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.
4,271
questions
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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:
$$
\begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}...
- 1,330
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120
views
Weighted logarithmic Sobolev inequality
$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\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 ...
- 5,163
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0
answers
89
views
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 ...
- 265
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Linking theorem
In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem:
$$
\begin{cases}
-\Delta u = \lambda u + |u|^{p-2}u, \Omega \\
u = 0, \...
4
votes
0
answers
197
views
Spectral problems with the wrong sign on the Poincaré disk
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
- 3,064
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86
views
Reference request: a PDE related to Kahler–Ricci flow
I was reading the survey by Imbert and Silvestre where I noticed the PDE
$$
\frac {\partial u} {\partial t} = \ln(\det (D^2u))
$$
for the study of the Kahler–Ricci flow (Eq (2.2) at page 10 in ...
user478492
4
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0
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PDE obtained while trying to construct a complex structure
Upon reading this answer to this question, the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this map." ...
- 1,735
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143
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Uniqueness of the "weak solution" to Fokker-Plank PDE
Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying
$$\...
- 1,230
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views
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}^...
- 729
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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 $(M,g)$...
- 4,089
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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
$$ \Delta \phi(x) ...
- 1,363
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Banach's fixed point theorem for quasilinear parabolic PDEs
I have recently started reading into PDE theory, and came across the following question. Consider the PDE $$
\begin{cases}
\partial_t \rho = \Delta (\rho + \rho^2) & \text{ on } (0,T) \times \...
- 213
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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 (...
- 819
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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 $\hbar \to 0$ ...
- 4,968
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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
$$\Delta^2 u = f(x) \mbox{ in } \Omega, \quad u=\partial_\nu u =0 \...
- 1,363
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views
Almost(?) elliptic operators
I would like some references concerning the following subject.
Suppose that $\Omega$ is a bounded subset in $\mathbb{R}^n$ with smooth boundary and consider the following PDE there stated
$$L(f)(x) = ...
- 1,774
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votes
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answers
166
views
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,...
- 653
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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?
- 395
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answers
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views
$L^\infty$ 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 $...
- 287
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answers
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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 $C^{k, \alpha}$ ...
- 191
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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 ...
- 517
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views
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 $A \in \Psi^{m}$ ...
- 309
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answers
272
views
Integral representation of solution of an elliptic PDE in divergence form
Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And ...
- 261
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answers
135
views
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 ...
- 1,641
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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 ...
- 211
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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, ...
- 620
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answers
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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, ...
- 1,245
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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 ...
- 1,641
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views
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 ...
- 2,169
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Global existence of $L^p$-solutions to a quasilinear diffusion equation
We consider the diffusion problem
$$\begin{cases}
\partial_t u = \nabla \cdot (a(u)\nabla u), \quad t>0, x \in \mathbb{R}^n \\
u(0) = u_0
\end{cases}$$
for functions $u \colon [0,T] \times \mathbb{...
- 101
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views
Determination of the nature of stationary values in variational calculus
In variational calculus, when we solve the Euler-Lagrange equation $\frac{d}{dx}L_p(u',u,x)-L_z(u',u,x)$, where $L=L(p,z,x)$, to find stationary inputs of the functional
$$
I[u]=\int_0^1 L(u',u,x)dx,
$...
- 1,641
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votes
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views
Auxiliary spaces/conditions for orbital stability of traveling waves
In the context of orbital stability, probably one of the most used theorem to show the orbital stability of traveling waves is the one from Grillakis-Shatah-Strauss "Stability theory of solitary ...
4
votes
0
answers
111
views
Looking for an electronic copy of Lebeau's paper
I would like to know if anyone has an electronic copy of the paper "Gilles Lebeau - Contrôle De L'Équation De Schrödinger"? This article appeared in Journal de Mathématiques Pures et ...
- 481
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views
Smooth dependence of solution to elliptic pde depending on parameter
I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-\Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ \partial B_1$ where $B_1$ is ...
- 1,363
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votes
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answers
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views
Existence results for Lagrangian solutions to the Incompressible Euler Equation?
It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
- 919
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views
Continuity of weak solutions to wave equation with time-dependent coefficients
Consider the following second-order wave equation
$$
u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega
$$
with boundary conditions
$$
u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
- 248
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votes
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answer
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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
$$
\partial_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}_+ \times \mathbb{R}...
- 562
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votes
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answers
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views
improved regularization for $\lambda$-convex gradient flows
It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
- 5,163
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votes
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views
A metric $w$ on a Kahler manifold is extremal if and only if the gradient vector field of the scalar curvature is holomorphic
I am trying to understand the calculation in An introduction to Extremal kahler metrics. On the fourth line of page 55 the author calculated that $\int_{M} - 2 S R^{\bar k j} \partial_{j} \partial_{\...
- 91
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votes
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answers
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views
Compactness of multiplication operators
Let $n \ge 3$ and $0 \le V\in L^p(R^n)$ for some $p \ge n/2$. Then the multiplication operator $$Tu=V^{1/2}u$$ is compact from $H^1(R^n)$ to $L^2(R^n)$. If $p>n/2$, this follows from the ...
- 5,255
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views
Inequality between Dirichlet and Neumann eigenvalue for Sturm Liouville problem
Consider the following Sturm Liouville problem on an interval $[a,b]$
$$\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 ...
- 593
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answers
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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 $|...
- 429
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votes
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answers
124
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$, ...
- 819
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votes
0
answers
167
views
Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$
Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...
user39481
4
votes
0
answers
107
views
Biharmonic heat flow on compact manifolds
Consider $\partial _t u (t,x) = -\partial _x ^4 u$ on a compact manifold, or even a special specific one like the torus.
Are there any estimates on the Green function (bihamornic heat kernel), for ...
- 3,554
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votes
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answers
609
views
Maximum Principles in Parabolic PDE with Neumann Condition
I am looking for some maximum principles and comparison results for parabolic equations. The most complete book I've found on this subject is: Murray Protter, Hans Weinberger - Maximum Principles in ...
- 1,330
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votes
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answers
92
views
Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian
What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
- 381
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votes
0
answers
178
views
Traces of manifold-valued Sobolev maps
Let $(M^m,g)$ be a compact Riemannian manifold with smooth nonempty boundary, and $N^n\subseteq \Bbb R^d$ a boundaryless isometrically embedded Riemannian manifold. For $1\le p<\infty$ we define as ...
- 728
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votes
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answer
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views
Numerics for continuity equation with Sobolev vector field
Has any work been done about numerical methods for the continuity equation
$$
\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,
$$
where $...
- 819
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votes
0
answers
206
views
How to prove that every weak solution is classic for a simple parabolic equation
Consider equation
$$
Lu=u_t-a(x)u_{xx}=0, \tag 1
$$
where coefficient $a$ is bounded and $a(x)\ge \delta>0$ for all $x\in \mathbb R$.
If $a$ is not smooth the conjugate operator $L^*$ can not be ...
- 2,645