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.
4,466 questions
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Derive elliptic maximum principle from weak derivatives
Let $U$ is a connected open set, and $a^{ij}, c^i \in L^\infty (U).$ $a^{ij}$ satisfies the uniform ellipticity condition. Suppose that $u\in H^1(U) \cap C(\overline U)$ satisfies the condition that
$$...
- 1,755
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2
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405
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Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
- 11.8k
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Elliptic PDEs in Finance
In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
- 5,405
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56
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Stability on manifold with boundary
Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that:
Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
- 395
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Using a theorem (which is originally set on 2-dim bounded domain in Euclidean space) on a torus
Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1}
-\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right)...
- 809
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1
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309
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Eigenvalues of a Schrödinger operator
I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator
$$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$
$$\varphi(0) = \...
- 337
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Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 825
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140
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Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\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 ...
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56
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Finding thin plate spline subjected to boundary conditions
I am trying to formulate a problem as a PDE. What I want to know is if my formulation is correct, if it admits solution and what am I missing.
This question is related to : Thin-Plate-Spline ...
- 115
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Adding a data-dependent term to the porous medium equation while retaining an explicit solution
I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by:
$
\frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1
$
For this equation, ...
- 11
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votes
2
answers
781
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Is there any bilinear Poincaré/Sobolev inequality?
Is the following, I call it bilinear Poincaré inequality, true?
Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
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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-...
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Techniques to estimate PDE which are elliptic in some directions and degenerate in others
I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some ...
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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 $\...
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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 ...
- 809
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Regularity of solution of $(-\Delta + w)f = 0$
I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
- 1,305
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260
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Interpretation of the singular integral for the definition of fractional Laplacian in classical case $s=1$
Fractional Laplacian is often defined via next principal value integral (assume dimension one for the simplicity)
$$(-\Delta)^su(x):=c_sP.V.\int_\mathbb{R} \frac{u(x)-u(y)}{|x-y|^{1+2s}}\, dy.$$
This ...
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Estimating a potential
Let $ \Omega$ denote a smooth bounded domain in $R^N$ where $N \ge 3$ and we let $u \in C^3( \overline{\Omega})$. Let $ \delta(x) = \operatorname{dist}(x, \partial \Omega)$. For $ x \in \Omega$ (but ...
- 1,385
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Looking for an interesting result on the Navier-Stokes equations
I am in my second year of master in Mathematics and one of my courses consists of a reading of Navier-Stokes Equations by Roger Temam. We have proven the existence for the weak Stokes and Navier-...
- 452
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99
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Two-sided estimates of fundamental solutions of second-order parabolic equations
I am reading the paper Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications by F.O. Porper and S.D. Eidel'man. Below, the cited paper is
[2] :S.D. ...
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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 ...
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94
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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 ...
- 16.2k
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63
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On an estimate in the paper by Donnelly and Fefferman
I was reading the following paper by Donnelly and Fefferman https://link.springer.com/content/pdf/10.1007/BF01393691.pdf which essentially deals with the Hausdorff dimension bound of the nodal sets ...
- 41
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100
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Rate of convergence of mollified distributions in Besov spaces with negative regularity
Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
- 293
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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 ...
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Understanding the effect of PDE solution on critical strip?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 383
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Better regularity than $C^{1,\alpha}$ for $p$-harmonic maps into certain target manifolds
This is a follow up question I had while reading through this question and "Representing Homotopy Groups and Spaces of Maps by $p$-harmonic maps" by Shihshu Walter Wei.
We know that ...
- 21
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Is Brascamp-Lieb inequality on the sphere applicable for these functions for some $1\leq p<2$
My question is on Brascamp-Lieb-inequality on the Euclidean sphere (which can be viewed as an analogue of Young's inequality on the sphere) obtained in [1].
(See also this question:
Brascamp-Lieb ...
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Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
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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
$$\tag{1}
G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
- 809
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Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?
Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider
\begin{align*}
\begin{cases}
u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
- 313
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An expansion for 2d Euler equation
Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$:
$$
-\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
- 471
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1
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384
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Continuity equation $\partial_t \mu_t+\operatorname{div} (v_t \mu_t)=0$: are these two notions of weak solution equivalent?
Let $\Omega$ be an open connected convex subset of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of Borel probability measures on $\Omega$. Let $C_0 (\Omega)$ be the space of real-valued ...
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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 $1$-...
- 1,231
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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 $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
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Questions about the results about $\Delta u + e^u=0$, $3 \le n \le 9$: no finite Morse index solution, $n \ge 10$: stable radial symmetric solution
I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$
$$\Delta u + e^u=0.$$
They proved that when $3 \le n \le 9$, there is no finite Morse index ...
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94
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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
$$
\partial_tu-\Delta ...
- 1,219
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309
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Reverse estimate on the Riesz potential $I_\alpha : L^{n/\alpha}\to \mathrm{BMO}$
$\newcommand\BMO{\mathrm{BMO}}$Consider the Riesz potential on $\mathbb{R}^n$ given by
$$
I_\alpha f(x) = c_{n,\alpha} \int_{\mathbb{R}^n} \frac{f(y)}{\lvert x-y\rvert^{n-\alpha}} dy.
$$
It is known ...
- 363
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votes
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496
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A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
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How using the standard Galerkin method
I am attempting to solve the following evolution problem using the standard Galerkin method
$$\begin{cases}
\dot y(x,t)=\Delta y(x,t) +b(t) \nabla y(x,t), \ (x,t)\in \Omega\times (0,T) \\
...
- 55
1
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1
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159
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Existence of solution to nonlinear first order PDE with C^1 bounds
I'm looking for general existence of a PDE of the form
$$ f: U \times [0, \delta) \to \mathbb{R}$$
$$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$
where $f(p,0)$ is prescribed and $F$ is non-linear ...
- 337
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49
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Galerkin scheme in $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$ ($s>0$)
What basis functions are usually choosen if one attempts to conduct a Galerkin finite element method given an evolution triplet $H^s_0(G)\subset L^2(G)\subset H^{-s}(G)$. Where $G$ is a sufficiently ...
- 163
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3
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314
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The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\...
- 852
3
votes
2
answers
355
views
General version of Weyl's lemma
The classical Weyl's lemma say, suppose $u \in L^1_{loc}(\Omega)$ satisfies
$$\int_{\Omega}u \Delta \phi dx=0\ \ \forall \phi\in C_c^{\infty}(\Omega),$$
then $u$ is harmonic in $\Omega.$ What I want ...
- 379
3
votes
1
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278
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{...
- 1,759
3
votes
0
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126
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On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,429
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1
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113
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An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$
Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e.,
$$
g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}},
\quad t>0, x \in \mathbb R^d.
$$
Let $f : \mathbb R^d \to \...
- 825
3
votes
0
answers
153
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Quasimode construction on a compact Riemannian manifold
Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
- 131
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63
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
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0
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65
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
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