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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Strong minimum principle for maximal plurisubharmonic functions
Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
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Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
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Open sets and Poincaré's inequality
In many references, Poincaré inequality is presented in the following way :
Let $\Omega\subset \mathbb R^d$ an open bounded set. We can find a constant $C$ which depend of $\Omega$ such that for ...
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Brezis-Nirenberg result compared to abstract bifurcation theory
Dear Mathoverflow'ers,
I am interested in the following equation:
$-\Delta u = u^{p-1} + \lambda u$ in $ \Omega$ with $ u=0 $ on $ \partial \Omega$.
1) My question is related to the Brezis-...
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Question on PDE
I am looking for a reference where the following situation or something similar could have been studied. As a foreword, my question may not be very technical since I am from an engineering background. ...
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separation of variables in differential euqtions and compact self-adjoint operators [closed]
For some partial differential equations in physics, people may separate the variables and get some eigenfunctions. And then for any solutions for that equation, people often suppose them to be a ...
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Estimates on the Green function of an elliptic second order differential operator.
Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free ...
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Showing a solution of elliptic PDe is non-degenerate
Dear Mathoverflowers:
I am interested in radial positive solutions of
$-\Delta u(r) = r^\alpha u(r)^p$ in the unit ball in $ R^N$ with $ u=0$ on the boundary.
Here $p>1$ and $ \alpha >0$. (...
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Solve |\nabla u|^2=1
I need all solutions of $(\partial_x u)^2+(\partial_y u)^2=1$ for the function $u(x,y)$. Of course I know simple solutions like $u=ax \pm \sqrt{1-a^2}y + c$, or $u=\sqrt{x^2+y^2}+c$; but what's the ...
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Poisson Equation:Why the boundary regularity of the domain is important for the regularity of the solution?
Dear all,
giving a support class for PDE lecture i am wondering is there an easy argument for :
Why the boundary regularity of the domain important for the regularity of the solution of the weak form ...
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Regularity of solutions to a linear degenerate parabolic pde
I've encountered the following problem which is causing me some trouble :
Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function $u:...
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How to prove that 1 is not an eigenvalue of $T'(x)$?
Given a compact continuous operator $T$ from a Banach space $V_1$ to itself and $T$ maps a convex closed bounded set $\mathcal{B}$ into itself, how can we show that 1 is not an eigenvalue of $T'(x)$ (...
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A variational problem involving a negative fractional Soboblev norm.
I've run into the problem of trying to evaluate the following:
$\max_{ \xi} \iint_{\partial B \times \partial B} \xi(x) \Phi(|x-y|) \xi(y) dS(x)dS(y)$
subject to $\int_{\partial B} \xi(y)dS(y) = 0$ ...
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Solutions to the eikonal equation
Theorem. Let V be a $C^\infty$ function on a riemannian manifold $M$ and $p$ be a nondegenerate local minimum with $V(p)=0$. Then there is a unique positive function $\varphi \in C^\infty(U)$ such ...
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Help with what is most likely an easy PDE
Hi! I ran into this PDE working on a question in cake cutting. Here it is:
$x\partial_1f(x,y)-(1-x)\partial_2f(y,x)=0$
for all $(x,y) \in [0,1]\times[0,1]$.
Thanks!
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Regularity theory for "nice" differential equations
My current research took me to the realm of PDE's (which for the long time used to be terra incognita for me as I'am a probabilist). Equations that I'am working with are mostly of second order or ...
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Oscillatory integral decay & sublevel set growth
I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in this article by M. Greenblatt it says on page 7:
By well-known methods ...
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Boundary regularity of the solution to the Beltrami equation
Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference:
Let us consider the orientation-preserving homeomorphic solutions $f: D \...
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Hard-sphere gases and the wave equation
I'm trying to bridge a basic gap in my own education:
Where can I find a written discussion concerning the derivation of the wave
equation (for the propagation of sound, say), assuming nothing (much?...
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Elliptic regularity in Sobolev spaces of negative order
Consider 1 < $p<\infty$ and an integer $k$. Does interior elliptic regularity for the Laplacian also hold in the Sobolev space $W^{k,p}$ of negative order?
More precisely I am interested in ...
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Asymptotics of the TBA equation
The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is
$$\epsilon(\beta)=R\cosh\beta-\int\frac{...
- 188k
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Elliptic estimates and regularity of the $\overline{\partial}$-operator with totally real boundary conditions in $W^{1,p},$ $1<p\le 2$
Let $\Omega$ be an open subset of the upper half-plane in the complex plane. I am considering the following problem:
(1) $\overline{\partial}u=f,$ $\textrm{Im} f=0$ on the real line for maps complex-...
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meromorphic family of pseudo-differential operators
Let $(M,g)$ be a closed, Riemannian manifold. Let $S(z)$ be a holomorphic family of pseudo-differential operators, with $z \in \Bbb{C}$. Let $u$ be a smooth function. Does it follow that $\lim_{y \...
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Reference for Neumann-Laplacian
Let $\Omega\subset R^d$ be a bounded, smooth domain. Consider $A=-\Delta$ subject to homogeneous Neumann boundary conditions in $L^p$-spaces. Does anybody know a good reference book on basic results ...
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Solving a simple PDE on a 2 dimensional manifold
On a 2 dimensional surface X is a smooth vector field with isolated zeros. When is there a vector field Y such that the Lie bracket [x,Y] is equal to fY for some given function f on the surface minus ...
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Regarding Discrete Eigenvalues
For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete.
But, ...
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what is the essence of the blowup technique and bubbling analysis in PDE or geometric analysis?
when we use the blowup technique and bubbling analysis? what are the essence and pespective? could you give some examples to explain them or some reference？ thank you
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Best Poincare constants on the surface of a ball
I'm considering specifically functions $\xi:\partial B(0,1) \to \partial B(0,1)$ in $\mathbb{R}^2$ and $\mathbb{R}^3$ satisfying $\int_{\partial B(0,1)} \xi(y) dS(y) = 0$. I would like to know first ...
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Why is the harmonic oscillator so important? (pure viewpoint sought). How to motivate its role in Getzler's work on Atiyah-Singer?
I'm in the process of understanding the heat equation proof of the Atiyah-Singer Index Theorem for Dirac Operators on a spin manifold using Getzler scaling. I'm attending a masters-level course on it ...
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Minimum set of subharmonic function in $\mathbb R^n$
Let $f :\mathbb R^n\to \mathbb [0, \infty)$ be a (continuous, $C^2$, or smooth) subharmonic function with minimum value $0$. Then we know the sublevel set $f^{-1}((-\infty, c])$ is mean convex for $c &...
- 3,709
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vibrations of higher dimensional drums
The solutions of the wave equation for a circular drum with fixed boundary are well known. What do the solutions look like for a spherical drum in three spatial dimensions? What about higher ...
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Functionals continuous with respect to weak convergence
It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...
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Jet bundles and partial differential operators
A geometric way of looking at differential equations
In the literature for the h-principle (for example Gromov's Partial differential relations or Eliashberg and Mishachev's Introduction to the h-...
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A nonlinear system with special structure
Suppose we have an $n \times n$ uniform grid, covering $[-1,1] \times [-1,1]$ (typically, n $\approx$ 500). We have a smooth, differentiable function $z(x,y)$ that we want to determine on the nodes of ...
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System of first order PDE
I have a system of first-order nonlinear partial differential equations.
$$A(x,t)\frac{\partial u}{\partial t}(x,t) + B(x,t)\frac{\partial u}{\partial x}(x,t) + c(x,t) = 0$$
$$x \in \mathbb{R}, \quad ...
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Definition of spectral gradient
Consider this differential operator
$$
\mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x}))
$$
where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow \...
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Lie $2$-groups and differential equations
I was reading the abstract of a recent preprint (Division Algebras and Supersymmetry III by Juhn Huerta), and I wondered if something much simpler than what he was talking about had been worked on: ...
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How to estimate isoperimetric constant?
Suppose $(X^m, g)$ is a closed Riemannian manifold of dimension $m$ with the following properties,
There is a constant $\kappa$ such that $\kappa r^m \leq Vol(B(x, r)) \leq \kappa^{-1} r^m$
for every ...
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regularity of solutions of fractional laplacian
Hello, I am looking for boundary regularity of solutions of $(-\Delta)^s u= f(x)$ in $\Omega$ with Dirichlet boundary conditions and where $f $ is nice enough say $f\in C^{1,\alpha}(\overline\Omega)$. ...
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Test function .
For a smooth test function \eta and some constant C is it possible to have an estimate like the following?
|grad \eta|^2 < C {\eta}^2 ?
Thanks.
- 13k
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optimality of energy estimates for non smooth metric
Consider the linear (geometric) wave equation in dimension (3+1) with non smooth background metric $g$ say $g \in L^\infty_t H^3_x$ and $\partial_t g \in L^\infty H^2_x$, then energy estimates enable ...
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Embeddings for spaces of maximal regularity
Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true
$W^{s_1,p}(0,T;L^...
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decay for spatially discrete parabolic equations with non-constant non-self-adjoint right hand side
Consider the following uniformly parabolic lattice differential equation
$ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & &...
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Units of time in the gradient flow equation?
From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for ``time'' in the gradient flow equation?...
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Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient?
Hello,
I am considering the following non-linear heat equation
$$
\left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R_+\times R^d
$$
where $F(t,x)$...
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Does there exist a global solution in L^2 for reaction diffusion equation with focusing nonlinearity?
It is known that the solution of equation
$$
u_t - \triangle u = \kappa|u|^{\sigma}u, u(0) = u_0
$$
blow up in finite times if $\sigma > 0$. That is, the $L^{\infty}$ norm of solution $u$ will goes ...
CommunityBot
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characteristic surface
Hello,
I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ :
(1) $G_{xy}=0$
(2) $G_{xz}=0$
(3) $G_{yz}=0$
(4) $G_{xx}-G_{yy}=0$.
It is not hard to see that the general ...
- 3
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Geometric description of Jacobi's theorem on complete integrals of HJ eqn.
I am not sure if this question is adapted to this site, if it is not, then I will delete it.
The Hamilton--Jacobi theory is about the connection between:
the solutions of an Hamilton--Jacobi ...
- 4,306
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1
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263
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Convergence of elliptic operators
Let $A_t$ be family of second order, positive, elliptic differential operator mapping Sobolev $H^2$ of a compact smooth manifold (or bounded domain) to L^2. Suppose that the coefficients of $A_t$ ...
- 27.5k
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424
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A free boundary problem by finite difference method
I wanna discretize the following free boundary problem
Find $u$ and $\Omega$ such that $\Delta u=1-\delta_0$ in $\Omega$ with the conditions $u=|\nabla u|=0$ on $\partial \Omega$.
I apply finite ...
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