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.
4,467 questions
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Gap to fill in the Aubin–Ekeland proof of the mountain-pass theorem
Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to ...
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Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In ...
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Does integration by parts formula hold in $H^1(0,T,L^2(\Omega))$?
Let $\Omega$ be an open set from $\mathbb{R}^N$. How can we prove that if $u,v\in H^1(0,T,L^2(\Omega))$ (in Bochner sense) then $(u\cdot v)'\in L^2(0,T,L^1(\Omega))$ with $(uv)'=u'v+v'u$ and the ...
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Properties of "potential vector field" in Helmholtz decomposition
It is a well known fact that given a vector field $F$ in $\mathbb{R}^3$, this can be decomposed as
$$ F= \nabla V+ \nabla \times R$$
with $V$ a potential and $R$ another vector field. These components ...
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141
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$L^{p}$ estimate for $\frac{|\nabla f|^{2}}{f}$
I’m trying to obtain an $L^{p}$ estimate under certain conditions. Suppose $f\in C^{2}(B_{2})$ is a function satisfying $0<f<1$ and $f\Delta f>\frac{1}{n} |\nabla f|^{2}$, where $B_{2}\subset\...
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The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
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Nitsche's method for p-Laplace equation
My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation.
The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the ...
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For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$
For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...
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Existence of solution to Cauchy boundary value problem in Lipschitz class of functions
For a research question I have run into the following problem that seems intuitively true but I do not know how to prove it and am not sure in which generality.
Let $\Omega\subset \mathbb{R}^2$ be a ...
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Can functions with "big" discontinuities be in $H^1$?
How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...
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System of equations with one integral equation
Start with a system of three equations such that two of the equations are ordinary or partial differential equations, but one of them is an integral equation as follows:
$C = \int_{0}^{\infty} X \: ...
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Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with external forcing
First, a quick summary of what to know about viscous (or dissipative) Burgers' equation
$$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$
Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev ...
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Calculating frequency of sound of ringing metal coin
I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
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Relative bounds for vorticity
Write the vorticity equation as
\begin{equation}\label{Eq20}
\begin{split}
\dfrac{\partial}{\partial t} v_i & = \biggl[|\textbf{v}|~|\nabla u_i|\cos(\beta_i)- |\textbf{u}|~|\nabla v_i|\cos(\...
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Algebraic normalisation of regularity structures: can there be a explicit expression of g?
This is related to Bruned, Hairer, and Zambotti - Algebraic renormalisation of regularity structures. In the method of re-normalization the functional $g$ shown in page 6 plays a major role. However, ...
CommunityBot
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170
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Extreme confusion over the tautology of the solution for PDEs
Let $u_0 : \mathbb{R}^3 \to \mathbb{R}^3$ be a smooth divergence free vector field. Then, it is well-known from the theory of Navier-Stokes equation that there exists some $T \in (0,\infty]$ and a ...
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
- 1,759
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What is kth vortex formula?
I want to study the kth vorticity equation. The NS equation is provided as
\begin{align}\label{eq1}
&\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \right)\textbf{u} = ...
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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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A question on the maximum principle of second order elliptic equations
Let $Lu=a^{ij}u_{ij} + b^i u_i$ be an elliptic operator of second order in a bounded domain $\Omega$. Assume that $a^{ij}$ is uniformly elliptic. Then it's well known that the following maximum ...
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Embeddings of the maximal domain for the Laplacian
Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...
- 6,035
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111
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Reversing heat transfer with respect to time
Fact: One can easily compute heat dispersion in a plane using the heat equation.
Question: Has any research been done on computing the process in the reverse time direction?
That is, given a heat map $...
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Parabolic Schwarz lemma
Trying to follow the computation in Song and Tian - The Kähler–Ricci flow on surfaces of positive Kodaira dimension, page 7, theorem 3.1 which proved a parabolic Schwarz lemma. Specifically, they ...
CommunityBot
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How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
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Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \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\} $$
...
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Wick rotation for Laplace and wave equations
I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...
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Proving that $\max_{w \in B(z)} e^{f(w)} \leq Ce^{f(z)}$
Let $f : \mathbb R^2 \to \mathbb R $ be a smooth function statisfying
$$
0 < \alpha \leq \Delta f(w) \leq \beta < \infty, \ \ \forall w \in \mathbb R^2
$$
where $\Delta$ denotes the Laplace ...
- 109k
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Solutions for a system of PDE's
Let $ \Omega_s(x) $ solve the system
$$s \dfrac{\partial^2}{\partial s^2}\Omega_s(x)=\pm x\dfrac{\partial}{\partial x}\Omega_s(x) $$
$$2\sqrt{s}\frac{\partial}{\partial s} \sqrt{\pm\Omega_s(x)}=\sqrt{...
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Spectrum of Laplace-Beltrami operator on tensors
Let $(M, g)$ be a complete Riemannian manifold diffeomorphic to $\mathbb{R}^n$. Under appropriate geometric assumptions concerning the geometry near infinity, but without any curvature sign ...
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Injectivity of div–curl operator
$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system
\begin{align*}
Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\
u &= 0 \...
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0
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Closed form ODE solutions for Jacobi field/eigenfunction of Laplacian on hyperbolic space
I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the ...
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Why $-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}$ type PDE is called 'mean-field equation'?
Why $$
-\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g}
$$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical ...
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Convex combination of cyclically monotone sets
I want to show the following statement, but I am not sure how.
Proposition(?):
Let $C \in \mathbb{R}^d$ be a compact convex set, and let $u, v : C \to \mathbb{R}$ be smooth convex functions.
Suppose
$$...
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Regularity of elliptic equation with Neumann boundary conditions
In the context of the regularity of the free boundary of the one-phase (a.k.a. Bernoulli) problem we want to show $C^{1,\alpha}$ regularity of the free boundary implies smoothness of the free boundary ...
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Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
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Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
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What's wrong with the Courant nodal domain theorem?
The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. ...
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Finiteness of theta vanishing in the KP direction for locally planar curves
I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...
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Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...
- 6,367
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Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
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1
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Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
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Nature of a certain invariant on smooth field of positive definite matrices
I initially asked on math.stackoverflow but have since come to understand this forum may be more appropriate, as this is indeed a question that arose in writing a research article.
Denote $g$ a ...
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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, ...
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The maximum in the Poisson problem on the cube with constant source
Question:
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial (...
- 1,446
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2
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260
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
- 2,816
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derivation of variational forms of PDE directly from conservation form
A lot of texts derive the variational form of a PDE as follows.
First, life begins with a conservation law for the field $q$:
$$\partial_t \int_\omega G(q)\;dx + \int_{\partial\omega} F(q, \nabla q, \...
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Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
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62
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Dispersive equations at low frequencies and time oscillations
It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
The Schrodinger dispersion relation is $\omega(...
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1
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Stochastic representation of Laplace equation with Neumann boundary condition
Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...
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0
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Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
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