Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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3
votes
2answers
152 views

Boundedness of Solutions to $\Delta u = f u$ on $R^2$

Consider the Laplacian $\Delta = d/dx^2 + d/dy^2$ on $\mathbb{R}^2$. This is true: Let $f$ be a nonnegative function, not identically zero. Then any positive solution of $\Delta u = f u$ is ...
2
votes
1answer
276 views

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 ...
1
vote
1answer
165 views

If a function is defined in terms of a solution to an initial value problem, is it also solution to an initial value problem?

Say $f:\mathbb R^{n+1}\to \mathbb R^p$ is a solution to an initial value problem, and $g:\mathbb R^{n+1}\to \mathbb R^q$, so that the components of $g$ can be expressed as polynomials in $f$, $f'$, ...
0
votes
2answers
266 views

Estimates for Green's function

Let $n$ - dimension $\geq 3$. Consider a compact manifold (M,g). Let $\epsilon_0$ denote the injectivity radius of $(M,g)$. Let $B_\epsilon(0)$ denote a geodesic ball of radius $\epsilon < ...
3
votes
1answer
242 views

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.
4
votes
2answers
247 views

Analytic dependence on the metric

It is often used implicitly that the maps which associate to metrics curvature quantities (Riemann, Ricci, scalar curvature) and Differential operators like the Laplacian are analytic maps between ...
0
votes
1answer
159 views

Solution formular for Laplace equation [closed]

I want to slove the Laplace equation on $R^3_+$ with Neumann boundary condition. The equation reads: $-\Delta u = f$ in $R^3_+$, $\partial_3 u|_{x_3=0}=g$ on $R^2$. If $f$, $g$ satisfy compatibility ...
10
votes
4answers
1k views

Is there anyway to rewrite a partial differential equation using language of differential forms, tensors, etc?

My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there anyway (or any algorithm) ...
4
votes
1answer
497 views

Exact solutions to nonlinear Klein-Gordon equation

The nonlinear pde $$ \partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0 $$ has the exact solution $$ \phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i) $$ ...
1
vote
0answers
70 views

h-oscillating function

I need help understanding the following condition: $u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
0
votes
1answer
180 views

Dual space of Bochner space: is there an easier proof to show they're isometric?

It is known that $[L^p(0,T;H)]^* = L^q(0,T;H^*)$. If $p=q=2$ and $H$ is a Hilbert space, is there an easier proof to show that the spaces are isometric? The proof that I know for the general case ...
0
votes
0answers
162 views

Gradient estimates for subsolutions of elliptic equations

Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and $\Delta u + \lambda u = 0,$ where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
0
votes
1answer
423 views

Strong convergence in the Bochner space L^p([0,T],X)

Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$. Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let $x_n(t)\in X_1$ be ...
1
vote
1answer
117 views

$C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$?

Is it true that the space $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? These are compactly supported functions that are $V$ valued, where $V$ is a Banach or Hilbert space.
3
votes
2answers
341 views

analysis question related to $L^p$ type inequalities

Dear mathoverflowers. Just wondering if the following inequality is true. For all $ p >1$ there is some $C$ such that $ | |x+1|^p-|y+1|^p -p(x-y)| \le C ( |x|+|y| + |x|^{p-1} + |y|^{p-1} ) ...
5
votes
1answer
253 views

Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$

How is the proof that $$[L^2(0,T;X)]' = L^2(0,T;X')$$ looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^*)$. Is the ...
1
vote
0answers
211 views

Laplacian type operator on compact Lie group

Consider the operator $S = \sum_{i,j} X_{ij}^2$ on $L^2(SO(n+1))$, where $X_{ij}$ generates the rotation of the sphere $S^n$ in the $ij$-plane keeping the $(n - 2)$ complementary directions fixed. ...
1
vote
1answer
173 views

A heat kernel for Schrödinger operator with low-order terms

In "Schrödinger Operator: Heat Kernel and Its Applications", Feng computes the heat kernels associated to Schrödinger operators with at most quadratic potentials. I am trying to see how these work ...
0
votes
0answers
86 views

$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$

Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in ...
3
votes
2answers
704 views

Regularization by mollifier sequence

A well-known feature used in PDE's is the regularization by convolution with a mollifier sequence $\rho_n$, i.e. $\rho_n(x) := n^d \rho(nx)$ with $x \in \mathbb R^d$, $\rho \in C^\infty_c(\mathbb ...
0
votes
3answers
455 views

I have this linear PDE…

Hi, The PDE in question is: $A P_{yy}(y,z) + B P_{zz}(y,z) + ( [ C y -D z] P(y,z) )_y + ( [ D y + C z ] P(y,z) )_z=0,$ where subscript $y,z$ indicates derivatives and $A,B,C,D$ are real. The PDE is ...
1
vote
0answers
115 views

optimal regularity for Laplace equation with inhomogeneous L^p Robin boundary condition

Consider the problem $$-\Delta u = 0 \mbox{ in }\Omega,\qquad \partial_\nu u+\tau u=g\mbox{ on }\partial\Omega,$$ where $\Omega\subset R^n$ is a bounded $C^2$-domain, $\tau>0$ is a constant, and ...
1
vote
0answers
113 views

(localized) L^2 norm of quasimode for Laplacian

Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$: $u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...
3
votes
1answer
745 views

integration by parts for the fractional Laplacian

Is there an integration by parts formula for fractional laplacians in $L^p(\mathbb{R}^N)$, something like $$ s\in(0,1),\qquad\int\limits_{\mathbb{R}^N}f[(-\Delta)^sg] ...
1
vote
0answers
91 views

null controllability of linear wave equation

Consider the linear wave equation : $$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$ Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...
33
votes
2answers
1k views

Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
1
vote
0answers
114 views

nodal lines in the dirichlet problem

In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues? Thanks for help.
-1
votes
1answer
201 views

Concerning Fritz John's article, The Ultrahyperbolic Differential Equation With Four Independent Variables [closed]

I am trying to read Fritz John's article, here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.dmj/1077490637 And for the proof of Thm 1.1 in page ...
3
votes
1answer
189 views

Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of differential forms ...
1
vote
1answer
279 views

maximum principle for a non-uniformly parabolic operator

Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator $$ P = \partial_t - \mathrm{e}^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x} \big( ...
3
votes
1answer
294 views

What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
1
vote
0answers
101 views

About definition of weak derivative in abstract PDE problems

I'm confused about weak derivative definition. $u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff $$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$ holds for all $\varphi \in ...
1
vote
0answers
58 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
7
votes
1answer
390 views

Representing immersions from a surface into 3-space

Let $\mathbb T^2=(S^1)^2$ be the 2-torus, for convenience. $\def\Imm{\operatorname{Imm}}$ Consider the Frechet manifold of immersion $\Imm(\mathbb T^2, \mathbb R^3)$ and the smooth mapping ...
2
votes
1answer
142 views

Asymptotic decay for the wave equation

Consider the wave equation $$ y_{tt} = \Delta y - \epsilon y_t $$ on $\Omega\subset R^n$, with Dirichlet boundary conditions. Where $\epsilon >0$. Is it possible to find an explicit value ...
2
votes
1answer
287 views

Trace theorem for manifolds with boundary

Can I get a reference to a trace theorem for a manifold $M$ with boundary $\partial M$, and I am hoping the inequality $$\lVert Tu \rVert_{L^2(\partial M)} \leq c\lVert u \rVert_{H^1(M)}$$ will hold. ...
12
votes
2answers
662 views

Applications of pseudodifferential operators to PDE

I am planning to build a PDE course centred around pseudodifferential operators. I know some important applications of pseudodifferential operators to PDEs, but I don't know enough to get the whole ...
2
votes
1answer
135 views

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
vote
1answer
155 views

Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then ...
2
votes
3answers
317 views

Nonharmonic solutions of Laplace's equation

Let $f \colon U \to \mathbb{R}$ be a twice differentiable function, where $U$ is an open subset of $\mathbb{R}^n$. Here twice differentiable means that all the second partial derivatives ...
1
vote
0answers
308 views

Solving a PDE involving a mixed derivative for a partial derivative

Consider a PDE of the form \begin{equation} \frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right) \end{equation} or \begin{equation} \frac{\partial^2u}{\partial ...
2
votes
1answer
256 views

Is there a lower bound for variance in terms of curvature?

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f)$ of $f$ is nonzero ...
1
vote
1answer
156 views

weak*closure of {f:||f||=1} in dual.

What is the weak* closure of {f:||f||=1}? I am sure this set is not closed in weak* topology. So what is the weak* closure of this set. Thanks.
0
votes
0answers
124 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
4
votes
1answer
236 views

Sharpness of the Sobolev embedding theorem

We know that $W^{k,p}\hookrightarrow C^{k-\lfloor\frac{n}{p}\rfloor-1,\gamma}(\bar{\Omega})$ with $kp>n,\gamma=\lfloor\frac{n}{p}\rfloor+1-\frac{n}{p}$, where $n$ is the dimension of $\Omega$, ...
2
votes
2answers
500 views

Using Galerkin method for PDE with Neumann boundary condition?

I am wanting to show existence of solutions to $$u_t +L(u) = f \;\;\text{on}\;\; \Omega$$ with initial condition $u|_{t=0} = u_0$ and Neumann boundary condition $\nabla u\cdot \nu = 0$ on ...
1
vote
2answers
524 views

$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
4
votes
1answer
625 views

Density of smooth functions in Sobolev spaces on manifolds

Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have ...
1
vote
0answers
124 views

Trace Inequality question

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
1
vote
1answer
478 views

Showing a singular integral operator takes Holder continuous functions to Holder continuous functions (of the same order)

I would like to show the following function is $\gamma$-Hölder continuous. Said function $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by a singular integral operator of convolution type as ...