudoconvexity related topics and in which respect to PDE theory ? One of the important results in PDE theory is the ...
2
votes
0answers
153 views

A contradiction to do with continuity? (involves chain rule)

Suppose for each $t$, $S(t) \subset \mathbb{R}^n$ is a domain (hypersurface). We have a diffeomorphism $D^0_t:S(0) \to S(t)$ for each $t$ such that it solves the ODE $$\frac{d}{dt}D^0_t(\cdot) = ...
2
votes
0answers
90 views

Elliptic equations with divergence-free drift terms

Given $\ \mathbf{u}\cdot \nabla c=\Delta c-a_{1}c+\rho \text{ on }\Omega $ with a $\Omega \subset %TCIMACRO{\U{211d} } %BeginExpansion \mathbb{R} %EndExpansion ^{2}$ bounded, $div$$(\mathbf{u})=0$, ...
2
votes
0answers
80 views

Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in ...
2
votes
0answers
119 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
2
votes
0answers
64 views

Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
2
votes
0answers
112 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
2
votes
0answers
111 views

linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator: $$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$ where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
2
votes
0answers
92 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
2
votes
0answers
312 views

Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real ...
2
votes
0answers
106 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 ...
2
votes
0answers
101 views

Existence of solutions to a reaction-diffusion problem.

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 ...
2
votes
0answers
162 views

Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation $$u_t-iu u_x-iu_{H,xx}=0$$ where $u_H(x,t)$ denotes the ...
2
votes
0answers
82 views

P-laplacian equation

Hi guys, in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ? Thank you!
2
votes
0answers
135 views

Extension divergence-free, curl-converging vector field

Hi. Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
2
votes
0answers
196 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
2
votes
0answers
99 views

Best constant of Gagliardo-Nirenberg inequality in exterier domain

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla ...