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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How to show this integral on boundary of Lipschitz domain is finite?
Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int_{\partial\Omega} \frac{1}{|y|^{...
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What fails when we try to extend existence and unique for parabolic PDEs for 'PDEs which are 'parabolic in two components''?
What I have in mind is that on $(0,T)\times\Omega$, we have a parabolic pde operator $L$, we have unique solution to
$Lu = f$ when f is Hoelder for some coefficient strictly between 0 and 1.
This ...
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Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
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Harmonic function defined on a cone
It's well known that: Given a continuous function defined on the boundary of the disk, then there exists a unique harmonic function in the interior of the disk. What if we replace the disk by a cone?
...
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Fractional Leibniz formula
Let $T=(-\Delta)^{1/2}$.
Can we have estimates, similar to the one below
$$
\| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1}g\|_p,
$$
hold in $L^p$, where $...
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Dirichlet energy of a harmonic function bounded above by the energy of the boundary function?
$\Omega$ is a domain in $R^2$ with sufficient smooth boundary. Given an absolutely continuous function f difined on $S^1$($[0,2\pi]$). Then there exists an unique harmonic function u defined on $\...
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Conditions for existence of $m$-th differentiable root of a non-negative definite matrix
In M.I. Friedlin's famous paper "On the Factorization of Non-Negative Definite Matrices", he shows that if a non-negative definite symmetric matrix $a(x)=\{a^{ij}(x)\}_{i,j=1}^n$ is in $C^2(\mathbb{R}^...
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Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
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Uniqueness of classical solution with degenerate boundary
Consider heat equation on the domain $\Omega = (0,1)\times (0,1)$
in the form of
$$ \partial_{t} u = \frac 1 2 x^{3} (1-x) \partial_{xx} u, \quad
(x,t) \in \Omega$$
with initial data
$u(x,0) = x$ for ...
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Space of solutions of nonlinear Helmholtz equation on a torus
On a unit torus $T^n$ (or equivalently, on $\mathbb{R}^n$ with periodic boundary conditions), the linear Helmholtz equation:
$\nabla^2 \phi + k^2 \phi=0$
will have no non-trivial solutions for ...
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K-Theory of Algebra of Zeroth Order Pseudo differential operators
Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators?
Thanx!
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Reverse Holder Inequality and the higher integrability of the gradient of a solution to Euler's equation for a certain functional
In Giaquinta-Giusti's (1978) paper "Nonlinear Elliptic Systems with Elliptic Growth" (thm 1.1) they consider the following system:
\begin{equation}
\sum_{i, j=1}^{n}\sum_{\alpha, \beta=1}^N\frac{\...
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Is Poisson's kernel integrable?
Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
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Deduce global estimate from scaling-invariant local estimate
Let $(M,g)$ be a non-compact Riemannian manifold, with finite volume (or compactly exhausted, or any nice condition you would like, except for compactness). Suppose I have a tensor $T$ on $M$ of which ...
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If $f \in H^{\frac 12}$ and $\varphi$ is Lipschitz, is $f\varphi \in H^{\frac 12}$ (on a Lipschitz hypersurface)?
Let $M$ be a bounded hypersurface. Let $f \in H^{\frac 12}(M)$ and let $\varphi\colon M \to \mathbb{R}$ be a Lipschitz function.
When $M=\Omega \subset \mathbb{R}^n$ an open domain, we know that the ...
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About a definition of quasi-conformal maps
A book I'm reading gives the following definition for quasi-conformal maps:
If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$:
$...
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In the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms
Can anyone help me and prove that in the case of the Ricci flow, the symmetries of the flow are scalings and diffeomorphisms?
Thanks for your time.
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An interpolation type inequality
Let $u(x),x\in R_+$ be a non-negative decreasing smooth function with compact support $[0,L]$, I want to know the following inequality is true? $a\in (0,1)$
$$\int_0^\infty \frac{1}{1+x}u^{1+a}dx \le \...
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Sobolev spaces on boundaries
Consider the Sobolev space $W^{s,2}=H^s$ for $s=\frac{1}{2}.$
Let $\Omega \subset \mathbb{R}^n$ be an open set with boundary $\partial\Omega$. I have seen two definitions of the space $H^s(\partial\...
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Maximum of the solution of a parabolic PDE
Let $u:\mathbb{R}\times [0, \infty) \rightarrow \mathbb{R}$ be defined by
$u_{xx} + u_x - u_t = u(u - 2)(u - 1)$
with $u \rightarrow 0$ as $|x| \rightarrow \infty$ and $u(x,0) = 3e^{-x^2}$. Now, let ...
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Upper bounds for the solution of an elliptic PDE depending on a parameter.
Suppose I have the following PDE on $[0,1]^n$
$$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$
with periodic boundary conditions and $\int f(x) dx =0$ . ...
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If $u \in W^1(0,T;L^2,H^1)$ and $\varphi \in C^1([0,T]\times \Omega)$ then $\varphi u \in W^1(0,T;L^2,H^1)$?
Let $\Omega \subset \mathbb{R}^n$ be an open bounded domain.
Define $$W^1 := W^1(0,T;L^2,H^1) := \{w \in L^2(0,T;H^1(\Omega)) \mid w' \in L^2(0,T;H^{-1}(\Omega))\}$$
where $w'$ means the weak ...
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Laplacians associated to symplectic cohomologies
I am reading the paper"cohomology and Hodge theory on symplectic manifolds I" by Tseng and Yau. In this paper they consider several cohomologies on symplectic manifolds $(M,\omega)$based on the ...
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Parametrised Hilbert spaces; can we put a norm on the following space of Hilbert spaces?
For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way.
Define $...
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Comparison principle for partial differential equation with singular coefficients
How (or if) a comparison principle works in the case of equations
singular at some point? For example, I am analyzing a partial
differential equation
$$
\partial_{t}u=\partial_{rr}u+\frac{2}{r}\...
- 2,494
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Stokes problem; naive question on the regularity of pressure term
I am attempting to recall some basic knowledge related to Stokes' problem.
In particular I am following along in Evans PDE book in section 8.4.
So lets assume that
$-\Delta u + \nabla P = f $ in $ \...
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Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
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Nearly elliptic equations [closed]
If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...
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Differences between parabolic operators of second order and higher order
Properties of parabolic operators of second order have been extensively studied, such as the existence or uniqueness theorem. In higher order case ($u_t-P(D)u$, where $P$ is a $2m$ order uniformly ...
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A question on the DeTurck trick
I am probably being obtuse here, but there is something in the DeTurck trick that I do not understand precisely. I was reading from Andrews Hopper, and they (on page 91) say that the equation $\frac{\...
CommunityBot
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Reference on a Monge-Ampère-like equation
We recently realized that a geometric questions of interest to us is strongly related to the regularity of solutions of the following simple equation on the unit disk in $R^2$:
$$ \det(Hess(w))=1~, $$
...
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functions of bounded variation and gradient vector measure
I want to prove a function of bounded variation on some domain $D\subset R^n$, $f\in BV(D)$, has the property that there is a constant $C$, such that
$$
\lim_{r\rightarrow 0}\frac{C}{r^{n+1}} \int_{...
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Extra regularity of Poisson problem having nonzero Neumann boundary condition in convex domain
Let $\Omega\subset\mathbb{R}^2$ be a convex simply connected domain having piecewise smooth boundary, $f\in L^2(\Omega)$ and $g\in H^{\frac 1 2}(\partial\Omega)$. Grisvard in [1] among others prove ...
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Subordination identity and heat operator
I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am ...
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Precise versions of "differential operators are unbounded but closed linear operators"
I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...
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What's a good resource for Hormander symbols of type (1/2, 1/2)?
I'm currently working with some pseudodifferential operators of Hormander class $L^{m}_{\frac{1}{2},\frac{1}{2}}$ and unfortunately many of the usual tools break down, due to difficulties with their ...
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Proof of Friedrichs inequality in a domain with simple geometry
Does exists a short, simple proof of the inequality
$ \|u\|_{L^{2}(\Omega)} \leqslant C \|Du\| _{L^{2}(\Omega)} + \|u\| _{L^2{(\partial{\Omega})}} $ for $u\in H^{1}=W^{1,2}(\Omega) $
(Sobolev ...
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Existence of harmonic maps between loops
Given a Riemannian manifold $M$ and two smooth loops $\gamma_0, \gamma_1: S^1 \longrightarrow M$ in it, I am looking for maps $\phi: [0, T] \times S^1 \longrightarrow M$ which minimize the energy
$$E[\...
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$L^2$ bounds for the gradient of subsolutions to parabolic equation
Suppose we have the differential inequality
$$
|\partial_t{u}+\Delta{u}|\leq C(|u|+|\nabla u|)
$$
in $\mathbb{R}^n\backslash B_R\times[0,1]$, where $B_R=\{x\in\mathbb{R}^n,|x|\leq R\}$. Then do we ...
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solving elliptic system of first-order linear PDE's
I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which ...
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Influence of Yau's solution to the Calabi Conjecture on the field of PDEs
I remember reading a long time ago(I can't recall where, unfortunately) that Yau's solution of the Calabi-Yau conjecture introduced new techniques that were very important for the field of partial ...
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Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?
On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
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A Poincare inequality for the Laplace-Beltrami operator [closed]
Suppose $w \in C^2 (S^{n-1}), \Lambda$ is Laplace-Beltrami operator on the sphere $S^{n-1}$, How can I prove follow Poincare inequality :
$\int_{S^{n-1}} w\Lambda w d\sigma \leq (1-n) \int_{S^{n-1}} |...
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Analyticity of one-dimensional PDE solutions with respect to the space variable
Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients
$$
u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0,
$$
in some neighborhood of a point $(x_0,...
- 2,494
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weak solution of viscous Burgers equation with non-homogeneous Dirichlet boundary conditions
I was wondering if anybody knows (and can give me a reference, please) if the PDE below has a unique weak solution. I can only find the result if we consider homogeneous Dirichlet boundary conditions, ...
CommunityBot
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Example for the Sobolev embedding theorem when p=n.
Let $\Omega$ be a bounded domain in $\mathbb R^2$. By the Sobolev embedding theorem, if $k>\frac np$ (in this case $k>\frac 2p$) then
$u\in W^{k,p}(U) \implies u\in C^{k-[\frac 2 p]-1,\gamma}(...
- 2,834
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A general inequality about spherical mean of a function
suppose $\overline u(r)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}u(r,w)dw,0<r<1,$ is the average of $u(r,w)$ on sphere $S^{n-1}$,where $(r,w)$ are the polar coordinates in $R^n$.
My question is ...
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On a conjecture of Lions for the wave equation
In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture :
Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider
$\phi'' - \triangle \phi = F$
$\...
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Analytic solution of Poisson's equation
Consider Poisson's equation $\nabla^2 u = 1$ on a square of side-length 1 centered at the origin. Cut out a circle of radius 1/3 at the center of this square.
Impose a von Neumann boundary condition $...
- 6,210
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Reference request: harnack inequality for distributional solutions of the heat equation
Dear Math Overflowers,
I'm looking for references on the parabolic Harnack inequality for distributional solutions of the heat equation on the whole space
$$
\partial_t u=\Delta u\quad\text{and}\quad ...
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