User dorian - MathOverflow

A Hölder continuous function which does not belong to any Sobolev space

2010-09-15T00:24:19Z

I'm seeking a function which is Hölder continuous but does not belong to any Sobolev space.

Question: More precisely, I'm searching for a function $u$ which is in $C^{0,\gamma}(\Omega)$ for $\gamma \in (0,1)$ and $\Omega$ a bounded set such that $u \notin W_{loc}^{1,p}(\Omega)$ for any $1 \leq p \leq \infty$. Take $\Omega$ to be bounded, open.

My first guess is to do a construction with a Weierstrass function. I know this is differentiable 'nowhere' but that doesn't convince me it isn't weakly differentiable in some bizarre way. Hopefully someone knows of an explicit example.

What kind of Lagrangians can we have?

2010-09-15T14:42:54Z

In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of non-interacting particles has a Lagrangian $\frac{1}{2}(m_1\dot{x}_1^2 + \cdots + m_n \dot{x}_n^2)$. It is then generally argued that $L=T-U$. I feel like something is missing here.

What exactly are the physical hypotheses that go into this? Can we have other forms of the Lagrangian? How do we know those are "right"? Do we always have to compare them to the form of the equations we derived previously? For example, the Lagrangian formalism seems to be justified usually in so far as it 'works' for a finite collection of particles. Then you can solve any dynamics problem involving a collection of particles.

I have been vague so let me try to be more precise in my question. Is the principle of least action an experimental hypothesis? Is it always true that $L=T-U$? When we don't know what the Lagrangian is, do we have to just guess and hope it is compatible with the dynamical equations we had already? Or can we perhaps start with the ansatz of a Lagrangian in some cases?

I hope this is sufficiently precise.

Bonnesen's inequality for non-simple curves

2012-06-29T15:10:54Z

Given a closed curve in the plane $\mathbb{R}^2$, it is well known that $L^2 \geq 4\pi A$ where $L$ is the length of the curve and $A$ is the area of the interior of the curve.

For a simple closed curve $\gamma$, the stronger inequality due to Bonnesen holds: $L^2 - 4\pi A \geq \pi^2 (R_{out}- R_{in})^2$, where, setting $\Omega =$ Int $(\gamma) $, $ R_{in}$ and $R_{out}$ denote the inner and outer radii of the sets:

$R_{in} = \sup_{B_r \subset \Omega} r$

$R_{out} = \inf_{\Omega \subset B_R} R$

Question: Does this inequality continue to hold without the assumption that the curve is simple? In particular, does it hold for any connected, rectifiable set?

If one adds components to the interior of a simple curve, it is clear that this increases the isoperimetric defecit. However while $R_{out}$ will remain unchanged, $R_{in}$ will necessarily decrease, making it not immediately clear that the inequality would continue to hold.

Update: I just realized this is false. Take a countable union of points in the interior of the unit ball. Around the kth point, make a circle of radius $\epsilon/2^k$. Distribute the points well enough and $R_{in}$ can be made arbitrarily small. Then if you make $\epsilon$ small enough, you don't change the length or area too much and so the desired inequality will be violated.

Generalization of an inequality due to Gage for curve shortening Part II

2012-06-06T21:20:45Z

I asked a question recently about generalizing an inequality due to Gage. This inequality asserts that given a convex domain $\Omega$ in $\mathbb{R}^2$ with support function $p(X) = \langle X, \nu \rangle$ where $X$ is the position vector with respect to the origin and $\nu$ is the normal on $\partial \Omega$ that

$\int_{\partial \Omega} p^2 dS \leq \frac{LA}{\pi}$, holds for some choice of origin where $L$ is the length of the curve and $A$ the area.

I have been trying to prove the following generalization for star shaped sets:

$\int_{\partial \Omega} p^2 dS \leq \int_{\partial \Omega^*} (p^*)^2 dS^* + C(L,A)(A^*-A)$ ,

where here $\Omega^*$ is the convex hull of $\Omega$, $p^*$ is the support function of the convex hull, $A$ is the area of the set and $A^*$ the area of the convex hull. Using Green's theorem I have been able to deduce the following: $\int_0^{2\pi} p(\theta)^2 \frac{dS}{d\theta}{d\theta} = \int_0^{2\pi} p^*(\theta)^2 \frac{dS^*}{d\theta} dS^* + \int_0^{2\pi} p^*(\theta)p(\theta)( \frac{dS^*}{d\theta} -\frac{dS}{d\theta}) d\theta + \frac{1}{2} \int_{0}^{2\pi} [p(\theta)^*-p^(\theta)][r^*(\theta)^2 - r(\theta)^2] d\theta$ ,

where $r^*$ and $r$ are the lengths to $\partial \Omega^*$ and $\partial \Omega$ respectively and we can paramateterize by $\theta$ since the set is star shaped. I thus need only control the term

$\int_0^{2\pi} p^*(\theta) p(\theta)(\frac{dS^*}{d\theta} - \frac{dS}{d\theta}) d\theta$

but this is not clear at all to me. My intuition suggests it should be negative.

I would appreciate any ideas or suggestions for alternative approaches to this.

An isoperimetric type maximization problem with a barrier.

2012-06-07T13:55:31Z

I'm trying to minmize a particular functional which depends on a curve with fixed endpoints which lies below a fixed line in $\mathbb{R}^2$. Here are the details:

Let $(r(\theta), \theta)$ be a smooth segment of a curve in polar coordinates satisfying the following for $0 < \theta_1, \theta_2 < \pi$.

1) $r(\theta_i) \sin (\theta_i) = \lambda$ for $i=1,2$ where $\lambda$ is a fixed constant.

2) $r(\theta) \sin(\theta) \leq \lambda$ for all $\theta \in [\theta_1, \theta_2]$.

Thus $(r(\theta),\theta)$ is a curve which touches the line $y=\lambda = $ constant at its two endpoints and lies beneath this line for all other value of $\theta$. Let $\mathcal{A}$ denote the class of curves $(r(\theta),\theta)$ satisfying conditions $1)$ and $2)$. Then consider the maximization problem:

$\max_{ (r,\theta) \in \mathcal{A} } \int_{\theta_1}^{\theta_2} \frac{1}{\sin^2(\theta) \sqrt{1+(\dot r/r)^2} } d\theta$ .

Geometrically speaking, the above can be written as $\int_{\theta_1}^{\theta_2} \frac{d\theta}{dS} \frac{r^2}{\sin^2(\theta)} d\theta$ , which makes it appear as a sort of inverse perimeter problem. Clearly the curve would like to follow the trajectory of a circle, but due to the constraint it cannot do this everywhere. My suspicion is that the solution should be attained by $r(\theta) = \frac{\lambda}{\sin \theta}$ but it's not clear that it would not be favorable for the curve to descend below this line if it can follow the path of a circle for some rangle of angle.

If this type of problem is known then references would also be appreciated.

Generalization of an inequality due to Gage for curve shortening

2012-06-03T17:22:22Z

There is a well known inequality due to Gage which asserts the following. Let $\Omega$ be a smooth, convex set in $\mathbb{R}^2$ and let $p = \langle X, \nu \rangle$ be the support function of $\Omega$, where $X = \langle x, y \rangle$ with respect to some origin $O$ and $\nu$ is the normal to the boundary.

Denoting $A$ and $L$ and the area and length of the curve, then it holds that $\int_{\partial \Omega} p^2 dS \leq \frac{AL}{\pi}$ for some particular choice of origin $O \in \Omega$.

Question/Conjecture: Given an arbitrary simply connected, smooth set $\Omega$, does it hold that $\int_{\partial \Omega} p^2 dS \leq \frac{LA^*}{\pi}$ where $A^*$ denotes the area of the convex hull of $\Omega$ and $L$ is the length of the original boundary $\partial \Omega$.

Update June 04/2012: There has been an answer to my original question so I would like to ask if a related although weaker assertion is true. Let $\partial \Omega$ be paramaterizable by the angle $\theta$ in polar coordinates, so that the curve is represented by $(r(\theta),\theta)$. Then $p = p(\theta)$ is obviously single valued. This means precisely that the domain $\Omega$ is star shaped. Let $p^*$ be the support function of the convex hull $\Omega^*$. Does it hold that $\int_{\partial \Omega} p^2 \leq \int_{\partial \Omega^*} (p^*)^2 dS$ ?

Any direction to references on related questions would also be greatly appreciated.

Different forms of Bonnesen's strong isoperimetric inequality in the plane.

2012-04-12T17:41:07Z

I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve:

$(R_{out} - R_{in})^2 \leq \pi^2 (L^2 - 4\pi A),$

where $R_{out}$ and $R_{in}$ are the outer and inner radii respectively, $L$ is the length of the curve and $A$ is it's area.

This form turns out to often be quite inconvenient for me, and I was wondering if there was a version which read as follows:

$R_{out}^2 - R_{in}^2 \leq C (L^2 - 4\pi A)$,

where now $C$ is a constant which may depend on the support of the curve or other quantities possibly. The main difference is that when $R_{in} - R_{out}$ is small, the second inequality I've written is strictly stronger if one assumes the curve is constrainted to a bounded domain. The sacrifice I make though is that I do not care what the constant is, or if it depends on the support of the curve.

This inequality seems quite reasonable but I have thus far been unable to prove it. Has anyone encountered such a version of this inequality?

Answer by Dorian for When is Sobolev space a subset of the continuous functions?

2010-09-05T02:48:36Z

I'll give you a hint for the first one $d=1$. Consider first the case that your function $f \in H^1([0,1])$ was smooth. Then we could say $f(x) - f(y) = \int_{x}^y f'(s)ds$. Apply Cauchy Schwartz now and you'll be able to see immediately that $f$ is $1/2$ Hölder continuous.

For higher dimensions you actually proceed similarly but you need to use the co-area formula.

What are the most general types of curves in $\mathbb{R}^2$ for which Gauss-Bonnet holds?

2012-05-24T17:23:38Z

I would like to know what is the most general form of the Gauss-Bonnet theorem in the plane for curves. It is well known for that for any piecewise $C^2$ simply connected curve with corners, one has

$\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$,

where $\kappa$ denotes the curvature of the boundary $\partial \Omega$. This formula continues to hold for any curve for which $\hat n \cdot (1,0)$ (for instance) defines a $BV$ function.

Question: I would like to know to what extent this generalizes. For example, if $\Omega$ is a simply connected, compact set in $\mathbb{R}^2$ whose boundary has finite length and has generalized curvature $\kappa \in L^{\infty}(\partial \Omega)$, does it hold that $\int_{\partial \Omega} \kappa(y) dS(y) = 2\pi$? If not, are there counter examples in the plane where this fails?

Is there a characterization of generalized constant mean curvature surfaces?

2012-04-21T10:48:39Z

It is a well known result of Alexandrov that the only compact, connected, constant mean curvature surface is the ball. There is a generalized notion of curvature known as generalized mean curvature which makes sense on rectifiable varifolds. In particular if $V$ is a recifiable varifold than the generalized mean curvature is defined as follows:

Let $\|\delta V\|$ denote the first variation of $V$. Then if this measure is absolutely continuous with respect to $\|V\|$ then by the Riesz representation theorem:

$ \delta V(g) = - \int g \cdot H d\|V\|$, for $H$ defined $\|V\|$ a.e.

H as defined here is the generalized mean curvature.

Question: Working in the plane, $\mathbb{R}^2$, assume that $\Omega$ is a set of bounded variation so $|\partial \Omega| < C < +\infty$ and define $H$ as above.

If $H=$ constant, what does this imply about $\Omega$? Must $\Omega$ be a ball? Are there examples of singular $BV$ sets which satisfy this condition but are not balls?
What is the generalized mean curvature of a square?

What is the constant in the rate of exponential convergence for mean curvature flow?

2012-03-01T18:25:45Z

Given a domain $\Omega \subset \mathbb{R}^d$ which is convex and smooth and $| \Omega|=1$, it is well known that the metric converges exponentially fast to that of the sphere under volume preserving MCF. I would like to know the following:

Question: Does the rate of convergence depend on the domain $\Omega$? If so, in what way? If I know in particular that $\frac{d}{dt} \|\kappa - \bar \kappa\|_{L^2(\partial \Omega)}^2 \leq -C \|\kappa - \bar \kappa\|_{L^2(\partial \Omega)}^2$ , can I say that $C$ is independent of $\Omega$? If not, can I see in which explicit way it does depend on $\Omega$? I suppose this is equivalent to asking if there is a particular $\Omega$ so that the convergence is slowest.

Thanks.

Stronger version of the isoperimetric inequality

2011-02-14T12:44:06Z

I have been searching for a version of the isoperimetric inequality which is something like:

$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind of sets this applies to (clearly connected and possibly simply connected). I was hoping somebody may recognize this inequality and be able to direct me to a source for it along with a proof.

Update: I'm curious if anyone can direct me to a some papers which relate the isoperimetric deficit to Hausdorff distance. Such as: $P(\Omega)^2 - 4\pi Vol(\Omega) \geq C d_H(\Omega,B)^2$ whre $B$ is a sphere in $\mathbb{R}^2$ which may be the inner or outer circle.

Update April 12: I would like to know if the first Bonnesen inequality written below is strictly stronger than the one in higher dimensions? In particular, if one considers the Fraenkel assymetry $\alpha(\Omega) = \min_B |\Omega \Delta B|$ where $|B|=|\Omega|$, does it hold on a bounded domain that

$ r_{out}^2 - r_{in}^2 \leq C \alpha(\Omega)$,

for some constant $C>0$? This seems like it should be true but I can't seem to find a concise proof of it.

What does convergence in the $L^2$ sense to a constant mean curvature surface imply?

2012-03-01T18:09:10Z

I have been thinking about the following question and have been unable to find any literature on the subject.

Question: Assume I have a sequence of smooth, simply connected, compact domains $\Omega_s \subset \mathbb{R}^d$ such that $|\Omega_s|=1$ and

$\int_{\partial \Omega_s} (\kappa - \bar \kappa)^2 dS(y) \to 0$ as $s \to +\infty$,

where here $\kappa$ is the mean curvature of the surface $\partial \Omega_s$ and $\bar \kappa$ denotes the average mean curvature over $\partial \Omega_s$. I can prove that the limit is in fact a ball in the following two cases:

All of the sets $\Omega_s$ are convex. or
I assume the uniform bound $\limsup_{s \to +\infty} |\partial \Omega_s| + \int_{\partial \Omega_s} \kappa^2 dS < +\infty$.

I would however like to remove these restrictions since they seem quite artificial. I have been able to rule out the standard "pinching" counter examples of a long rod with capped ends, but am not sure if there could exist other pathologies. Any direction to results in this direction would be appreciated.

Classification of limits under volume preserving mean curvature flow?

2012-03-08T12:45:17Z

It is well known that if you start with a domain $\Omega \subset \mathbb{R}^d$ which is uniformly convex, then it converges exponentially fast to the ball when evolved under volume preserving mean curvature flow (VPMCF).

In $\mathbb{R}^2$, if one starts with a smooth simply connected domain, then after a finite amount of time it will become convex, and converge to the ball.

Question: If $\Omega \subset \mathbb{R}^d$ is smooth and connected, does it converge (perhaps after surgery) to a constant mean curvature (CMC) surface? Is it possible to classify the type of CMC surface it converges to based on the intial domain $\Omega$? I am hoping mostly for direction to the appropriate literature in this case, as mathscinet is flooded with papers on the subject.

Update: I would like to know in particular, on the torus $\mathbb{T}^2$, if I start with a set $\Omega \subset \mathbb{T}^2$ which satisfies $|\Omega|=1/2$, and I evolve it under VPMCF, does it eventually converge to the stripe pattern?

How to minimize the length of a graph connecting n points in $\mathbb{R}^3$

2012-01-30T11:23:06Z

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. When $n=4$, the solution would be a tetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

Update: Thanks for the helpful comment and the answer. This answers my question.

A variational problem involving a negative fractional Soboblev norm.

2011-11-08T04:06:27Z

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 $\int_{\partial B} \xi(y)dS(y) = 0$ and $\int_{\partial B}\xi^2(y)dS(y)=1$ where $B \subset \mathbb{R}^3$ is a ball of radius $1$ and $\Phi(|x-y|)=\frac{1}{|x-y|}$ is the Newtonian potential.

This seems to resemble an inverse fractional Soblev norm such as $H^{-1}$ and moreover appears to be related to the problem of finding an optimal Poincare constant.

My guess is that the maximum is obtained for $\xi=+1$ on the upper half and $\xi=-1$ on the lower half. Given this however, I still cannot do an explicit calculation to determine this quantity. Is there a standard reference for such problems arising in Potential Theory perhaps which will allow one to evaluate (even approximately) such expressions?

For instance I know I can rewrite the above as: $\int_{\partial B} |\nabla w|^2$ where $-\Delta w = \mu$ and $\mu(x) = \xi(x)dS(x)$ but I'm not sure how this can help me to evaluate such an expression.

To summarize, I would like to try to evaluate the above double integral for the particular function $\xi = +1$ on the upper half of the ball and $\xi=-1$ on the lower half. Being able to solve explicitly the above maximization problem would be a bonus.

Elliptic regularity for the Neumann problem

2010-09-06T20:06:38Z

I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data.

For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$. Interior regularity works the same as always.

When proving boundary regularity, for the dirichlet boundary case we first consider some ball $B(0,1) \cap \mathbb{R}_+^n$ and let $\xi = 1$ on $B(0,1/2)$, $\xi = 0$ on $\mathbb{R}^n - B(0,1)$ and then estimate all derivatives $\frac{\partial^2 u}{\partial x_i \partial x_j}$ except $\partial^2 u/\partial x_n^2$. Two main points are needed

1) $\xi$ vanishes on the curved part of $B(0,1) \cap \mathbb{R}_+^n$\

2) $u=0$ on ${x_n=0}$.

This allows us to let $-\partial_{x_i} (\xi \partial_{x_j} u)$ (with derivatvies replaced by difference quotients) be an admissible test function for our weak definitoin of a solution.

I presume the main difficulty in neumann boundary data is making your test function be admissible. In other words, we would need $\int v = 0$ since our existence was established on $H^1(\Omega)$ restricted to mean value zero functions.

So in order to proceed, can we just subtract off a constant from our original $-\partial_{x_i}(\xi \partial_{x_j}u)$? Is there some more natural way to establish regularity in this case? I do not want to take advantage of the fact that we have a green's function in this case however as I only chose the Laplace equation for simplicity.

Best Poincare constants on the surface of a ball

2011-10-11T22:55:27Z

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 of all if there is a Poincare inequality of the form:

$\int_{\partial B(0,1)} |\nabla^{-1}\xi(y)|^2dS(y) \leq C_{B} \int_{\partial B(0,1)}|\xi|^2 dS(y).$

Secondly (and more importantly) I'd like to know if there are any resources I can be directed to regarding the optimal constant in such a case.

Answer by Dorian for Functionals continuous with respect to weak convergence

2011-09-10T13:31:45Z

I am not going to try to find the most general conditions under which lower semi-continuity holds but for that I suggest the standard reference for all of this is "Direct methods in the Calculus of Variations" by Bernard Dacorogna which covers all of this in full detail. I will give a brief outline of the answer however.

In order for $\int f(x,u,du)dx$ to be lower semi-continuous with respect to weak convergence one does not in general need any sort of convexity in the $u$ variable and what is more important generally is

1) Coercivity (see below) of the functional $f$ in the $du$ variable.

2) The right embedding along with continuity of $f$ in the $u$ variable with respect to the topology of strong convergence in this space.

Example: A good example is $E(u) = \int_0^1 |\nabla u |^2 + (1-u^2)^2$ with $u = 0$ on $\partial \Omega$ in $\mathbb{R}^d$. Observe that when $d \leq 4$ one has $H^1(\Omega) \subset \subset L^4(\Omega)$ and consequently the non-convex term depending on $u$ is lower order. Therefore one can pass to the limit in any minimizing sequence even though $(1-u^2)^2$ is very non convex (but it is continuous with respect to strong convergence). Notice however that for the energy $\bar E(u) = \int_0^1 |\nabla u|^2 + u^4$ in $\mathbb{R}^5$ with $u$ prescribed on $\partial \Omega$, the second term is not lower order but here we may use convexity to conclude lower semi-continuity of the second term.

The function $p \mapsto f(x,z,p)$ is coercive if there exists some constant $C > 0$ so that $f(x,z,p) \geq C|p|^q$ for some range of $q$. It then depends on what spaces one is working in but the goal is to use an embedding theorem such as $L^q \subset\subset W^{1,p}$ for $1 \leq q < p^*$ and to conclude that for a minimizing sequence $u_n$, there is in fact a strongly convergent subsequence in $L^q$ for some $q$. Then one will generally expect some sort of continuity of $f$ in the $u$ variable.

There are many more interesting examples but in almost all cases in practice the goal is to show that one has strong convergence of the $u_n$s in your minimizing sequence in some $L^p$ space. The book by Braides focuses mostly on asymptotics of functionals which depend on some large (or small) parameter and I'm not sure how much he talks about the assumptions needed in the direct method.

How does electric potential relate to mean curvature?

2011-09-08T14:49:01Z

Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.

Now consider the electric potential generated by this uniform mass distribution: $\phi = \int_{\Omega} \frac{1}{|x-y|} dx$. Question: I would like to know if there is a relationship between mean curvature $H$ and $\phi$. What I have in mind is an inequality of the form $\|\phi - \bar \phi\|_{L^p(\partial \Omega)} \leq C \|H - \bar H\|_{L^p(\partial \Omega)}$ where $\bar \phi$ and $\bar H$ denote the averages over $\partial \Omega$ of $\phi$ and $H$ respectively and $p$ can be anything for the time being.

Although one term is much more 'non-local' than the other, for a convex body it seems reasonable that the closer to being an surface of constant mean curvature, the closer one is to being an equipotential surface. This is related to a previous question of mine which ended up going unanswered regarding whether the only convex, compact equipotential surface in $\mathbb{R}^3$ was a sphere or not.

Stuck on a convergence argument in $H_0^1(\Omega)$.

2010-09-10T14:04:41Z

I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.

However I've encountered this step along the way which seems clear to me but I'm second guessing whether or not it is true.

Question: If $u_k \to u$ in $L^{p+1}$ for $p + 1 < 2^*=\frac{2n}{n-2}$ then I would like to see that $\Delta^{-1}(|u_k|^{p-1}u_k) \to \Delta^{-1}(|u|^{p-1}u)$ in $H_0^1(\Omega)$. This is of course equivalent to showing that $|u_k|^{p-1}u_k \to |u|^{p-1}u$ in $H^{-1}(\Omega)$.

My idea: Since I have convergence in $L^{p+1}(\Omega)$ it follows that I have convergence in all $L^q(\Omega)$ for $p+1 \geq q \geq 1$. By Sobolev embeddings I believe that it's true that $||w||_{H^{-1}} \leq ||w||_{L^q}$ for any $q$ with $1/q + 1/r = 1$ for $1 \leq r \leq 2^*$. So this should imply the needed $H^{-1}(\Omega)$ convergence if I knew that $|u_k|^{p-1}u_k \to |u|^{p-1}u$ in some $L^q$ within this range. The best however I can say is that I have convergence in $L^{\frac{p+1}{p}}$ since $u_k \to u$ in $L^{p+1}$. But then $1 + 1/p > 1 + \frac{n-2}{n+2} = \frac{2n}{n+2}$ which is the conjugate exponent to $2^*$.

This appears to work but is quite technical and messy and all of the 'proofs' I've seen hint at some "simply energy argument". This doesn't appear simple at all! Therefore I would appreciate any suggestions about a better approach or if someone could point out something wrong with how I've thought about it. I hope this fits within the paramaters of the website.

Spectral Galerkin method for a semi-linear parabolic PDE

2010-09-10T04:19:37Z

I'm trying to understand how to apply the Galerkin method to $u_t - \Delta u = u^3$. I understand how to obtain all of the a-priori estimates using Sobolev embeddings and such but my question concerns the actual discretization procedure where we project onto the finite dimensional subspace spanned by the eigenfunctions of $-\Delta$.

In the linear case we may simply set $u_N = \sum \phi_n c_n$ and plug this in to obtain a set of $N$ O.D.Es which we then show satisfy the same energy bounds. In the non-linear case though we may not just substitute directly because of the $u^3$ term. How can this be dealt with? Is there perhaps a better way to do the approximation?

Addition: In this example if we let $u_N = \sum \phi_n c_n^N(t)$ then when we plug this into the weak form of our PDE we obtain and then choose our test function to be one of the basis elements $w_k$ we obtain $d/dt c_k^N(t) + \sum_{i=1}^n e^{ki}(t) c_k^N(t) = \int (\sum_{n=1}^N c_n^N(t) \phi_n)^3w_k$, for some coefficients $e^{ki}(t)$. My question is, how do I deal with the integral on the right? I would like to be able to solve this ODE and then say I have a solution.

Existence of rearrangements of functions in $L^p([0,1])$ when given a measure preserving map.

2011-03-27T10:27:47Z

Given a funtion $f \in L^p([0,1])$ (take $p=\infty$ if you'd like), and also a measure preserving map $s:[0,1] \to [0,1]$ (meaning $s$ pushes Lebesgue measure forward to itself) I would like to know if there exists some $f^* \in L^p([0,1])$ such that $f^*\circ s = f$. If $s$ is invertible this is of course obvious but measure preserving maps need not be invertible (although must be onto).

Recall that given $f$ there exists a monotone rearrangement of $f$, and a measure preserving map $t:[0,1] \to [0,1]$ which yields this rearrangement.

However my question is in some sense the reverse question (with no monotonicity added).

Somehow it seems intuitive that such a map should exist but I'm not able to prove it directly. It seems like something which may be well known however.

A counter example to Hahn-Banach separation theorem of convex sets.

2010-09-02T22:32:37Z

I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed and the other one compact. My goal is to find a counter example when these hypotheses are not satisfied but the sets are still convex and disjoint. So here is my question:

Question: I would like a counter example to the Hahn-Banach separation theorem for convex sets when the two convex sets are disjoint but neither has an interior point. It is trivial to find a counter example for the strict separation but this is not what I want. I would like an example (in finite or infinite dimensions) such that we fail to have any separation of the two convex sets at all.

In other words, we have $K_1$ and $K_2$ with $K_1 \cap K_2 = \emptyset$ with both $K_1$ and $K_2$ convex belonging to some normed linear space $X$. I would like an explicit example where there is no linear functional $l \in X^*$ such that $\sup_{x \in K_1} l(x) \leq \inf_{z \in K_2} l(z)$.

I'm quite sure that a counter example cannot arise in finite dimensions since I think you can get rid of these hypotheses in $\mathbb{R}^n$. I'm not positive though.

Failure of regularity up to the boundary for a linear elliptic PDE

2010-09-11T23:11:29Z

I asked a question before where I wanted a simple example where regularity up to the boundary fails for a linear elliptic PDE. I was presented an example with $\Omega = B(0,1) \backslash {0}$ (ball minus a point) which is nice but I would like something less pathalogical. I would like an example where my domain is at least Lipschitz (so a rectangle is an example). In the case where we subtract off a point, a boundary value problem doesn't really even make sense in a weak formulation to begin with.

More precisely,

Question: I would like an example where an elliptic PDE with smooth coefficients satisfies $Lu = f$ in $\Omega$ for smooth $f$ (or zero) and $u = 0$ on $\partial \Omega$ (which is of Lipschitz class) but where $u$ somehow fails to be "regular" up to the boundary (I'm being vague on purpose here as any failure of regularity will do for the most part).

My guesses have been try try looking at the upper right quadrant $[{ (x,y) : x, y > 0}]$ but nothing has come from this so far. Any suggestions/ideas are welcome and appreciated.

A simple example where elliptic boundary regularity fails due to a kink in the domain

2010-09-08T13:44:47Z

I'm seeking a simple example of where elliptic (preferably linear) boundary regularity fails due to a simple kink in the domain.

So far my gueses were to look at $-\Delta u = f$ on $[0,2\pi] \times [0,2\pi]$ with $0$ Dirichlet boundary conditions and choose an $f$ which was far from $0$. This hasn't seem to produce any results (I was checking regularity directly by the method of Fourier series).

So more precisely, I would like an example where

1) $Lu = f$ in $\Omega \subset \mathbb{R}^n$ with $f$ smooth

2) $L$ is elliptic and $u = 0$ on $\partial \Omega$

3) $\Omega$ is not smooth and consequently $u$ is not smooth up to the boundary.

Compatibility conditions for parabolic regularity

2010-10-07T14:41:40Z

I'm trying to understand the compatibility conditions for regularity of second order parabolic equations. Let's consider the equation $u_t - Lu = f$ with $u(0)=g$ on $\Omega \times [0,T)$ with $u = 0$ on $\Gamma_T$, the parabolic boundary of $\Omega \times [0,T)$ .

We need the assumptions that $\frac{d^{m-1}f}{d^{m-1}}(0) - Lg_{m-1} \in H_0^1(\Omega)$ for the $L^2$ theory to work to conclude smoothness up to the boundary (I'm presuming $f$ and $\Omega$ are smooth). As far as I can tell this is required so that we don't have the following situation:

$u_t - u_{xx} = 0$ on $[0,1] \times [0,1]$ with $u(0,x)=1$ and $u(t,0)=u(t,1)=0$. Here we clearly can't have differentiability up to the boundary since $1 \neq 0$. However after time $t = 0$ it would appear to me that this problem goes away, ie. that the solutions are smooth up to the boundary except at $t=0$. Is this the case? Can people perhaps share insightful examples?

Fourier series of non-periodic functions

2010-09-30T03:58:57Z

It's well known that the Fourier series converges uniformly when a function is $C^2$ and periodic on say $[0,2\pi]$. If the function is not periodic you can have lack of uniform convergence near the endpoints due to "Gibbs" phenomenon. However I would like to understand if and when you still have uniform convergence on compact subsets of $[0,2\pi]$? This seems to be geometrically obvious but I can't find a good source for it. Any suggestions would be appreciated. For instance $f(x) = x$ on $[0,1]$. It would appear that the Fourier series converges uniformly on compact subsets of $[0,2\pi]$ but this doesn't seem to follow from analysis of the Fourier coefficients. Isn't it the same as if I had done an odd extension of $f(x)$ so that I have $-|x|$ on $[-2\pi,2\pi]$ and find the fourier series for this new domain? In the latter case I clearly get uniform convergence and I don't think this should depend on the fact that I reflected and chose a different domain.

Inconsistency in definition of characteristics for a linear PDE? Folland versus Fritz John.

2010-09-26T16:54:17Z

There seems to be a major inconsistency (perhaps due to my lack of understanding) between what Folland calls a "characteristic" and what I had previously thought was a characteristic.

For example, Folland says that the characteristics of the equation $\partial_x u = 0$ are $\{ \xi : \xi_1 = 0\}$ (in $\mathbb{R}^2$). This confuses me to no end since I thought the characteristic hypersurface was the $x_1$ axis here? According to this definition it's the orthogonal compliment to this. The same issue arises with the heat operator $L=u_t-u_{xx}$ where he says the characteristics are $\{\xi \neq 0 : \xi_x = 0\}$ . Aren't the characteristics $t=$ cosntant which are orthogonal to these?

Sorry if this question is elementary but it's given me a real headache and I'm not sure what it is I'm missing here.

Answer by Dorian for Can a self-adjoint operator have a continuous set of eigenvalues?

2010-09-25T02:50:12Z

You are confusing two notions. First of all, the point spectrum just means eigenvalues; there is no assumption that these form a discrete set. The shift operator is a simple example where the spectrum is "continuous".

The condition for the eigenvalues to be discrete is precicsely that the operator $A:H \to H$ is compact. It is however possible for non-compact self adjoint operators to have a discrete spectrum. The simplest example of this is the orthogonal projection operator $P:H \to Y$ where $Y$ is a closed subspace of the Hilbert space $H$. Here the spectrum is $0$ and $1$.

Bounded self adjoint operators have no residual spectrum but they do indeed have a continuous spectrum. Take any compact operator $A:H \to H$ where dim$H=+\infty$. Then $0$ belongs to the continuous spectrum because otherwise $A:H \to H$ would be invertible, implying that dim$H <\infty$. Continuous = "exists a set of approximate eigenvectors".

If you want a continuous range of spectrum take $Af = x f(x)$ on $L^2([0,1])$. Then the range of the spectrum is just $[0,1]$. There are no eigenvalues for this operator and moreoever since the residual spectrum is empty for self adjoint operator, $[0,1]$ is the spectrum and it is equivalent to the continuous spectrum.

So a final point, continuous just means that $R(A-\lambda I)$ is not dense but that $\lambda$ is not an eigenvalue. It has nothing to do with the actual spectrum being discrete or continuous.

So I think you were mixing up two notions but hopefully I've provided examples for both.

Comment by Dorian

2012-06-24T11:04:51Z

This type of question is not appropriate for math overflow. Math overflow is reserved for research level math questions. Try stackexchange.

Comment by Dorian

2012-06-20T23:25:56Z

This kind of question is not suitable for mathoverflow. Try stackexchange. This site is reserved for research level questions.

Comment by Dorian

2012-06-07T10:18:00Z

By improves, is a strictly weaker inequality. Sorry.

Comment by Dorian

2012-06-07T09:55:39Z

Thank you for this interesting example, but if $C(L,A) > 0$ this improves the inequality no? If you see the last term that I added in my calculation, the term $\int_0^{2\pi} [p^*(\theta)-p(\theta)][(r^*)^2(\theta)-r(\theta)] d\theta$ , this term indeed is controlled by $C(A^*-A)$ where $C>0$. Thus your example shows that what I want is certainly false for $C(L,A)=0$ but the question remains if adding such a term compensates.

Comment by Dorian

2012-06-05T22:53:33Z

Thanks for your answer. Do you think the result will hold if one assumes that the curve is star shaped and symmetric with respect to reflections across the origin? This seems quite reasonable.

Comment by Dorian

2012-06-05T19:00:27Z

Are you familiar with the Brenier theorem for existence of optimal transport plans for the $L^2$ Wasserstein metric? This may be useful.

Comment by Dorian

2012-06-04T14:10:12Z

My rough idea has been the following: Take a ray of angle $\theta$ where $\theta$ is very small. By Green's theorem, for the piece of the curve below the convex hull, we will have $p^* \Delta L^* = p^2 \Delta L + A_{\theta}^* - A_{\theta} + o(\theta)$ as $\theta \to 0$ where $A$ and $A^*$ here denote the area contained in the ray intersected with the original set, and convex hull respectively. Then take the limit as $\theta \to 0$ so that we get something pointwise and go from there. Haven't been able to solidify anything though.

Comment by Dorian

2012-06-04T13:51:54Z

Interesting, however with many wiggles, observe that p will be very close to zero very frequently since the oscillating segments will be almost parallel to the normal vector, makign $X \cdot \nu \sim 0$ on these pieces.

Comment by Dorian

2012-06-04T13:22:05Z

Yes, corrected this. Thanks.

Comment by Dorian

2012-06-04T13:06:18Z

Ok I have hopefully clarified the question. Yes I believe that the name for what I'm saying is star shaped.

Comment by Dorian

2012-06-04T11:12:45Z

Hopefully clarified this. I mean that the curve is $(r(\theta),\theta)$ in polar coordinates.

Comment by Dorian

2012-06-02T16:05:21Z

I was just thinking about this inequality. As written I think it is incorrect. Take a circle which is not centered at the origin. Then r_1-r_2 is not zero but the left side is zero..

Comment by Dorian

2012-05-29T10:42:13Z

I am looking for a class of curves, in paritcular rectifiable verifolds with generalized mean curvature in $L^{\infty}(\partial \Omega)$..

Comment by Dorian

2012-04-13T09:54:35Z

Very nice counter example. Thank you Rbega.

Comment by Dorian

2012-04-07T18:18:23Z

This is for regular mean curvature flow, not volume preserving mean curvature flow.