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

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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.
0
votes
1answer
181 views

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

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
183 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
277 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
281 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
98 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
387 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
140 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
272 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
641 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
154 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
312 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
299 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
253 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
154 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
122 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
232 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
491 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
609 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
468 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 ...
0
votes
1answer
180 views

On the fractional Schrödinger equation

I wonder if there is any theory about what we can call the fractional Schrödinger equation: $$ \mathrm{i}\frac{\partial \psi}{\partial t} = (-\Delta)^s \psi + g(|\psi|^2)\psi \quad\hbox{in ...
3
votes
1answer
364 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...
3
votes
1answer
168 views

Does $\partial\overline{\partial}f=0$ imply $f\equiv c$ for particular kind of $f$?

Hi! Let $f\in C^{2,\alpha}\left( \mathbb{C}^{m}\setminus \overline{B_{R}},\mathbb{R} \right)$ with $m\geq 2$, $R>0$ and s.t. $f$ has an expansion of type $$f=1+\mathcal{O}\left( \frac{1}{|z|} ...
1
vote
0answers
180 views

Galerkin method for existence for PDE with nonsymmetric bilinear form

Suppose we have a PDE $$\langle u', v \rangle + a(u,v) = 0$$ where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in ...
2
votes
2answers
323 views

Replacing large-dimensional ODE systems with one PDE [closed]

Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?
2
votes
1answer
171 views

Manifold structure for the set of solutions to a first order elliptic system?

Consider a bounded domain $S\subset R^2$ and an elliptic system of two first order PDEs, namely a generalization of the Cauchy-Riemann system allowing nonconstant coefficients and lower order terms. ...
1
vote
0answers
180 views

Stationary Phase and Propagation Speed (Reference)

I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if ...
23
votes
3answers
1k views

“Wild” solutions of the heat equation: how to graph them?

It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional assumptions. In ...
18
votes
10answers
4k views

Open problems in PDEs, dynamical systems, mathematical physics

(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.) I am an undergrad in math ...
1
vote
1answer
220 views

In which way is this a linearization of the Gross-Pitaevskii-Equation?

In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation $0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - ...
0
votes
1answer
77 views

Integral of a harmonic function on a manifold with two non-parabolic ends

Let M be a complete Riemannian manifold.Suppose there are two non-parabolic ends on M with respect to $M\backslash {B_p}\left( {{R_0}} \right)$Then there is a harmonic function f on M.Is it right that ...
2
votes
0answers
87 views

P-laplacian equation

Hi guys, in what sense the p-Laplacian is degenerate for p greater than 2 and singular for p smaller than 2 ? Thank you!
10
votes
1answer
537 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
2
votes
0answers
187 views

Extension divergence-free, curl-converging vector field

Hi. Consider a smooth open Set $\Omega\subset\mathbb{R}^3$ and a bounded sequence of vector fields $(u_n)_n \in L^2(\Omega)$ having $0$ divergence. I know how to extend this sequence to the whole ...
22
votes
1answer
1k views

What goes wrong for the Sobolev embeddings at $k=n/p$?

For $u\in W^{k,p}(U)$, where $U\subseteq\mathbb{R}^n$ is open and bounded with $C^1$-boundary, we have the celebrated Sobolev inequalities: If $k < n/p$ then $u\in L^q(U)$ for $q$ satisfying ...
1
vote
0answers
82 views

Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$. Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
2
votes
2answers
243 views

Regularity of parabolic equation; divergence free drift

I am curious about weak solutions of the parabolic problem $ u_t - \Delta u + a(x,t) \cdot \nabla u = f(x,t)$ in $ \Omega \times (0,\infty)$ with $ u=0$ on $ \partial \Omega$. Here $a(x,t)$ is ...
1
vote
2answers
179 views

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 $ ...
1
vote
0answers
164 views

multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\left | \sigma \right |\leqslant ...
0
votes
1answer
133 views

Higher dimensional analogue of Kellog's theorem? (Holder continuity of solution to Dirichlet problem with Holder continuous boundary data)

Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following Theorem: ...
8
votes
1answer
302 views

Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear ...
0
votes
0answers
144 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
3
votes
1answer
156 views

Coarea formula in a subelliptic context

Consider smooth vector fields $X_1,..,X_k$ in ${\mathbb R}^n$, satisfying the H\"ormander condition, i.e. for all $x$, the Lie algebra generated by $X_1(x),...,X_k(x)$ is ${\mathbb R}^n$. Do you know ...
2
votes
1answer
151 views

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?

How to compute the first eigenvalue of hyperbolic space ${H^2}$and ${H^n}$?
0
votes
1answer
318 views

Existence of a function

[also asked here http://math.stackexchange.com/questions/307197] All arguments are in $\mathbb{R}^3$. Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. ...