User yakov shlapentokh-rothman - MathOverflow

Common Zeros of Holomorphic Functions

2012-11-17T17:22:59Z

The following assertion appears plausible to me: Let $f(z,w,u)$ and $g(z,w,u)$ be holomorphic in $z$, $w$, and $u$. Suppose that $f(z_0,w_0,u_0) = g(z_0,w_0,u_0) = 0$ and that $f$ and $g$ are non-degenerate in the sense that none of $f(z,w_0,u_0)$, $f(z_0,w,u_0)$,...,$g(z_0,w_0,u)$ are identically zero. Then there exists $w(u)$ and $z(u)$ defined near $u_0$ such that $f(z(u),w(u),u) = g(z(u),w(u),u) = 0$. Note that I am not imposing any uniqueness or regularity requirements on $z(u)$ and $w(u)$.

Is this true? What is a good reference for this?

Reference Request: Holomorphic Dependence on Parameters to Solutions of Complex Linear ODEs Near an Irregular Singularity

2012-05-10T23:03:25Z

I'm looking for a reference which discusses the following:

Consider the ODE

$\frac{d^2H}{dz^2} + f(z,x)\frac{dH}{dz} + g(z,x)H(z) = 0$

where

1) $f$ and $g$ depend holomorphically on $x$ and $z$

2) $f/z$ and $g/z^2$ are holomorphic at $\infty$.

As discussed in many books one can find a pair of formal solutions to these which turn out to be asymptotic expansions of actual solutions in appropriate sectors of the complex plane. Presumably, these actual solutions depend holomorphically on $x$ under suitable assumptions, but I cannot find this written down anywhere at this level of generality. Does anyone know a reference or paper which discusses this?

Answer by Yakov Shlapentokh-Rothman for Physical Interpretation of Robin Boundary Conditions

2012-04-30T17:08:11Z

Here is an example where $\Omega = \mathbb{R}^3$. One way to establish dispersion for the wave equation involves taking a temporal Fourier transform. In order to do this one has to multiply by a cutoff function supported in $t \in [0,\infty)$. You then get the equation

$(\Delta+\omega^2)\psi = F$

where $\psi$ is the temporal Fourier transform of the product of the original solution with the cutoff, $\omega$ is the Fourier variable, and $F$ is a function controllable by initial data via a finite time energy inequality. If this plan of attack is going to work, we need to make sure that $\psi$ is uniquely determined by $F$. This of course requires appropriate boundary conditions at $\infty$. These turn out to be

1) $\psi = O\left(|x|^{-1}\right)$

2) $\frac{\partial\psi}{\partial r} - i\omega\psi = O\left(|x|^{-2}\right)$

This is a sort of Robin condition at infinity. See http://terrytao.wordpress.com/2011/04/21/the-limiting-absorption-principle/ for more details.

How to go from a potential resolvent to the associated operator

2011-08-17T19:07:51Z

I am reading http://www.springerlink.com/content/l76542r216362714/. The author appears to use the following fact:

Let $H$ be a Hilbert space. For every $\zeta \in \mathbb{C}\setminus\mathbb{R}$ we have a bounded operator $R(\zeta): H \to H$. We also know that

(1) $R(\zeta)$ has nullity $0$

(2) $R(\zeta)$ has dense range

(3) $R(\zeta)$ satisfies the first resolvent identity $R(\zeta) - R(\zeta') = (\zeta-\zeta')R(\zeta)R(\zeta')$.

Then we claim that there exists a densely defined operator $T: H \to H$ such that $\sigma(T) \subset \mathbb{R}$ and for $\zeta \in \mathbb{C}\setminus\mathbb{R}$ the resolvent at $\zeta$ is $R(\zeta)$.

How is this proved? Alternatively, does anyone know a reference where this is proved?

Answer by Yakov Shlapentokh-Rothman for Textbooks for PDE between Strauss and Folland

2011-08-08T01:11:39Z

Fritz John's book "Partial Differential Equations" is one of the more elementary but still substantial PDE books I have run across.

Answer by Yakov Shlapentokh-Rothman for Counterexamples in PDE

2011-06-24T01:04:22Z

Lewy's Example gives a PDE where local solvability fails.

http://en.wikipedia.org/wiki/Lewy%27s_example

Fritz John's PDE book has a detailed discussion.

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II

2010-05-25T18:06:18Z

For some context see http://mathoverflow.net/questions/25846/random-walks-in-z2-z2-intrinsic-characterization-of-euclidean-distance

As per Noah's answer and JBL's comment this was false as stated. However, I think the following reformulation is interesting.

As before we consider a random walk on $\mathbb{Z}^2$ where a particle either stays at its vertex or moves to a neighbor with probability 1/5. We start the process with a particle at the origin. For $x \in \mathbb{Z}^2$ we let $p_n(x)$ denote the probability that we find the particle at $x$ after $n$ iterations. Let $|\cdot|$ denote the Euclidean distance of two points in $\mathbb{Z}^2$ via the standard embedding of $\mathbb{Z}^2 \subset \mathbb{R}^2$.

Now for the reformulated question: For each $n$, let $C_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| \leq |y| \Rightarrow p_n(x) \geq p_n(y)$

Does $\lim_{n\to\infty} C_n = \infty$? If so, how fast does this diverge?

EDIT: As per George Lowther's comment, I now find it quite probable that $\lim\inf_{n\to\infty} C_n \leq 5$ if not $C_n = 5$ for all large $n$.

A natural attempt to salvage the question is the following: For each $n$, let $\tilde{C}_n$ be the supremum over all $C > 0$ so that for all $x,y \in \mathbb{Z}^2$ we have

$|x|,|y| \leq C$ and $|x| < |y| \Rightarrow p_n(x) > p_n(y)$

Again we ask if $\lim_{n\to\infty} \tilde{C}_n = \infty$ and if so, how fast this diverges.

Answer by Yakov Shlapentokh-Rothman for Extension theory with bump function

2011-06-09T19:18:47Z

I believe that the fractional Sobolev spaces can be defined as a complex interpolation space between the integer Sobolev spaces (see this http://www.scribd.com/doc/45316527/Sobolev-Spaces-2ed-Robert-a-Adams-John-J-F-Fournier for example). As noted in the comments, your question is easily seen to hold on the integer valued spaces. Then we can interpolate to get the result for the fractional spaces.

Answer by Yakov Shlapentokh-Rothman for Characterize where the Dirichlet Problem for the Laplacian is always solvable

2011-05-25T20:47:34Z

http://eom.springer.de/r/r080680.htm contains some characterizations of domains where the Dirichlet problem is solvable. I believe that a key term in searching for references is "Wiener's criterion."

Answer by Yakov Shlapentokh-Rothman for Proof of L^p Elliptic Regularity

2011-04-11T01:45:01Z

I stumbled across the book Second Oder Elliptic Equations and Elliptic Systems by Yah-Ze Chen which appears to contain an answer to my question. It is available on google books here

http://books.google.com/books?id=eQcbiPQPweQC&pg=PA49&dq=strong+solution+dirichlet+problem+Lp&hl=en&ei=byKiTeXXO6GG0QGG7dGgBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwADgK#v=onepage&q=strong%20solution%20dirichlet%20problem%20Lp&f=false

To save time for those who are interested, here is the relevant argument:

For large $\lambda > 0$ we want to show that $L_{\lambda} = L - \lambda I$ is injective on $W_0^{1,p}$.

Claim: Let $L^T_{\lambda}$ be the transpose of $L^{\lambda}$ with respect to the paring that defines weak solutions. Then we claim that $L^T_{\lambda}$ inective on $W_0^{2,p}$ implies that $L_{\lambda}$ is injective on $W_0^{1,p}$

Proof: Suppose that $L^T_{\lambda}$ is injective on $W_0^{2,p}$. Then, by an argument contained in the original post above, for every $f \in L^p(\Omega)$ we can find $u \in W_0^{2,p}(\Omega)$ such that $L^T_{\lambda}u = f$. Now, suppose that $L_{\lambda}v = 0$ for some $v \in W_0^{1,p}$. After an integration by parts and the definition of weak solution, we see that $\varphi \in W_0^{2,q}$ implies that

$\int_{\Omega}uL^T_{\lambda}\varphi = 0$.

Now choose $\Omega'' \subset\subset \Omega' \subset\subset \Omega$ and a bump function $\rho$ identically one in $\Omega''$ with support in $\Omega'$. $\rho\text{sgn}(u)$ is in $L^q$, and we can find $g \in W_0^{2,q} $ such that $L^T_{\lambda}g = \rho\text{sgn}(u)$. Plugging this $g$ into the above equality gives

$\int_{\Omega''}|u| = -\int_{\Omega\setminus\Omega''}\rho |u|$

Due to the arbitrariness of $\Omega''$, this implies that $\int_{\Omega} |u| = 0$ and hence $u$ is $0$ a.e.

Claim: For $\lambda$ large enough, $L_{\lambda}$ is injective on $W_0^{2,p}$.

Proof: Suppose $L_{\lambda}u = 0$ for $u \in W_0^{2,p}$. Let $\tilde{\Omega} = \Omega \times (-1,1)$, and $\tilde{\Omega'} = \Omega \times (-1/2,1/2)$. Let $(x,t)$ be the coordinates on $\Omega \times (-1,1)$. Then define $v(x,t) = \cos(\sqrt{\lambda}t)u(x)$. Let $\hat{L_{\lambda}} = L_{\lambda} + \partial_t^2$. We have $\hat{L_{\lambda}}v = 0$. The strong solution estimates give

$\vert\vert v\vert\vert_{W^{2,p}(\tilde{\Omega'})} \leq C\vert\vert v\vert\vert_{L^p(\tilde{\Omega})} \leq C\vert\vert u\vert\vert_{L^p(\Omega)} \Rightarrow $

$\vert\vert \partial_t^2v\vert\vert_{L^p(\tilde{\Omega'})} \leq C\vert\vert u\vert\vert_{L^p(\Omega)} \Rightarrow $

$\lambda\vert\vert u\vert\vert_{L^p(\Omega)}(\int_{-1/2}^{1/2}|\cos(\sqrt{\lambda}t|^p)^{1/p} \leq C\vert\vert u \vert\vert_{L^p(\Omega)} \Rightarrow$

$\lambda^{1 - \frac{1}{2p}}\vert\vert u\vert\vert_{L^p(\Omega)}(\int_{-1/2}^{1/2}|\cos t|^p)^{1/p} \leq C\vert\vert u\vert\vert_{L^p(\Omega)}$

Now taking $\lambda$ large enough implies that $u = 0$ almost everywhere.

Proof of L^p Elliptic Regularity

2011-02-27T15:51:42Z

Let $L = \sum_{i,j=1}^n -\frac{\partial}{\partial x^i} (a^{ij}(x)\frac{\partial}{\partial x^j}) + \sum_{i=1}^n b^i(x) \frac{\partial}{\partial x^i} + c(x)$ be a second order elliptic operator with smooth coefficients, $\Omega$ a bounded open domain with smooth boundary in $\mathbb{R}^n$, and $f$ be a function in $L^p(\Omega)$. We say that $u \in W_0^{1,p}(\Omega)$ (one weak derivative in $L_p$ and vanishing boundary values) is a weak solution of $Lu = f$ if for all $g \in W_0^{1,q}(\Omega)$ $(q = p^*)$ we have

$\int_{\Omega} \sum_{i,j=1}^n a^{ij}(x)\frac{\partial u}{\partial x^i}\frac{\partial g}{\partial x^j} + \sum_{i=1}^n b^i(x)\frac{\partial u}{\partial x^i}g(x) + c(x)u(x)g(x) = f(x)$.

The standard result is of course that all such weak solutions $u$ actually belong to $W_0^{2,p}(\Omega)$.

I am trying to complete the following proof of this statement:

(1) First we establish an a priori estimate for strong solutions $v \in W_0^{2,p}(\Omega)$ of $Lv = f$:

$\vert\vert v\vert \vert_{W_0^{2,p}(\Omega)} \leq C(\vert\vert f\vert\vert_{L^p(\Omega)} + \vert\vert v\vert\vert_{L^p(\Omega)})$

This is non-trivial but can be established by proving the relevant estimate for the Laplacian with a Newton Potential argument and then using the freezing coefficients technique.

(2) Next we observe that if $L$ is injective on $W_0^{1,p}$, then we are done. This is because $L$ injective implies

$\vert\vert v\vert\vert_{L^p(\Omega)} \leq C\vert\vert Lv\vert\vert_{L^p(\Omega)}$

One proves this by assuming it was false and then using Rellich compactness to produce a non-zero solution to $Lv = 0$.

Having established this estimate, we consider the smooth mollifications $f_{\epsilon}$ of $f$. By $L^2$ theory we can find smooth $v_{\epsilon}$ strong solutions of $Lv_{\epsilon} = f_{\epsilon}$. Since we have

$\vert\vert v_{\epsilon} - v_{\epsilon'}\vert\vert_{W_0^{2,p}} \leq C\vert\vert f_{\epsilon}-f_{\epsilon'}\vert\vert_{L^p(\Omega)}$

The $v_{\epsilon}$ converge to some $v \in W_0^{2,p}(\Omega)$ which will solve $Lv = f$ strongly. Since strong solutions are clearly weak solutions, by the injectivity of $L$ on $W_0^{1,p}(\Omega)$ we conclude that $u = v \in W_0^{2,p}(\Omega)$ and we are done.

(3) We have no way to guarantee that $L$ is injective, for example $0$ might be an $L^2$ eigenvalue of $L$. However, if $p=2$ then we could guarantee that $L_{\lambda} = L + \lambda I$ is injective for some large $\lambda$. If we could establish this fact in the general case we would be done since $L_{\lambda}u = f + \lambda u \in L^p$ and $L_{\lambda}$ injective imply that (2) is applicable.

Question: What is the simplest way to prove that $L_{\lambda} = L + \lambda I$ is injective on $W_0^{1,p}(\Omega)$ for large $\lambda$? Do there exist weak $L^P$ maximum principles?

Of course, I would prefer that the proof not use $L^p$ regularity.

Removable Singularities for Elliptic Equations

2011-04-06T13:09:39Z

The following fact is quite standard and does not have a very long proof:

$(*)$ If $u$ is harmonic on $B_1(0)\setminus {0}$ and uniformly bounded, then $u$ in fact extends to a harmonic function on the whole ball.

Some googling reveals that such statements are in fact true for large classes of elliptic operators with much more general singularity sets and growth conditions. There appears to be a vast literature on the subject.

I would like to know if the exact analogue of $(*)$ has a "simple" proof for linear elliptic operators. "Simple" is slightly ambiguous but is meant to mean tools present in standard elliptic theory textbooks, i.e. Gilbarg and Trudinger or Jost.

Answer by Yakov Shlapentokh-Rothman for Removable Singularities for Elliptic Equations

2011-04-08T02:28:31Z

Here are the details behind my original interpretation of Deane Yang's comment:

By Schauder estimates, $\vert\vert \nabla u\vert\vert_{L^{\infty}} \leq C\vert\vert u\vert\vert_{L^{\infty}}$, so we also know that the gradient of $u$ is uniformly bounded. Next, we claim that $u$ is a weak solution of $Lu = 0$ on $B_1(0)$. We have $Lu = \sum_{i,j=1}^n \partial_j(a^{ij}(x)\partial_i u) + \sum_{i=1}^nb^i(x)\partial_i u + c(x)u$. To show that $u$ is a weak solution on $B_1(0)$, we need to show that for every $\varphi \in C_c^{\infty}(B_1(0))$,

$\int_{B_1(0)} \sum_{i,j=1}^n a^{ij}(x)\partial_{i}u\partial_j\varphi + \sum_{i=1}^nb^i(x)(\partial_i u) \varphi+ c(x)u\varphi = 0$

Since all of the relevant terms are uniformly bounded, the left hand side is equal to

$\lim_{\epsilon\to 0}\int_{B_1(0)\setminus B_{\epsilon}(0)} \sum_{i,j=1}^n a^{ij}(x)\partial_{i}u\partial_j\varphi + \sum_{i=1}^nb^i(x)(\partial_i u) \varphi+ c(x)u\varphi$

After an integration by parts, this gives

$\lim_{\epsilon\to 0}\int_{B_1(0)\setminus B_{\epsilon}(0)}(Lu)\varphi + \int_{\partial B_{\epsilon}(0)}\sum_{i,j=1}^n a^{ij}(x)(\partial_{i}u)\varphi \nu_j = $

$\lim_{\epsilon\to 0}\int_{\partial B_{\epsilon}(0)}\sum_{i,j=1}^n a^{ij}(x)(\partial_{i}u)\varphi \nu_j$

Since everything involved is uniformly bounded, this goes to $0$. (EDIT: The dimension should be at least $2$ for this to work)

Now $L^2$ elliptic regularity implies that $u$ is smooth on the whole ball and we are done.

Answer by Yakov Shlapentokh-Rothman for Localization of Laplacian eigenfunction on the unit square?

2011-04-08T13:35:46Z

My offic mate and I believe this is true. By separation of variables the eigenfunctions are just $Csin(\pi kx)sin(\pi ly)$ for some fixed constant $C > 0$. Using the trig identity $\sin^2(x) = (1-\cos(2x))/2$, we see that

$\int_a^b \sin^2(kx)\ dx = (1/k)\int_{ak}^{bk} \sin^2(x)\ dx = (b-a)/2 - (1/2k)\int_{ak}^{bk}\cos(2x)\ dx \geq (b-a)/2 -1/2k$

The integral of $|u_k|^2$ on a small square is just

$\int_a^b\int_a^b \sin^2(\pi kx)\sin^2(\pi ly)\ dxdy$

so we can apply the previous line twice.

Answer by Yakov Shlapentokh-Rothman for What is Symplectic Area?

2011-01-22T16:37:18Z

For one interpretation of the area inside a curve in phase space, see Arnold's Mathematical Methods of Classical Mechanics page 20. In case you do not have a copy of the book, he defines a function $S: (E_0 - \epsilon, E_0 + \epsilon) \to \mathbb{R}$ which gives the area inside the curve associated to an energy level $E$ (assuming this is well defined). The problem on page 20 asserts that $T = \frac{dS}{dE}$ where $T$ is the period of motion along the curve.

Note: The relevant page is available on google books.

Derivative of Exponential Map

2010-12-30T09:23:21Z

Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector field along $\gamma$. Define $\varphi: (a,b) \to M$ by $t \mapsto \exp_{\gamma(t)}(E(t))$. Is there a "nice" expression for $\varphi'(t)$?

This question originates in an attempt to understand the proof of corollary 1.36 in Cheeger and Ebin's "Comparison Theorems in Riemannian Geometry."

Answer by Yakov Shlapentokh-Rothman for A simple example where elliptic boundary regularity fails due to a kink in the domain

2010-09-08T16:48:50Z

You might consider this cheating, but at some point this tripped me up: What is the first Dirichlet eigenvalue and eigenfunction for $\Delta$ on the ball minus the origin? Well, since points have measure $0$, from the min-max principle it is the same as the first eigenvalue and eigenfunction for $\Delta$ on the ball. However, the eigenfunction certainly doesn't vanish at the origin. What went wrong -> The boundary of the punctured ball is a sphere and a point, which is not smooth.

EDIT: I forgot to note that the dimension should be 2 or more for this to make sense (see comments below)

Discrete Hairy Ball Theorem

2010-08-07T02:58:12Z

This question is inspired by

http://mathoverflow.net/questions/29323/math-puzzles-for-dinner/29399#29399

The arrow compatibility conditions in that problem can be considered an attempt to discretize the notion of a continuous vector field.

The Hairy Ball Theorem states that there is no continuous nowhere vanishing vector field on the sphere, http://en.wikipedia.org/wiki/Hairy_ball_theorem.

We are led to the following formulation of a discrete Hairy Ball Theorem:

Instead of a sphere we will consider the squares lying on the surface of a 3 x 3 x 3 cube (Rubik's cube). Instead of searching for a continuous nowhere vanishing vector field, we ask if there exists a "legal configuration" of arrows on the squares of the cube?

A "legal configuration" consists of the following:

1) For any non-corner square, an arrow pointing in one of the eight cardinal directions is placed.

2) For any corner square, there is one cardinal direction which does not point to an adjacent square. For these squares, the placed arrow should point in one of the other seven directions.

The following conditions are modified from the original formulation:

3) Orthogonally adjacent non-corner squares should be compatible in the sense that if they are flattened to lie in a plane, the arrows should be at most 45 degrees apart.

4) Orthogonally adjacent corner squares should be compatible in the sense that if they are flattened to lie in a plane, the arrows are one rotation away from each other within the seven allowed directions. For example, if northeast is a prohibited direction on a corner square, then a northward pointing arrow on the square and an eastward pointing arrow on an adjacent square are compatible.

So, can you comb a hairy Rubik's cube? Does a legal configuration of arrows exist? What about an n x n x n cube?

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance

2010-05-25T05:21:09Z

Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start the random walk with one particle at the origin. For each $n \geq 1$ and $x \in \mathbb{Z}^2$ let $p_n(x)$ be the probability of finding the particle at $x$ after $n$ iterations. For two points $x,y \in \mathbb{Z}^2$ let $|\cdot|$ denote the Euclidean distance of $x$ and $y$ via the standard embedding $\mathbb{Z}^2 \subset \mathbb{R}^2$.

For what $n$ is it true that $|x| \leq |y| \Rightarrow p_n(x) \geq p_n(y)$? What kind of techniques are available to prove statements like this? Barring arithmetic mistakes I have verified this up to n=6 via explicit computation.

Please forgive me if this is actually a trivial question (I know very little about random walks). I would also be very happy with suggested approaches or references.

A Little Motivation/Another Problem: Suppose we list the elements of $\mathbb{Z}^2$ is ascending order by Euclidean distance from the origin, $z_1 \leq z_2 \leq \cdots$, and then set $D_n = \cup_{i=1}^n z_i$. For various reasons I have been dealing with these $D_n$ and would like to consider analogues in other groups. Hence I would very much like to have a "$\mathbb{Z}^2$-intrinsic" characterization of these ${D_n}$, i.e. it would be nice to have a characterization of $D_n$ that only used group or graph theoretic statements about $\mathbb{Z}^2$. Most importantly I do not want to mention the specific embedding of $\mathbb{Z}^2$ into $\mathbb{R}^2$.

Note: The $D_n$ are not exactly well defined since there are choices involved in the list $z_1 \leq z_2 \leq \cdots $. So I am actually interested in characterizing them up to the forced ambiguity.

Comment by Yakov Shlapentokh-Rothman

2013-01-21T18:02:15Z

This might interest you books.google.com/…

Comment by Yakov Shlapentokh-Rothman

2012-11-26T19:02:40Z

@Nik: I should have restricted my comment to classical relativity. However, I'm puzzled about your statement about the Klein-Gordon equation. It surely satisfies finite speed of propagation (one can establish this with energy estimates), see e.g. tinyurl.com/cw5d6be and wiki.math.toronto.edu/DispersiveWiki/index.php/…. This implies that compactly supported solutions stays compactly supported. Maybe I'm misunderstanding the claim.

Comment by Yakov Shlapentokh-Rothman

2012-11-26T17:48:06Z

In relativistic/hyperbolic problems the non-locality of analyticity is fundamentally incompatible with "finite speed of propagation," i.e. the fact that information cannot travel faster than light.

Comment by Yakov Shlapentokh-Rothman

2012-11-17T18:06:58Z

@robot: In the application I have in mind it is not easy to check the hypothesis for the implicit function theorem.

Comment by Yakov Shlapentokh-Rothman

2012-11-17T18:06:02Z

@Julian Rosen: Thanks, that's indeed a relevant counter-example.If you want to add that as an answer I can accept it.

Comment by Yakov Shlapentokh-Rothman

2012-09-07T21:31:19Z

I would guess that $S^{\alpha}$ denotes a symbol class a la Hormander

Comment by Yakov Shlapentokh-Rothman

2012-05-04T13:39:36Z

I'm not really sure what this says about "why" there is dispersion, but I'm really the wrong person to ask. One nice thing about this approach is that it generalizes to Riemannian manifolds with a potential naturally, i.e. you can end up reducing certain dispersive statements about the wave equation to some geometric assumptions about the manifold (like behavior of trapped geodesics and asymptotic flastness) and/or spectral assumptions about $\Delta_g + V$. From what I can tell, there is a large literature on this. A more recent paper is arxiv.org/abs/1105.0873. See Proposition 1.38.

Comment by Yakov Shlapentokh-Rothman

2011-08-17T20:13:59Z

Thanks, I'm not sure why it didn't occur to me to set $A=R(i)^{−1}+i$...

Comment by Yakov Shlapentokh-Rothman

2011-06-29T13:54:25Z

The maximum principle is still the same. It just might be harder to apply if one does not know the boundary values exactly...

Comment by Yakov Shlapentokh-Rothman

2011-06-24T13:33:36Z

What does formally integrable mean in this context?

Comment by Yakov Shlapentokh-Rothman

2011-06-15T17:52:08Z

Thanks, that makes sense.

Comment by Yakov Shlapentokh-Rothman

2011-06-15T13:50:40Z

Is approach (b) supposed to work for $\Omega \neq \mathbb{R}^n$? It is not clear to me how the Fourier transform will be useful unless $\Omega = \mathbb{R}^n$, since that is the only case in which the Sobolev spaces are defined directly in terms of the Fourier transform.

Comment by Yakov Shlapentokh-Rothman

2011-06-08T16:52:04Z

Ah, thanks. I was sure that I was missing something

Comment by Yakov Shlapentokh-Rothman

2011-06-08T15:55:39Z

In the above comment, the 2 should be replaced by $s$ where $s > \tau$.

Comment by Yakov Shlapentokh-Rothman

2011-06-08T15:53:50Z

I think I am missing something obvious, but why can you not pick any $\phi \in C^2$ and set $C = C(\vert\vert \phi\vert\vert_{C^2(B_{r+2}(0))})$, i.e. pull out the derivatives of $\phi$ in $L^{\infty}$?