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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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem

This is the equation given ($n\geq2$) $$ \begin{cases} u_{tt}=a^{2}\left(\Delta u\right), \\ \left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\ \left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) . \end{...
Zydragon's user avatar
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Continuity of the constant in maximal Sobolev regularity

Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
Keba's user avatar
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How to understand the unique continuation result

Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$ Suppose $K(x) \in C^1\left(\mathbf{R}^...
Davidi Cone's user avatar
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Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?

Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$: \begin{equation} u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1} \end{equation} Assume that $...
Guohuan Zhao's user avatar
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Elliptic regularity for Dirichlet problem

Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective ...
user505117's user avatar
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Linearized operator of higher order $p$ Laplacian

The $p$th Laplacian is defined as $-\Delta_pv= \text{div}(|Dv|^{p−2}|Dv|)$. My question is whether there are any analogous notions of $p$th $m$-Laplacian for $m$ even and odd. For the $p$th bi-...
Sarthak's user avatar
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$L^\infty$ estimate for elliptic PDE with mixed boundary conditions

Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint. Consider the problem $$\Delta u = f \quad\text{in $\...
BBB's user avatar
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3 votes
1 answer
181 views
+100

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
Daniel Castro's user avatar
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Specific type of PDE

While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices): $$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
Gennaro Marco Devincenzis's user avatar
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Convergence of metric implies convergence of eigenvalues?

Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions: Does $g_\varepsilon$ converge to the flat metric on ...
Student's user avatar
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If a weighted Laplacian's eigenfunction is zero in an open set, when is it identically zero?

Let $m, s \in ([0, 1]^d \rightarrow \mathbb{R}_{\geq 0}$). Define a weighted Laplacian $\Delta_{m, s}f$ evaluated at $x \in [0, 1]^d$ to be: $m(x) \cdot \text{div} ( s(x) \nabla f(x))$. What ...
Timothy Chu's user avatar
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Any theory on the elliptic operator $Lu=\Delta u + b_iu_i + cu$ when $c>0$

I wonder if there are theories on elliptic operator $$Lu=\Delta u + b_iu_i + cu$$ when $c>0$, when $c<0$, we are glad to have maximum principle, so the bijectivity can be easily analyzed, but I ...
Elio Li's user avatar
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?

I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying $$\Delta_c u = f(x,u),$...
Elio Li's user avatar
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Uniqueness of constructed solutions to the Helmholtz equation

My question is regarding the inhomogeneous Helmholtz equation on $\mathbb{R}^3$ with real wavenumber $k$ and outgoing radiation condition \begin{equation} \Delta u + k^2 u = - f \quad \text{and} \quad ...
confused postdoc's user avatar
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Does the Poincaré inequality hold on annular domains?

Does the following Poincaré inequality hold $$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$ where $B_r$ denotes a ball of radius ...
Student's user avatar
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A simple bilinear estimate

Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that $\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$. Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$. What is the optimal value of $t=t(\...
Medo's user avatar
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Non-selfadjoint operators and physical systems

There are plenty of examples of non-selfadjoint operators modelizing physical phenomena: to name a few, let's quote the the heat equation (\ref{HEAT}, see below), the Navier-Stokes system for ...
Bazin's user avatar
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Galerkin’s Method for hyperbolic PDEs: proving convergence without using compactness

Lawrence Evan's PDE book prove the existence of solution to the following problem where $L$ is an elliptic operator: $$ \begin{cases} u_{tt} = -Lu+f,\\ u|_{t=0} = u_0,\\ u|_{\partial U} = 0 \end{cases}...
Ma Joad's user avatar
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An expansion for 2d Euler equation

Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$: $$ -\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
Davidi Cone's user avatar
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An asymmetric quadrilinear estimate

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
Medo's user avatar
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann,...
Elio Li's user avatar
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What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
Julian Chaidez's user avatar
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233 views

Convergence of metric and eigenvalues on a tubular neighbourhood

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
Student's user avatar
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Growth/Decay of conformal Killing fields in cone metrics

Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$. Does there exist a nontrivial conformal Killing field vanishing ...
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On the existence of a complicated fractal-like set of finite perimeter

Let $f\in BV(\Bbb R^n)$ be an integer-valued function that maps into $\{0, 1\}$ and is identically $0$ outside some bounded set in $\Bbb R^n$. In particular, $f$ determines a bounded Caccioppoli set $...
BigbearZzz's user avatar
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2 votes
1 answer
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Gradient flows and particle representations

I was looking into gradient flows and their particle representations, mostly in the context of probability. A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
CComp's user avatar
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68 views

Solution to the Eikonal equation with almost everywhere continuous derivative

Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE? $$|...
Nate River's user avatar
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2 answers
336 views

Nontrivial invariant transformations for heat equations

It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by $$ v(r,\theta) = u(\frac{1}{r},\theta)$$ is also harmonic for $r>0$. Note that the Kelvin ...
Ali's user avatar
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1 vote
1 answer
79 views

Sufficient initial conditions for "Non-local" PDE

I am studying a problem of the form $$i\, \partial_t \psi(t) = L \psi(t) + \int_0^t U(t-r) \psi(r) \, dr, \qquad \psi(0) = \psi_0,$$ where the evolution occurs in some Hilbert space, $L$ is a self-...
DerGalaxy's user avatar
2 votes
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88 views

Harmonic heat flow, formal and rigorous

Let $ (M,g) $ be a smooth Riemann manifold without boundary, $ S^{n-1} $ is an $ n $-dimensional sphere, and $ T>0 $. Consider a weak solution $ u:M\times[0,T]\to S^{n+1} $ of $$ \partial_tu-\Delta ...
Luis Yanka Annalisc's user avatar
2 votes
2 answers
171 views

A Inequality in the paper by Kenig, Ponce and Vega

I was trying to read the appendix of the paper by Kenig, Ponce and Vega, "Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle", ...
Sarthak's user avatar
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3 votes
3 answers
304 views

Generalized Fuchsian-type PDE?

Consider $$ \big(1+ t\partial_t\big) \left(\partial^3_x+ {6\over x}\partial^2_x + {6\over x^2}\partial_x\right)A(x,t)+ {t\over (1-x)^3} A(x,t)=0 $$ with the initial condition $A(x,0)=1$. In a small $t$...
Math2024's user avatar
1 vote
0 answers
94 views

How using the standard Galerkin method

I am attempting to solve the following evolution problem using the standard Galerkin method $$\begin{cases} \dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\ ...
elmas's user avatar
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0 answers
22 views

Metric entropy of mixed norm spaces with exponent-free bounds

Suppose $\mathcal{F}\subset L^p([0,1]^d)$ is a subset with the following property: The $L^q$-covering number of $\mathcal{F}$ is independent of $q$, for all $1\le q\le\infty$. An example of $\mathcal{...
chrisv's user avatar
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About the naturality of Krasnoselskii genus on Variational Methods

I have recently watched a seminar about Variational Methods from Mónica Clapp and she gave a very interesting motivation of why the Lusternik–Schnirelmann category (click on the link for the ...
Jacaré's user avatar
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1 answer
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About the polynomial characterization of $C^{1,\alpha}(\bar{\Omega})$ Hölder space in Lipschitz domain

I have trouble proving the following statement regarding a characterization of $C^{1,\alpha}$: Let $\Omega$ be a Lipschitz domain. $u$ is pointwise $C^{1,\alpha}$ at all points with the same constant $...
Stack_Underflow's user avatar
3 votes
0 answers
69 views

Norm estimate for parabolic SPDE solution

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
0xbadf00d's user avatar
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3 votes
0 answers
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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
G. Blaickner's user avatar
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4 votes
0 answers
412 views

A 4th-order linear PDE

I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $x^3 f_{xxxt}+ f =0$ Does anyone know if this type of PDE already appeared in the literature? ...
Math2024's user avatar
2 votes
0 answers
50 views

Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of the ...
BBB's user avatar
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4 votes
1 answer
259 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
0 votes
1 answer
117 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar
6 votes
0 answers
195 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
  • 517
1 vote
0 answers
47 views

Using connection form for unknown frame field

I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
yak's user avatar
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2 votes
0 answers
102 views

Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it. We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
Akira's user avatar
  • 815
7 votes
1 answer
494 views

Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
Laithy's user avatar
  • 865
1 vote
1 answer
93 views

Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator

I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time. Consider the following ...
vent de la paix's user avatar
3 votes
1 answer
143 views

PDE: compactness vs blowup

There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following: Solve (easier) approximate problems, show some form of compactness for the approximate ...
Sebastian Bechtel's user avatar
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0 answers
85 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
  • 493
2 votes
0 answers
167 views

Question about the formula of Green function of Laplacian on sphere

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
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