# 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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questions

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### Gradient $L^p$ estimates for heat equation

I'm looking for a proof of the gradient estimate associated to the heat equation with Dirichlet boundary conditions, to see if I can express the constant $\color{red}{C}$.
$$\|e^{t\Delta_d}f\|_{W^{1,...

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28 views

### Commonly used metrics to compare two Young measures

Let $\Omega\subset \mathbb{R}^n$ be a bounded open set, $K\subset \mathbb{R}^d$ be a compact set, and $M_1(K)$ be the set of probability measures on $K$. Then a Young measure is defined as a $\textrm{...

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22 views

### A little push in a test function

Let $M$ be an $n\times n$ matrix of real entries, which has at least one positive eigenvalue. Now consider $f$ a function of class $C ^{2}$ in a domain of $\mathbb{R} ^ n$, and $g$ a continuous ...

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92 views

### Inequality between Dirichlet and Neumann eigenvalue for Sturm Liouville problem

Consider the following Sturm Liouville problem on an interval $[a,b]$
$$\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{d} x}\right]+q(x) y=-\lambda w(x) y$$
for given ...

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**1**answer

204 views

### Regularity on the boundary for the heat equation with linear source

This is probably a known problem but I was not able to find exactly what I am looking for.
I have the following linear heat equation with zero-flux boundary conditions:
\begin{equation}
\begin{cases}...

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**1**answer

218 views

### Metric and particular system of PDE

I have a big problem to solve this system:
$\Delta f−hf^2=0$
$p|\nabla f|^2+hf^3=0$
where $h$ and $p$ are constants (with $h \neq 0$ and $p \neq 0$, $p \neq -1$), $f$ is a scalar function defined ...

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67 views

### Reference for the following flow equation

I'm looking for reference on the following partial differential equation. $\partial_tF(t,x) = G(t)((\partial_xF(t,x))^2 + \partial_x^2F(t,x))$, where G(t) is a fixed Schwartz function. If possible, I ...

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54 views

### What's the essential definition of resonance of Schrodinger operator?

Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|...

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213 views

### Proof of the du Bois-Reymond lemma “by approximation” [closed]

I'm curious about the following argument in Morrey ("Multiple integrals in the calculus of variations", Lemma 2.3.1). Suppose $f\in L^1[0,1]$ and $$\int_0^1 fg\,dx=0$$ for every test function $g\in C^\...

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132 views

### Regularity of Neumann eigenfunctions at vertices of polygons

Given a bounded polygonal domain $D$ in $\mathbb{R}^2$, the Neumann eigenfunctions have continuous version on $\overline{D}$. The eigenfunctions also have critical points at vertices of $D$ (I have ...

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122 views

### References for Neumann eigenfunctions

I am looking for references on eigenfunctions with Neumann boundary condition.
In an article, the author wrote in introduction that when a domain is planar polygon, the second eigenfunctions on it ...

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26 views

### Reference request: numerical methods for HJB free boundary problems

Suppose $r: \mathbb{R}^{d+1}\to \mathbb{R}, \ g: \mathbb{R}^d \to \mathbb R,\ b: \mathbb{R}^{d+1}\to \mathbb{R}^d$ and $ \sigma: \mathbb{R}^d \to \mathbb R^d, d \ge 2$, and consider an optimal ...

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59 views

### Logarithmic Sobolev growth of time-space-periodic Schrödinger solutions

Consider the following Schrödinger equation
$$i\partial_t \psi (t,x) + \Delta \psi - V(t,x) \psi = 0 \, ,$$
where $x\in \mathbb{T}^d$ and $V(t,\cdot)$ is real, smooth, and periodic (with a diophantine ...

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204 views

### How can I prove this Weitzenbock formula

Update: It is almost sure that the expression of $\kappa$ in coordinates given in the book, namely
$$\kappa(u_t)=h_{\alpha\beta}\frac{\partial u_t^\alpha}{\partial x^i}\frac{\partial u_t^\beta}{\...

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1k views

### A question on trig series

Assume $\{a_k\}_{k\ge1}$ is a real sequence such that $u(x) = \sum_{k\ge 1}a_k\sin(kx)$ is a smooth function, and for every $x \in [-\pi, \pi]$
$$\left(\sum_{k\ge 1}\frac{a_k}{k}\sin(kx)\right)\left(\...

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**1**answer

121 views

### References for systems of elliptic PDEs

I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity'...

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46 views

### What's the state of the art on stochastic representations of hyperbolic PDE?

I saw this paper: https://arxiv.org/abs/1306.2382
Chatterjee gives a representation of many solutions of the wave equation in terms of Brownian motion. I haven't seen much other than this. Is there ...

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34 views

### Estimates on density for Stokes equation

Consider a bounded, smooth domain $\Omega\subset \mathbb{R}^3$ and in there the Stokes equations
$\nabla p(\rho)-\Delta u=\rho f\\
\operatorname{div}(\rho u)=0\\
u\restriction_{\partial \Omega}=0$
...

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89 views

### Topology on function spaces for pointwise convergence

Suppose I have some collection of maps $T_\lambda: C^\infty(\Omega)\rightarrow C^\infty(\Omega)$ which are linear and parameterized by a parameter $\lambda >0$. (Perhaps more generally take $C^\...

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151 views

### Representing a nonlinear elliptic PDE as an energy minimization problem

I need to solve a PDE in 2D representing a (time-independent) nonlinear diffusion process. The unknown function is $\phi(x,y)$ and its gradients create fluxes $\vec J$ through a nonlinear relation:
$$\...

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**1**answer

102 views

### Continuity of solution of a parabolic PDE w.r.t. system parameters

If we have a system of PDE of the form:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)=D\Delta y+F(t,x,f(x),y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)=0,\ (t,x)\in (...

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70 views

### solutions of a pde smooth with respect to a parameter

I have a generic type question, but I will pose it with a specific example. Suppose $1<p<\frac{N+2}{N-2}$ and $B_1$ is the unit ball in $ R^N$ for $N \ge 3$. Consider the pde
$$-\Delta u(x) ...

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286 views

### Regularity of conformal maps

In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...

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56 views

### Reference for kinetic theory on manifolds

I am looking for a reference for kinetic theory on (Riemannian) manifolds. (In particular for mean-field limits for the Vlasov equation in this setting.)
In 'A review of the mean field limits for ...

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**1**answer

95 views

### Non-trivial examples of regular Lagrangian flow in BV case

What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant?
With concrete I mean that we can compute the flow ...

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**1**answer

122 views

### Finite energy solution for Allen -Cahn equation

I am interested in the Allen-Cahn equation in $ R^N$ and one can consider the related energy functional
$$ E(u):= \frac{1}{2}\int_{R^N}| \nabla u(x)|^2 dx + \frac{1}{4} \int_{R^N} (u^2-1)^2dx.$$ ...

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134 views

### Log-concavity of function

Consider the function
$$f_{n}(x)=e^{-x^2}x^n.$$
My goal is to show that
$$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$
is log-concave.
Let us ...

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168 views

### Non real eigenvalues for elliptic equations

I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...

**1**

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**1**answer

135 views

### Some properties of fractional Dirichlet heat kernel

Let $\Omega\subset \mathbb{R}^n (n\geq 2)$ be a bounded open domain with smooth boundary $\partial\Omega$. Consider the fractional heat equation with Dirichlet boundary condition：
\begin{equation}
\...

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73 views

### Reference (foundamental sol. and grad estimate, etc.): a particular elliptic PDE

In $\mathbb{R}^d$, consider the following equation
$$\Delta u -x\cdot \nabla u = f $$
where $f$ can be $C^\infty$ and decay like $e^{-\frac{c|x|^2}{2}}$.
I would like to know fundamental sol. to this ...

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66 views

### Vlasov Poisson: linear momentum conservation [closed]

The 3-dimensional Vlassov -Poisson equation I am studying at university is
$$ \partial_t f (t,x,v) + v\cdot \nabla_x f (t,x,v) - \nabla_x \phi (t,x) \nabla_v f (t,x,v) =0,$$
where $$\Delta \phi = 4\pi\...

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64 views

### WKB expansion for NLS

We consider the equation (NLS)
\begin{eqnarray}\label{gnls}
i \epsilon\partial_t u^{\epsilon} + \frac{\epsilon^2}{2}\Delta_{\eta}u^{\epsilon} = \epsilon |u^{\epsilon}|^{2}u^{\epsilon}, \quad x \in \...

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115 views

### Courant nodal domain theorem for fractional Laplacian

Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\...

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299 views

### Angle of analyticity of semigroup

Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?
For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. ...

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146 views

### Help to understand a limit $\varepsilon\rightarrow 0$ computation on a fluid mechanic paper

In Córdoba and Gancedo - Contour dynamics of incompressible 3-D fluids in a porous medium with different densities (page 4) I read that if
$$ v (x_1,x_2,x_3,t)=-\frac{\rho_2-\rho_1}{4\pi} \...

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84 views

### Explanation for the energy method used here

I am reading a paper where the authors prove
$$ \frac{d}{dt} \Vert f \Vert_{H^4} (t)\leq C\Vert f \Vert^5_{H^4}(t) $$
Where $f=f(x,t)$ and $H^4$ stands for the usual Sobolev space. Using Gronwall's ...

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123 views

### Existence of weak solutions of a parabolic PDE

Assume that $\Omega\subset\mathbb{R}^n(n\geq3)$ is a bounded open set with smooth boundary, $\varphi\in H^1_0(\Omega)$, $F(t)$ is differentiable in $\mathbb{R}$ and $F'$ is bounded. Given a PDE
$$
\...

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**1**answer

95 views

### finding Morse index for the following functional

not sure if this meets the standards here in this forum. I was dealing with the following functional $I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$ for $p \geq ...

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64 views

### Derivative estimates for Laplace eigenfunctions on Riemannian manifolds

In $\mathbb{R}^2$ (or more generally in $\mathbb{R}^n$), if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies $\Delta f+ f=0$, then we can write the following regularity/derivative estimates for $f$:...

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38 views

### Quantitative estimate on continuity with respect to parameter of ODE

Let $V:\mathbb{R} \rightarrow \mathbb{R}$ be a bounded, $\text{Holder}$ continuousfunction with degree $\delta\in (0,1]$ such that $\inf_{\mathbb{R}} V = 0$ but $V(x) > 0$ for all $x\in \mathbb{R}$....

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267 views

### An inequality for abstract Cauchy problem

Consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates an analytic $C_0$-semigroup on a Banach space $X$. How we can prove an inequality of ...

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72 views

### Regularity and normal trace of “Hdiv” measures

In order to fix the ideas let me consider an open, smooth, bounded domain $\Omega\subset \mathbb R^d$.
I am wondering what can be said about a vector-valued measure $v\in \mathcal M^d(\Omega)$ with ...

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43 views

### Dirichlet-to-Neumann map's estimate for mixed boundary value problems

The study on DtN or NtD maps for Dirichlet or Neumann boundary value problems (or PML for Helmholtz exterior problems) is pretty mature and there are tons of papers on this topic, yet I couldn't find ...

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58 views

### Prove that the solution belong to ${L^2}\left( {0,T;{L^2}\left( \Omega \right)} \right)$

Let $k \in {L^\infty }\left( {0,T} \right)$ and we assume that
$$\phi :t \mapsto u\left( t \right) + \int_0^t {k\left( s \right)u\left( s \right)ds} \in {L^\infty }\left( {0,T;{L^2}\left( \Omega \...

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215 views

### Inequality for initial data

I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,...

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157 views

### A basic question about the Spectral Theorem

Let $\Omega$ be a bounded open region in $\mathbb{R}^n$ and $\phi_i $ be the eigenfunctions of $-\Delta$ with Dirichlet boundary condition, i.e.
$$-\Delta \phi_i=\lambda_i \phi_i, \ \ \phi_i|_{\...

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196 views

### Does small Laplacian imply small oscillation?

Let us consider a compact Riemann manifold $(M,g)$ and a function $u$ on $(M,g)$. $u$ also satisfies that $|\triangle_g u|\leq \varepsilon$ where $\varepsilon$ is a small constant. Can we obtain that $...

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115 views

### Reference request: Schauder estimate in the space variable for parabolic equations

Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.
Let $$u\...

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106 views

### Relationship between three different definitions of solutions for ODE with irregular coefficient

What is the difference between the notions of
Regular Lagrangian flow
Filippov solution
Caratheodory solution
of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...

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63 views

### Behavior of the principal eigenfunction of fractional Laplacian

How does the first eigenfunction $\phi_{1}$ behave near the boundary of $\Omega$ where $$(-\Delta)^s\phi_{1}=\lambda_{1}\phi_{1},\text{ in } \Omega; \phi_{1} =0 \text{ in } \Omega^c$$ in a n-...