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.
1,733
questions with no upvoted or accepted answers
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Regularity of reflection coefficients (or more generally the scattering transform)
Consider the Schrodinger operator $L(q) = -\partial_x^2 + q(x)$ where the potential $q$ is a real-valued function of a real variable which decays sufficiently rapidly at $\pm \infty$.
We define the ...
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68
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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?
$$|...
4
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0
answers
412
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A 4th-order linear PDE
I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$):
$x^3 f_{xxxt}+ f =0$
Does anyone know if this type of PDE already appeared in the literature? ...
4
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90
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Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$
In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
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Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
4
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1
answer
235
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If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
4
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120
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Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
4
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238
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Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
4
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168
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Reference request: $ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $
I have a Riemannian manifold $M$ of dimension 2 on which I am considering the following equation:
$$ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $$
on some patch $U$ of the manifold which is &...
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75
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Reference/Help request for formula $[A,e^{-itB}]$ found in physics thread
I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are ...
4
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121
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Distinguishing the Besov and Triebel-Lizorkin spaces
Theorem 2.3.9. in Triebel's Theory of Function Spaces states that the Besov space $B^{s_1, p_1}_{q_1} (\mathbb R^d)$ coincides with the Triebel-Lizorkin space $F^{s_2, p_2}_{q_2} (\mathbb R^d)$ if and ...
4
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Reference request: "stacked traveling waves" or "wave trains" in PDEs
I am looking for general reference on "stacked traveling waves" or "wave trains", or perhaps wave superpositions. They are a bit like multi-soliton solutions to the KdV equation, ...
4
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123
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Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
4
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105
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Continuity of solutions of Elliptic PDE with respect to parameters
Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy
$$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$
where $f$ is a fast decaying smooth function.
I would like to know how the ...
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Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
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58
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Pairs of functions with commuting Hessians
Let $f,g:\mathbb{R}^2\to\mathbb{R}$ be smooth functions and let $H_f, H_g$ be their Hessians. Is anything known about the differential equation $H_f H_g=H_g H_f$? Or, in higher dimensions, let $F=(f_1,...
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Effective way for studying PDEs
I am new to this stack, and thought my question belongs here.
I am a first-year graduate student currently taking my second course on PDEs (basically covering Evans ch. 5 and onwards). I am planning ...
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Pohozaev identity for linear equations
For $-\Delta u =0$, the Pohozaev identity on say $B_1$ says
$$ \int_{S_1} |u_T|^2 \,d\sigma = \int_{S_1} |u_N|^2 \,d\sigma + (n-2) \int_{B_1} |\nabla u|^2 \ dx$$
Here $u_T$ are the tangential ...
4
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102
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Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators
I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
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141
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Approximation by gaussian mollification in Sobolev spaces
I have been trying to find a good estimate on the constant in the inequality (in dimension $d=3$ to simplify)
$$\label{0}\tag{0}
\|(1-e^{t\Delta})f\|_{L^{5/3}(\Bbb R^3)} \leq C_0\,t^{2/5}\, \|\nabla \...
4
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68
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Representing homotopy classes of Kähler manifolds by harmonic maps
Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...
4
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294
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Topology of the soliton manifold unchanged by symmetry breaking potential and nonlinearity
For the NLS equation with a power non-linearity and no potential in $\mathbb{R}^d$
$$
i\partial_t \Psi = -\Delta \Psi + |\Psi|^{\sigma}\Psi,
$$
the soliton manifold, due to symmetries of the NLS, is $\...
4
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0
answers
100
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The logistic elliptic equation
Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
4
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answers
117
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Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
4
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answers
87
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Characteristic of Sobolev space generated by Hörmander vector fields
Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
4
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90
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Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
4
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answers
197
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Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
4
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85
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Reference request: a PDE related to Kahler–Ricci flow
I was reading the survey by Imbert and Silvestre where I noticed the PDE
$$
\frac {\partial u} {\partial t} = \ln(\det (D^2u))
$$
for the study of the Kahler–Ricci flow (Eq (2.2) at page 10 in ...
4
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answers
296
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PDE obtained while trying to construct a complex structure
Upon reading this answer to this question, the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this map." ...
4
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141
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Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
4
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answers
209
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Problems arising from the Trudinger's paper in 1968 "Remarks concerning the conformal deformation of riemannian structures on compact manifolds"
I'm reading the paper Remarks concerning the conformal deformation of riemannian
structures on compact manifolds by NEIL S. TRUDINGER.
I'm stuck with the Theorem 3, which says that let $u$ be a $W_{2}^...
4
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0
answers
78
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On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
4
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102
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Gradient bounds on a solution of a linear elliptic problem
Take $\Omega$ to be a bounded domain in $N$ dimensional Euclidean space with smooth boundary and we assume $\Omega$ contains the origin. I am interested is the following equation
$$ \Delta \phi(x) ...
4
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304
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Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
4
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0
answers
153
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An estimate for the Benjamin-Ono equation from T. Tao's well-posedness paper
In https://arxiv.org/abs/math/0307289 (eq. (8)),
for a (smooth) solution of the equation $$u_t - uu_x + Hu_{xx} = 0$$
(where $H$ denotes the Hilbert transform) the following estimate is stated (...
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What role do semiclassical methods play in the study of Ginzburg--Landau-type equations?
As far as I understand, semiclassical limits are used in quantum mechanics to analyse equations that depend on a small parameter $\hbar$. Apparently studying properties of the PDE as $\hbar \to 0$ ...
4
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90
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Biharmonic operator and maximum principle (PPP)
I have a question related to the Positivity Preserving Principle (PPP) for $ \Delta^2$ and related topics. Recall if $u$ solves
$$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \...
4
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79
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Almost(?) elliptic operators
I would like some references concerning the following subject.
Suppose that $\Omega$ is a bounded subset in $\mathbb{R}^n$ with smooth boundary and consider the following PDE there stated
$$L(f)(x) = ...
4
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answers
166
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Techniques for showing non-degeneracy results (PDE)
Motivation:
Consider the equation,
$$-\Delta u = u^p$$
in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
4
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answers
101
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Minimal regularity for domains in Green's formula
The Green formula is well-known for smooth bounded domains of $\mathbb R^d$. My question is:
What is the minimal regularity known for domains where Green's formula still holds?
4
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answers
173
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$L^\infty$ solutions for parabolic Neumann problem (heat equation)
Consider the heat equation on a (smooth) domain in $\mathbb{R}^n$ with homogeneous Neumann BCs:
$$u_t - \Delta u = f$$
$$\partial_\nu u = 0$$
$$u|_{t=0} = u_0$$
where $f \in L^p(0,T;L^r(\Omega))$ and $...
4
votes
0
answers
189
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How to use blow-up to prove the boundary regularity for a harmonic function
While reading the book Regularity Theory of Elliptic PDE I’m confused with a theorem:
Thm. 2.30.
Let $\alpha \in (0,1)$ and $k \in N$ with $k \leq 2$, and let $\Omega$ be a bounded $C^{k, \alpha}$ ...
4
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119
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Algebra properties regarding Gevrey spaces: closed under multiplication
In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a ...
4
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136
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If theorem valid for compactly supported distribution then is it also valid for tempered distribution?
I have seen many theorem which Author wanted to prove for tempered distribution, but without saying anything proves for compactly supported distribution.
For instance,
Theorem: Any $A \in \Psi^{m}$ ...
4
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0
answers
133
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Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
4
votes
0
answers
323
views
Estimates for the heat equation with inhomogeneous boundary condition
EDIT2: I believe the estimate required is false. There is some evidence that I have added to this post. I only believed it was true because it seemed like it was used in certain papers. However, in ...
4
votes
0
answers
85
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Non-separable Laplace-Beltrami eigenfunctions have isolated critical points (reference request)
Consider the Laplace-Beltrami operator on a compact manifold. Generically, Uhlenbeck has shown that eigenfunctions of the Laplace-Beltrami operator are Morse functions.
But there are some manifolds, ...
4
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0
answers
143
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Estimating the size of $\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \}$
Let $\Omega$ be a bounded domain in $\Bbb R^n$. Define
$$
\Omega_r=\{x\in\Omega: \text{dist}(x,\partial\Omega)<r \},
$$
i.e. it the ring of thickness $r$ at the boundary of $\Omega$. Intuitively, ...
4
votes
0
answers
133
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Uncertainty principle, Sobolev embedding, and norm estimates
Terence Tao offers a nice discussion of different function spaces in this blog. In the blog there is an explanation of the tradeoff between regularity $s$ and integrability $p$, where $s,p$ are ...
4
votes
0
answers
115
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Umbilic points of minimal hypersurfaces and distributional Simons inequality
Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel ...