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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Heat equation bounds
I am interested in the following damped heat equation on $\mathbf{R}$, $u_t = u_{xx} - 1_{x \in [-1,1]} u$ with initial data $u(0,x) = \delta(x-x_0)$ for some $x_0 \in \mathbf{R}$.
In particular I am ...
CommunityBot
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On a family of $C^0$-convergent Riemann metrics
I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself.
Suppose that $M$ is a smooth compact manifold of dimension $m$ and $...
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22
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1
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Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
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the inverse for the trace theorem
The trace theorem says that the restriction of a $W^{1,p}(\Omega)$ function $u$, $Tu$ belongs to $W^{1-1/p,p}(\partial\Omega)$ if $\Omega$ satisfies some smooth condition, for example, $\Omega$ is ...
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Lower bound on the solution of a Schrödinger-type equation
Consider the equation
$-\Delta u + Vu=f$,
on a closed manifold (or on a bounded domain with homogeneous Neumann condition). Here one can assume whatever integrability or smoothness conditions on $V$ ...
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2
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1
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The perturbed KdV Equation
I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...
CommunityBot
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2
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Seminorms in sharp Garding's inequality
When working with symbols of limited regularity, what is exactly the number of seminorms of the symbol $Re (a) \geq 0$ that one needs in the sharp Garding's inequality:
$Re \langle a(x, D) u, u \...
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1
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1
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$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?
I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
...
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2
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Does there exists a necessary condition for Lp multiplier?
Let $1 \leq p \leq 2$. A measurable function $m(\xi)$ is called a $L^p(R^n)$ ($L^p$ for convenience) multiplier, if $$\|m(D)\varphi\|/\|\varphi\|_{L^p} \leq C , \varphi \in L^p
$$ for some constant $C$...
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Is $C_c(\mathbb{R}^2,\mathbb{R}^2)$ dense in the irrotational square integrable functions?
Let $L_D(\mathbb{R}^n)^n$ be the set of square integrable functions which are the weak derivative of a locally square integrable function. That is
$$L_D(\mathbb{R}^n)^n=\{Du\colon u\in H^1_{loc}(\...
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Existence of cut-off functions in metric spaces
Let $X_1,\ldots,X_m$ be Lipchitz continuous vector fields in an open set $\Omega \subset \Re^n$, Let $d(\cdot, \cdot)$ denote the control distance associated to $X$. With respect to this control ...
CommunityBot
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partial differential equations with mixed boundary conditions
hi,
does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ?
actually I am intrested in the following: Let $f(x)=(f_{1}(x),...,f_{n}(...
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2
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1
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Regularity of harmonic functions with robin data up to the boundary
I want to prove that if $u$ is a solution of
$\Delta u = 0$ in $\Omega$ with Robin boundary conditions $\frac{\partial u}{\partial n} = \lambda u$, where $\Omega \subset \mathbb{R}^n$ has analytic ...
CommunityBot
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2
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Does these commutator estimates bound in $L^{2}$
According to the basic rules of symbolic caculus,$[a(x,D),x_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x_i]=\partial_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded.
It's also ...
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Smooth Sobolev extension from $W^{1,p}(U)$ to $W^{1,p} (\mathbb{R}^n) $
The question I would be asking is roughly : do the smooth Sobolev functions defined on an open bounded domain extend to smooth Sobolev functions on the Euclidean space ?
For detail :
Fix $p \geq 1. ...
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Density of $H^{1/2}(\partial \Omega)$ in $L_2(\partial\Omega)$
Hi,
i know that the following statement is used extensively, but i cannot find a proof anywhere:
For $\Omega$ a Lipschitz domain with boundary $\Gamma$, the space $H^{1/2}(\Gamma)$ is dense in $L_2(\...
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1
answer
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sobolev embedding theorem in the smooth metric measure space
we know the sobolev embedding theorem of Saloff-Coste
$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu
$
wtih $Ric\ge-(n-1)K$, for ...
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0
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elliptic system; bounds on $v$ when $u$ is small
I am interested in the following system
$-\Delta u = f(u,v) $
$-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary.
The solutions are smooth and positive. ...
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Pólya's conjecture on the spectra of the Laplacians
Recently I've learned something about the spectra of the Laplacians. Given a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ smooth, we can consider eigenfunctions of Dirichlet ...
CommunityBot
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2
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Monge Ampere equations (concavity)
The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\...
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Which PDE from physics (and geometry) are supercritical?
I am currently trying to understand the notion of criticality (as discussed, e.g., in Terence Tao's book on nonlinear dispersive equations) from a physical viewpoint. That's why i'm interested in the ...
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2
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hodographic transformation
Let $\phi(x,t)$ be smooth function.
Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$.
Let $u$, $v$ satisfies following equations:
1-
$$u_\eta- v_\zeta= 0$$
$$v^2u_\zeta-(1+2uv)u_\...
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3
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Distributional derivative is locally integrable - then the distribution as well?
Given a distribution $T \in D'(\mathbb{R})$ such that the distributional derivative $\partial T \in L^1_{loc}(\mathbb{R})$. Can one deduce that $T \in L^1_{loc}(\mathbb{R})$ as well? Or can anyone ...
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Strichartz estimates over cones
I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality:
$\|...
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Moser regularity proof avoiding John-Nirenberg lemma
I heard a rumor that there exists a proof by Moser-style iteration of the $C^{0,\alpha}$-regularity for $W^{1,2}$-solutions $u$ to elliptic equations with measurable coefficients which does not rely ...
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Weak maximum principle / comparison principle for parabolic equations with Neumann conditions
Hello everyone. I'm about to use a comparison principle that I belive is true, but I can't find any precise reference to be sure of it. Here is what I would like to say :
I have a parabolic equation $...
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Eliminating 1st order terms in elliptic partial differential equation
Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves the 2nd derivatives,...
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ellipticity independent of metric?
I am new to the theory of pseudo-differential operators on compact manifolds, but I need to use a result related to this theory in a proof I'm working on. The problem is as follows: Let $(M,g)$ be a ...
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Mean value property with fixed radius
Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e.
$$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ ...
CommunityBot
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a question about Lp norm of curvature on convex curves
Suppose we have two strictly convex closed curves $C_{1}$ and $C_{2}$, $C_{1}$ contains $C_{2}$,
then can we conclude $\int_{C_{1}} \kappa_{1}^{p} ds\geq \int_{C_{2}} \kappa_{2}^{p} ds$, $\kappa_{1}$ ...
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A limit involving a regularizing kernel
I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#
...
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Methods for determining domains of influence
Given a hyperbolic PDE, the domain of influence of a spacetime point $x$, say $I_x$ though $x$ could be replaced by any set, can be defined in two ways. Lets call one of them geometric ($I_x^G$) and ...
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A slightly subcritical elliptic equation on the ball; blow-up behavior near zero
I am interested in positive ground state solutions of the following elliptic pde:
$-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ p:=\...
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Seeking scalar functions in n>=2 variables (preferably as solution to PDE) with limited regularity.
I have bumped into a phenomenon in the geometry of jet space $J^r(\mathbb{R}^n,\mathbb{R})$ for $r,n\geq 2$ that I think might help one measure and understand the failure of regularity of functions, ...
CommunityBot
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Asymptotics of quasilinear elliptic equations with Dirac right hand side
On a small open neighborhood $U$ of $0 \in \mathbb{R}^n$, consider the quasilinear (possibly monotone) elliptic, scalar PDE in divergence form
$$\nabla \cdot(a(x,u,\nabla u)\nabla u) = \alpha \...
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Maximum principle for heat equation on infinite domain
Let $u(x, t)$ be a solution of $u_t=u_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in ...
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Integrability of ground state solution for elliptic equation
For the solution of semi-linear elliptic equation, for example I'm considering the 2D cubic nonlinear Schroedinger equation, the correspongding elliptic equation is $\Delta u+u^3=u$, with $u>0$. By ...
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ellipticity and invertible differential operators
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $P$ be a $2r$th order pseudo-differential operator, where $r \in \Bbb{R}_+$. Suppose that the differential equation $Pu=f$ has a unique $H^r(M)...
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Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
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regularity of solution of linear elliptic PDE
I am interested in the boundary regularity of solutions of $ L(u) = f(x) \ge 0$ in $ \Omega$ with zero Dirichlet boundary conditions, here $L(u) = (-\Delta)^\frac{\alpha}{2}$ where $ 0 < \alpha &...
CommunityBot
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Is this Stefan-type problem an open problem?
I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
1
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1
answer
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Reparametrizing characteristic curves for PDE's
I'm looking for solutions for a PDE that looks like this
$$
\nabla u(\vec x) \cdot f(\vec x) = k.
$$
For some clarification, $u$ is a log-probability. And this arises from a Fokker-Plank-like equation,...
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The ground state is signed and symmetric
Background
In Berestycki and Lions it is asserted that (on page 316), if I am not misreading, that the "ground state", i.e. action minimizer among nontrivial solutions, corresponding to the action
$$...
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Nonlinear PDE ${u_{tt}}^2u_{ttxx} = 1$
I have been trying to solve this equation during fortnight
$$
{u_{tt}}^2u_{ttxx} = 1.
$$
But I still here. The only thing is change of variables $u_{tt}(t,x) = y(t,x) $ and solved the ODE $y'' = \frac{...
- 1,687
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420
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Variational Formulation of Boundary Value Problems With Unknown on the boundary
Suppose that we have a linear operator equation on $\Omega$ with Lipschitz boundary $\partial \Omega$,
\begin{eqnarray}
Lu &=& \frac{\partial u}{\partial t},
u(x,0) &=& u_0 \; \; \...
CommunityBot
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3
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Imaginary exponential functional of Brownian motion
Thanks to the work by M. Yor and colleagues, much is known about the following exponential of Brownian motion:
$X= \int_0^{\infty}{\rm d}t \ e^{-t + g \ B(t)}$
where $g$ is a real scale parameter.
...
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0
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Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
CommunityBot
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3
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Is there a PDE for this phenomenon?
At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface.
Is there a physical law - like the heat equation - that describes the flow?
Will ...
- 8,336
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0
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Compactness of solutions to parabolic equations (parabolic regularity)
I am working on a Ricci flow $(\mathcal{M} , g(t))$, for which the conjugate heat operator is $\Box ^* := - \partial _t - \Delta + R$, where $R$ is the scalar curvature.
For each $s>0$, I have a ...
- 1,123
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Results about existence/uniqueness of solution to Euler-Lagrange equations?
While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
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