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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0answers
16 views

Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation: $(\omega_0 +i \partial \bar \partial \varphi)^...
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0answers
39 views

Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries

Could anyone please suggest related papers or article about the topic related to my one question below? Reduce PDE to ODE by dilation symmetry I also cite a paper in the link above. We know that ...
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110 views

Minimal assumption for an elliptic equation

On the disc $\mathbb{D}$ on the disc with a metric $g=e^{2\lambda} \vert dz \vert^2$(let assume $\lambda$ is smooth on $\overline{\mathbb{D}}$) and I consider either $$\newcommand{\Div}{\operatorname{...
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55 views

A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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0answers
37 views

Well-posedness in modified H2 space

Can I modify the $H^2$ space such that: $$\tilde{H}^2 := \left[ u(\Omega): ||u||^2_{L_2(\Omega)} + ||\nabla u||^2_{L_2(\Omega)} + ||\Delta u||_{L_2(\Omega)}^2 < \infty \right]$$ and then use the ...
3
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0answers
102 views

Reference on noncommutative PDE

I would like to ask if there is reference on semi-linear parabolic PDE (or more generally any kinds of PDE) with non-commutative unknown variable. For example, assume $u$ is a matrix-valued function (...
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0answers
50 views

Trace inequality normal derivative

For $v(\Omega) \in W^1_2$ and $\Omega \in C^1$ we have a trace inequality: $$\Vert v \Vert _{L_2(\partial \Omega)} \leq C_\Omega \Vert v \Vert _{W_2^1},$$ which can be found in many places in the ...
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104 views

An inequality for the Hessian of eigenfunctions of the Laplacian on compact manifolds

Let $(M,g)$ be a compact Riemannian manifold, and let $\Delta$ be the Laplace-Beltrami operator. Let $\lambda_1 >0$ be the first positive eigenvalue. That is, there exists a non-trivial function ...
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53 views

Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator

I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why ...
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1answer
36 views

Computing the fractional Laplacian of power function

Is it possible to compute explicitly the fractional Laplacian (in $\mathbb R^n$) of a power function $|x|^p$?
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1answer
117 views

Reduce PDE to ODE by dilation symmetry

I am reading Rodrigues, Henrion, and Cantwell - Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems, p.753. Consider the following PDE: ...
2
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1answer
74 views

Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
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3answers
176 views

Uniqueness of solution of the wave equation

Consider the wave equation $$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$ with initial conditions $$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$ Does ...
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1answer
124 views

Is there any nontrivial characterization of weakly differentiable functions?

When $f\in L_\text{loc}^1$, it's distributional derivative can be defined as $D_{f'}\in\mathfrak{D}'$, such that $D_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$...
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33 views

A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
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1answer
60 views

fractional Laplacian estimates

Suppose $(-\Delta)^s u=f \geq 0$ in a ball $B_2$ and $u=0$ in $ R^N \setminus B_2.$ Also suppose $u$ is $C^{s}$ non-negative and $(-\Delta)^s u=0$ in $B_2 \setminus B_1$ and $u\leq a$ on $\partial B_1$...
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59 views

Inhomogeneous wave equation - a reference

Consider the inhomogeneous wave equation $$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$ where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
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39 views

Understanding weak formulation of a linear elliptic pde [closed]

hey I'm working on weak formulation on an elliptic pde and I have these questions: Is there any difference between: $\Delta u=f$ in $\Omega$ and $\Delta u=f$ almost everywhere in $\Omega$ when $\...
3
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1answer
94 views

Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
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1answer
59 views

Poincaré-type Inequality

In Lieb's paper "On the lowest eigenvalue of the Laplacian for the intersection of two domains" one finds the following remark: Let $u\in L_{loc}^p(\mathbb{R}^N)$ with $\nabla u \in L^{p}$ and $\|\...
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1answer
115 views

Decay of Fourier coefficients of real analytic functions

I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, ...
2
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1answer
83 views

What is the example of non-regular boundary point?

I am studying in PDE and I have next definition : Definition. Let $\Omega\subset\mathbb{R}^n$ open, connected. Then $\xi\in\partial\Omega$ is regular if there exists a superharmonic function $p$ in ...
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1answer
68 views

A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$. Consider the problem of finding $u:\Omega\times[0,T]\to\...
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1answer
119 views

Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?

Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...
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1answer
57 views

Initial value problems on manifolds around submanifolds (reference)

I am looking for a reference on initial value problems formulated on smooth manifolds with initial conditions on submanifolds. More precisely, let $X$ be a smooth manifold and $Y\subset X$ a embedded ...
6
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1answer
383 views

Solutions of PDE under changing topology

Let suppose we have a PDE on a manifold. I'm interested in the following question. How does the space of solutions of this PDE change when the topology of the manifold changes? For example in 2D we ...
2
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1answer
67 views

On $s$-harmonic functions

Is this statement true? A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...
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0answers
96 views

Existence theory with an integral equation

I am reading a paper in which it is proposed that one can solve a problem from mathematical physics by establishing an existence theory for a system of equations. One of the equations in the system ...
6
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0answers
269 views

A question in elementary differential geometry

Let $M$ be a finite dimensional manifold of constant curvature $\kappa$. Consider a solution of the Hamilton--Jacobi equation $$ \partial_t u + |\nabla u|^2 = 0. $$ Can we give a precise estimate of a ...
2
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2answers
102 views

inequality involving the fractional Sobolev space

Let $X_{0}$ be the Sobolev space defined on $(1, 2)$ by $X_{0}(1,2)= \{u\in H^s(\mathbb R): u=0 \text{ in } \mathbb R-(1, 2)\}.$ Is it possible to determine the constant $C$ of the inequality $$|u(x)...
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1answer
124 views

How to solve the integral equation $f(x)=\frac{\lambda\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_1}{\int_{x}^1 \sqrt{1+(f(t))^2}dt+c_2}$?

Recently, I have asked a question about variational analysis (First moments of uniform distribution on a curve from (0,0) to (1,1) in two-space). Such the question can be addressed in some cases by ...
4
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0answers
160 views

Uniqueness condition for Hamilton-Jacobi equation?

Let $f= f(t,x) : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function such that $$ \partial_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R}...
3
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1answer
83 views

Space of holomorphic functions multiplied by smooth functions taking real values

Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \...
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0answers
45 views

Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
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0answers
36 views

A nonlinear PDE on a matrix domain involving a maximum eigenvalue operator

In a research problem in optimal control $^{\ast}$ I came across the following nonlinear first-order PDE: $$\frac{\partial V}{\partial t}=\max\text{eig}\left[\sum_{i=1}^m P_i\frac{\partial V}{\partial ...
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2answers
1k views

Can learning Riemann surfaces be more beneficial than numerical analysis for an analyst?

I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, ...
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0answers
91 views

Spectrum of Laplacian-like operator

Let $\kappa_1, \kappa_2>0$ be fixed. Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by $$ A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
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0answers
76 views

Compatibility between the source and the boundary condition for an Helmholtz-type equation

Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
2
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1answer
292 views

Is there a diffeomorphism of the disk with constant sum of singular values?

This question is a relaxed version of this question. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$. Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...
4
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1answer
119 views

Heat flow, decay of the Fisher information, and $\lambda$-displacement convexity

In the whole post I will work in the flat torus $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $\rho$ will stand for any probability measure $\mathcal P(\mathbb T^d)$. This question is strongly related to ...
2
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0answers
41 views

improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
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35 views

When does the solution to the Fokker-Planck equation admit a density wrt Lebesgue measure?

Given a Markov process $(X_t)_{t\geq 0}$ on $(\mathbb R^n, \mathcal B_{\mathbb R^n})$, under which conditions does the solution to the Fokker-Planck equation $$\frac{\partial u(t,x)}{\partial t}= \...
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0answers
68 views

Strichartz estimate for the Schrödinger equation

Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP: $$i\partial_t ...
2
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0answers
76 views

Wave equation for smooth Schwartz kernels

Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus ...
2
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1answer
69 views

Approximating functions in $H^1_0(U) \cap H^2(U)$ via $H^1$ norm and $L^2$ projection

Let $U$ be a bounded domain in the Euclidean space with sufficiently smooth boundary. Let $\{f_i\}$ be a orthonormal basis of $H^1_0(U)$ satisfying $-\Delta f_i = \lambda_i f_i$ where $\lambda_i \leq \...
3
votes
0answers
60 views

Bound on number of linearly independent eigenvectors of adjoint of composition operator

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\...
7
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0answers
150 views

Smoothness of solution map for PDE

I am wondering what sort of results are available for the following sort of problem, or where to look in the literature for work dealing with such problems, especially in the degenerate elliptic ...
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0answers
60 views

Solution existence for two-dimensional parabolic PDEs

I am looking for a solution $(f,g) \in C^{1,2}([0;T]\times\mathbb R;\mathbb R^2)$ to the following PDE system $$ f_t(t,x) + a_1(t,x) f_x(t,x) + a_2(t,x) f_{xx}(t,x) - b_1 f(t,x)^2 + c(t) g(t,x) - d(t,...
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0answers
32 views

Are there maximum principles related to the third boundary condition?

I am working on this problem \begin{equation} \begin{cases} u''(s)+\frac{2}{s} u'(s)=R^2 f(u) \quad \text{ for } \eta<s<1, \\ u'(\eta)=0, \ u'(1)+\beta R (u(1)-\bar{\sigma})=0, \end{cases} \end{...
5
votes
1answer
128 views

Backward uniqueness for a heat equation with a drift

Consider heat equation with a drift (=reaction-diffusion equation) $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}+f(t,u(t,x)), \quad t\ge0,\, x\in [0,1] $$ with periodic or ...

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