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Questions tagged [ap.analysis-of-pdes]

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

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55 views

Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
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0answers
43 views

Energy estimates involving test functions for weak solutions of PDE problems

I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page: "A weak (in the PDE sense) solution satisfying the energy inequality ...
2
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1answer
159 views

Underdetermined system of linear PDEs

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$. I ...
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0answers
58 views

Conditions for the embeddig of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$. If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
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0answers
27 views

Local well-posedness of the quadratic NLS on the 1D torus

What is the proof of the local well-posedness of the quadratic nonlinear Schrödinger equation $\mathrm{i} \,\partial_t u + \Delta u \pm \left|u\right| u = 0$ on the 1D torus in $H^s$ for $s > 1$ (a ...
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1answer
40 views

Exact solution of two coupled transport equations

I want to solve the following system $$\eqalign{ & y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr & z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr & y(0,x) = y_0,\,\,z(...
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1answer
61 views

Reference request: Moving source to initial condition and vice versa in PDE problem

I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws. Inhomogeneous Cauchy problem: $$(1) \hspace{1cm} ...
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120 views

Level Sets of Harmonic Maps

Can anybody point me in the direction of some references in which the level sets of harmonic maps between Riemannian manifolds are studied. (Sorry I am unfamiliar with the area and would like some ...
10
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2answers
292 views

Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?

Good morning, I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
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3answers
298 views

fundamental solution of an elliptic PDE in divergence form with non-symmetric matrix

I am looking for the fundamental solution of the following PDE $$\partial_i (a^{ij}\partial_j u)=f$$ where $a^{ij}(x)$ is a non-symmetric matrix with possibly non-constant coefficients. I could ...
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1answer
89 views

Interpretation of the Schouten bracket as an integrability condition

The Schouten bracket is an extension of the Lie bracket to multivector fields. Given a multivector field $\Lambda$ the vanishing of the Schouten bracket $[\Lambda,\Lambda]=0$ is referred to as a sort ...
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0answers
62 views

$|\nabla \omega|^2= \alpha \omega^n+ \beta$ - References

I'm interested to know if anyone can give me some information (name, references, etc.) about this PDE $$|\nabla \omega|^2= \alpha \omega^n+ \beta,$$ where $\omega$ is a scalar function of two or ...
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0answers
53 views

Strichartz estimates for fractional Schrodinger equations

A pair $(q,r)$ is $\alpha-$fractional admissible if $q\geq 2, r\geq 2$ and $$\frac{\alpha}{q} = d \left( \frac{1}{2} - \frac{1}{r} \right).$$ We take fractional Schrodinger propagator $U(t)=e^{it (...
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0answers
71 views

Linearization around a traveling wave

In [1], Remark 2.1., the authors say the following: ".. in moving coordinates, the linearization of (37) around a traveling wave profile is given by \begin{equation} v_t = x \partial^2_x v + \frac{2}{...
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1answer
119 views

“Overdetermined” Poisson equation

Consider the PDE $-\Delta u = f$ on a bounded domain $\Omega \subset \mathbb{R}^n$, where $f \in C^\infty(\bar{\Omega})$. I wish to consider both the boundary conditions $u = 0$ and $\frac{\partial u}{...
4
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1answer
225 views

Existence of solution for the PDE $p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q)$

Consider the following PDE: \begin{equation} p \frac{\partial f(p, q)}{\partial p}-q \frac{\partial f(p, q)}{\partial q}=g(p, q),\tag{$\star$} \end{equation} where $g$ is a flat function at the point (...
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1answer
55 views

Is $H_0^1$ a redundant assumption in the 2D Agmon inequality?

The Wikipedia article on Agomon's inequality states the following: Let $u\in H^2(\Omega)\cap H^1_0(\Omega)$ where $\Omega\subset\mathbb{R}^2$. Then Agmon's inequality in 2D states that there ...
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0answers
53 views

Vector fields whose divergence is Gaussian

Let f be the pdf of a $n$ dimensional $N(0,C)$ distribution i.e up to a multiplicative constant, $f(x) = \exp(-\frac{1}{2} x'C^{-1}x)$. Which vector fields $F$ are so that ${\rm div} (F)= f$ ?
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59 views

A Generalized Bernstein's Problem

Suppose $\Sigma\subset \mathbb{R}^n$ be an area-minimizing hypersurfaces, the standard results in GMT tell that when $3\leq n\leq 7$, $\Sigma$ is a hyperplane; while for $n\geq 8$, there are examples ...
3
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2answers
201 views

Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability

We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth ...
2
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1answer
92 views

Optimal control theory of PDEs

This is a somewhat openly phrased question because I am not quite sure what has been done in that direction. Imagine one has two evolution equations $$\partial_t u = p(x,\partial_x,f)u$$ $$\...
2
votes
1answer
130 views

Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\ &\Phi(0,x) = x, & x \in \...
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0answers
54 views

Smooth dependence of convex functions on Monge-Ampère measure

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as ...
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votes
2answers
76 views

Reference request for fractional Poincare inequality

Suppose we consider in $\mathbb R^n$, then how to show $\Vert f \Vert_{L^{p}} \leq C\Vert \nabla^{s}f \Vert_{L^{q}}^{\alpha}$, where $s>0$ is noninteger and $\alpha \in (0,1)$?
3
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1answer
68 views

Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$

Is there any Bernstein type theorems for CMC hypersurfaces in $\mathbb{R}^{n+1}$ in the literature? More precisely I would like to know if there is an answer to the following QUESTION: Let $f : \...
0
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1answer
85 views

Separation of variables for PDE

Consider the PDE $$ \partial_t f(t,x) = \Phi(x) f(t,x)+ \Psi(x)f(t,x-a)$$ $$f(0,x)=1$$ $$f(t,0)=1$$ where $a$ is a constant and $\Phi$ and $\Psi$ are some differentiable functions. To solve this I ...
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1answer
216 views
+50

Meaning of Alberti rank-one theorem

Heuristically what does Alberti's rank-one theorem imply about the structure of a $\mathrm{BV}$ vector field $\boldsymbol{b}$? Is it rigorously fair to say that the level lines of $\boldsymbol{b}$ ...
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1answer
104 views

Liouville theorem for fractional Laplacian

Is there any Liouville type theorem for the half space problem \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s v &= 0 &&\text{in } \mathbb R^N_+\\ v & =0 &&\text{...
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0answers
43 views

linear fractional laplacian problem

If $U(x)$ is a classical solution of \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s U+ U &=m(r)U &&\text{in } B \\ U &= 0 &&\text{in } \mathbb R^N \...
1
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1answer
74 views

Extended Global approximation theorem

In Evans, $\textbf{Theorem} $ (Global Approximation Theorem) Assume $U$ is bounded, and $\partial U$ is $C^1$. Suppose as well that $u \in W^{k,p}(U)$ for some $1\leq p < \infty$. Then, there ...
3
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1answer
211 views

Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator

For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$. It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
4
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1answer
132 views

Interpolation of some Lebesgue spaces

When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p ...
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1answer
97 views

Exponential decay of resolvent kernel

For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
3
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1answer
46 views

Difference between linear and parabolic velocity profiles in Stokes flows of two fluids

For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a). ...
4
votes
1answer
99 views

Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...
1
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1answer
173 views

Boundary regularity for the Monge-Ampère equation $\det D^2u=1$

Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying $$ \det D^2u=1,\quad u|_{\partial\Delta}=0. $$ Classical ...
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0answers
58 views

Unique continuation from the boundary for inhomogeneous elliptic pde

Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...
3
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0answers
142 views

Stationary Navier-Stokes solutions

Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
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0answers
70 views

asymptotic behaviour of principal eigenfunctions and Large Deviations

Dear Math Overflowers, I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
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0answers
47 views

On the non-existence of weak solutions to nonlinear evolutionary PDEs without time derivative

Imagine we have a nonlinear PDE, e.g. some velocity model $$\begin{aligned} -\Delta v + (v \cdot \nabla) v + \nabla p = f \\ \text{div} \, v = 0 \end{aligned}$$ which has a weak solution for some ...
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0answers
53 views

The space $H_\Delta(\Omega)$

I'm looking for good references for the study of the following space $$H_\Delta(\Omega)=\{u\in L^2(\Omega) : \Delta u \in L^2(\Omega)\},$$ especially the trace theorems with proofs, where $\Omega \...
1
vote
1answer
83 views

Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: \begin{equation} C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T). \end{equation} what condition should be put ...
1
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1answer
109 views

Existence and regularity for fractional Poisson-type equation

According to Theorem 2.7 in the paper https://arxiv.org/pdf/1704.07560.pdf, we have the following classical results. Let $s \in (0,1)$ and $1<p<\infty$. Then for any $F \in L^p(\mathbb{R}^n)$ ...
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2answers
434 views

Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
14
votes
3answers
311 views

What is known about sufficient conditions for the rigidity of a convex surface?

A convex surface is a connected open subset of the boundary of a convex body in $\mathbb{R}^3$. An "infinitesimal bending" of a convex surface $S$ is a deformation of $S$ given by a velocity field $v:...
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2answers
382 views

Klein Gordon equation - references

The Klein Gordon equation of the form: $\Delta u+ \lambda u^p=0$ is been studied for $p = 2$? (i.e.$\Delta u+ \lambda u^2=0$) If yes are there references?
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0answers
74 views

geodesic balls in the conformal change

Consider a compact smooth Riemannian manifold $(\mathcal{M}, g)$. We consider a conformal change of the metric. Let $\tilde{g}=\phi g$, where $\phi$ is smooth and positive. Moreover, it satisfies $...
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0answers
38 views

What are tools we need to prove scattering result for NLS when propagator is unitary?

Consider nonlinear Schrödinger equation (NLS) $$i\partial_t u + \Delta u =F(u) , u(x,0)=u_0$$ where $F$ is some nonlinearity. Assume that there exists Banach space $X$ so that $\|e^{it\Delta} f\|_{X}...
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0answers
53 views

Global interior estimate complex Monge-Ampere equation

Suppose u is a smooth strictly plurisubharmonic function satisfying $(i\partial \bar{\partial} u)^n =f dV_{Euc}$ on the unit ball centered at the origin, where $f>0$ is a smooth function. Let $C$ ...
3
votes
1answer
337 views

Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...