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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2answers
270 views

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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0answers
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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1answer
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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1answer
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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2answers
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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1answer
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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1answer
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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1answer
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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1answer
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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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1answer
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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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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2answers
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: $$\...
2
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1answer
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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0answers
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) ...
14
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1answer
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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1answer
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 ...
2
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1answer
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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1answer
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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0answers
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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1answer
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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0answers
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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1answer
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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1answer
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 \...
2
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1answer
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, $\...
4
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2answers
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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0answers
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 ...
2
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1answer
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 $$ \...
0
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1answer
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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0answers
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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0answers
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 $x\in \mathbb{R}$....
2
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1answer
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 ...
2
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1answer
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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0answers
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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0answers
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 \...
3
votes
1answer
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,...
2
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0answers
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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0answers
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 $...
2
votes
1answer
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\...
3
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0answers
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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0answers
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-...

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