Stability of eigenfunctions - Revision history

Teorth: /* Another Sobolev bound */

2012-07-14T22:42:00Z

Another Sobolev bound

← Older revision		Revision as of 15:42, 14 July 2012
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	This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation1.pdf this set of notes].		This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation1.pdf this set of notes].

	~~== Another Sobolev bound ==~~


	The Bessel function <math>J_\nu(x)</math> for <math>\nu \geq 0</math> has the Weierstrass factorisation		The Bessel function <math>J_\nu(x)</math> for <math>\nu \geq 0</math> has the Weierstrass factorisation
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	:<math>J_\nu(x) = \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})</math>		:<math>J_\nu(x) = \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})</math>

	where <math>j_{\nu,i}</math> are the positive zeroes of <math>J_\nu</math>; see e.g. [http://books.google.com/books?id=jEmhV5xzdZsC&pg=PA254&lpg=PA254&dq=weierstrass+factorization+of+bessel+function&source=bl&ots=3k8JK3Vsfg&sig=Ggr_ZnkgTXVva2mqrlQIszVviOo&hl=en&sa=X&ei=OrcBULr5JefO2gXo1NCUCw&ved=0CE4Q6AEwAw#v=onepage&q=weierstrass%20factorization%20of%20bessel%20function&f=false this link]. ~~Thus if we define the normalised Bessel function~~		where <math>j_{\nu,i}</math> are the positive zeroes of <math>J_\nu</math>; see e.g. [http://books.google.com/books?id=jEmhV5xzdZsC&pg=PA254&lpg=PA254&dq=weierstrass+factorization+of+bessel+function&source=bl&ots=3k8JK3Vsfg&sig=Ggr_ZnkgTXVva2mqrlQIszVviOo&hl=en&sa=X&ei=OrcBULr5JefO2gXo1NCUCw&ved=0CE4Q6AEwAw#v=onepage&q=weierstrass%20factorization%20of%20bessel%20function&f=false this link].

	~~:<math> \tilde J_\nu(x) := \frac{2^\nu \Gamma(\nu+1)}{x^\nu} J_\nu(x) = \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})</math>~~

	~~then we have~~

	~~:<math>0 \leq \tilde J_\nu(x) \leq 1</math>~~		In particular, we have

	~~for~~ <math>0 \~~leq~~ x \~~leq j_~~{\~~nu,~~1}~~</math>. Also, it is known that <math>~~j_{\nu,i}</~~math> are non~~-~~decreasing in <math>~~\nu</math> (page 508 of WATSON, G.N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944; one can also use here the Sturm comparison theorem). Thus we see that we in fact have		:<math>J_\nu(x)/J_\nu(y) = (x/y)^\nu \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})/(1 - \frac{y^2}{j_{\nu,i}^2}).</math>

	:<math>0 \leq \~~tilde J_~~{\nu_0}(x) \~~leq \tilde J_~~\nu(x) \~~leq 1~~ </math> ~~(1)~~		It is known that <math>j_{\nu,i}</math> are non-decreasing in <math>\nu</math> (page 508 of WATSON, G.N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944; one can also use here the Sturm comparison theorem). Thus if <math>0 \leq x \leq y \leq j_{\nu_0,1}</math> and <math>\nu \geq \nu_0</math>, then
			<math>(1 - \frac{x^2}{j_{\nu,i}^2})/(1 - \frac{y^2}{j_{\nu,i}^2})</math> is non-increasing in <math>\nu</math>, and thus

	~~whenever~~ <math>0 \~~leq~~ \~~nu_0~~ \leq \nu~~</math> and <math>0~~ \~~leq~~ x ~~\leq j_~~{\nu_0,1}</math>.		:<math>J_\nu(x)/J_\nu(y) \leq (x/y)^{\nu-\nu_0} J_{\nu_0}(x)/J_{\nu_0}(y) </math>. (1)

	Now, consider a solution to the eigenfunction equation <math>-\Delta u = \lambda u</math> in the sector <math>\{ (r,\theta): 0 \leq r \leq R; 0 \leq \theta \leq \alpha\}</math> for some acute <math>0 < \alpha < \pi/2</math> and some radius <math>R>0</math>, obeying the Neumann conditions <math>\partial_\theta u = 0</math> for <math>\theta=0,\alpha</math>. For technical reasons we will also need the upper bound		Now, consider a solution to the eigenfunction equation <math>-\Delta u = \lambda u</math> in the sector <math>\{ (r,\theta): 0 \leq r \leq R; 0 \leq \theta \leq \alpha\}</math> for some acute <math>0 < \alpha < \pi/2</math> and some radius <math>R>0</math>, obeying the Neumann conditions <math>\partial_\theta u = 0</math> for <math>\theta=0,\alpha</math>. For technical reasons we will also need the upper bound

	:<math>R \leq j_{\pi/\alpha,1}</math> (2)		:<math>\sqrt{\lambda} R \leq j_{\pi/\alpha,1}</math> (2)

	which seems to hold in practice.		which seems to hold in practice.
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	:<math>u(r,\theta) = \sum_{k=0}^\infty c_k J_{\pi k/\alpha}(\sqrt{\lambda} r) \cos(\pi k \theta/\alpha)</math>		:<math>u(r,\theta) = \sum_{k=0}^\infty c_k J_{\pi k/\alpha}(\sqrt{\lambda} r) \cos(\pi k \theta/\alpha)</math>

	for some coefficients <math>~~c_k</math>. We may renormalise this as~~		for some coefficients <math>c_k</math>. From (2) and the triangle inequality we thus have the bound

	~~:<math>u(r,\theta) = u(0) J_0(\sqrt{\lambda} r) + \sum_{k=1}^\infty \tilde c_k \tilde J_{\pi k/\alpha}(\sqrt{\lambda} r) r^{\pi k/ \alpha} \cos(\pi k \theta/\alpha)</math>~~

	~~for some renormalised coefficients <math>\tilde~~ c_k</math>. From (1) and the triangle inequality we thus have the bound

	:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq \sum_{k=1}^\infty \|~~\tilde~~ c_k\| ~~r_0^~~{\pi k/\alpha}</math>		:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq \sum_{k=1}^\infty \|c_k\| J_{\pi k/\alpha}(\sqrt{\lambda} r_0)</math>

	for any <math>0 \leq r_0 \leq R</math>. To bound the coefficients, we assume an L^2 normalisation		for any <math>0 \leq r_0 \leq R</math>. To bound the coefficients, we assume an L^2 normalisation
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	The left-hand side may be expanded as		The left-hand side may be expanded as

	:<math> \alpha \|u(0)\|^2 \int_0^R J_0(\sqrt{\lambda} r)^2 r dr + \frac{\alpha}{2} \sum_{k=1}^\infty \|~~\tilde~~ c_k\|^2 \int_0^R ~~\tilde~~ J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r~~^{1+2\pi k/\alpha}~~ dr</math>		:<math> \alpha \|u(0)\|^2 \int_0^R J_0(\sqrt{\lambda} r)^2 r dr + \frac{\alpha}{2} \sum_{k=1}^\infty \|c_k\|^2 \int_0^R J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r dr</math>

	and thus		and thus

	:<math> \sum_{k=1}^\infty \|~~\tilde~~ c_k\|^2 \int_0^R ~~\tilde~~ J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r~~^{1+2\pi k/\alpha}~~ dr \leq A</math>		:<math> \sum_{k=1}^\infty \|c_k\|^2 \int_0^R J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r dr \leq A</math>

	where A is the quantity		where A is the quantity
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	where B(r_0) is the quantity		where B(r_0) is the quantity

	:<math> B(r_0) := \sum_{k=1}^\infty \frac{~~r_0^{2(k-~~1~~)\pi/\alpha}~~}{ \int_0^R ~~\tilde~~ J_{\pi k/\alpha}(\sqrt{\lambda} r)~~^2 r^~~{~~1+2~~\pi k/\alpha} dr }.</math>		:<math> B(r_0) := \sum_{k=1}^\infty \frac{1}{ \int_0^R (J_{\pi k/\alpha}(\sqrt{\lambda} r)/J_{\pi k/\alpha}(\sqrt{\lambda} r_0))^2 r dr }.</math>

	We may upper bound B(r_0) as follows. By (1) and (2), we may replace <math>\tilde J_{\pi k/\alpha}</~~math> here by <math>\tilde~~ J_{\pi/\alpha}~~</math>. We also select an intermediate radius <math>r_0~~ \~~leq r_1~~ \~~leq R~~</math> ~~and reduce the range of integration from [0,R] to~~ <math>[r_1~~,R]<~~/~~math>. Then one can lower bound <math>r~~^{~~1+2~~\pi k/\alpha}~~</math> by <math>r^~~{~~1+2~~\pi/\alpha} ~~r_1^~~{~~2(k-1~~)\pi/\alpha}</math>~~, and so~~		We may upper bound B(r_0) as follows. We may reduce the range of integration from [0,R] to <math>[r_1,R]</math> for some intermediate radius <math>r_0 \leq r_1 \leq R</math>.
			By (1) and (2), we may replace <math>\tilde J_{\pi k/\alpha}(\sqrt{\lambda} r)/J_{\pi k/\alpha}(\sqrt{\lambda} r_0)</math> by
			<math>(r_1/r_0)^{\pi(k-1)/\alpha} \tilde J_{\pi/\alpha}(\sqrt{\lambda} r)/J_{\pi/\alpha}(\sqrt{\lambda} r_0)</math>. Thus

	:<math> B(r_0) \leq \sum_{k=1}^\infty \frac{(r_0/r_1)^{2(k-1)\pi/\alpha}}{ \int_{r_1}^R ~~\tilde~~ J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r~~^{1+2\pi/\alpha}~~ dr }.</math>		:<math> B(r_0) \leq \sum_{k=1}^\infty \frac{(r_0/r_1)^{2(k-1)\pi/\alpha} J_{\pi/\alpha}(\sqrt{\lambda} r_0) }{ \int_{r_1}^R J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r dr }.</math>

	One may then sum the geometric series and conclude that		One may then sum the geometric series and conclude that

	:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq ~~r_0^~~{\pi/\alpha} A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}</math> (5)		:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq J_{\pi/\alpha}(\sqrt{\lambda} r_0) A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}</math> (5)

	for any <math>r_0 \leq r_1 \leq R</math>, where		for any <math>r_0 \leq r_1 \leq R</math>, where

	:<math>C(r_1) := \int_{r_1}^R ~~\tilde~~ J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r~~^{1+2\pi/\alpha}~~ dr.</math>		:<math>C(r_1) := \int_{r_1}^R J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r dr.</math>

	One can use (5) to exclude extrema for small values of r_0. For instance, (5) shows that <math>\|u(r_0,\theta)\|</math> cannot exceed <math>\|u(0)\|</math> unless		One can use (5) to exclude extrema for small values of r_0. For instance, (5) shows that <math>\|u(r_0,\theta)\|</math> cannot exceed <math>\|u(0)\|</math> unless

	:<math> \|u(0)\| (1 - \|J_0(\sqrt{\lambda} r_0)\|) \geq ~~r_0^~~{\pi/\alpha} A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}.</math>		:<math> \|u(0)\| (1 - \|J_0(\sqrt{\lambda} r_0)\|) \geq J_{\pi/\alpha}(r_0) A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}.</math>

	The left hand side behaves like r_0^2, and the right-hand side like <math>r_0^{\pi/\alpha}</math>, so for acute <math>\alpha</math> this inequality becomes impossible for small enough <math>r_0</math> (and "small enough" can be made numerically explicit if one is willing to compute with Bessel functions). Things become bad when <math>\alpha</math> is close to a right angle, but it seems in those cases that the extremum is obtained elsewhere.		The left hand side behaves like r_0^2, and the right-hand side like <math>r_0^{\pi/\alpha}</math>, so for acute <math>\alpha</math> this inequality becomes impossible for small enough <math>r_0</math> (and "small enough" can be made numerically explicit if one is willing to compute with Bessel functions). Things become bad when <math>\alpha</math> is close to a right angle, but it seems in those cases that the extremum is obtained elsewhere.

Teorth: /* An alternate approach to the H^2 theory */

2012-07-14T19:59:51Z

An alternate approach to the H^2 theory

← Older revision		Revision as of 12:59, 14 July 2012
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	This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation1.pdf this set of notes].		This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation1.pdf this set of notes].

			== Another Sobolev bound ==


			The Bessel function <math>J_\nu(x)</math> for <math>\nu \geq 0</math> has the Weierstrass factorisation

			:<math>J_\nu(x) = \frac{x^\nu}{2^\nu \Gamma(\nu+1)} \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})</math>

			where <math>j_{\nu,i}</math> are the positive zeroes of <math>J_\nu</math>; see e.g. [http://books.google.com/books?id=jEmhV5xzdZsC&pg=PA254&lpg=PA254&dq=weierstrass+factorization+of+bessel+function&source=bl&ots=3k8JK3Vsfg&sig=Ggr_ZnkgTXVva2mqrlQIszVviOo&hl=en&sa=X&ei=OrcBULr5JefO2gXo1NCUCw&ved=0CE4Q6AEwAw#v=onepage&q=weierstrass%20factorization%20of%20bessel%20function&f=false this link]. Thus if we define the normalised Bessel function

			:<math> \tilde J_\nu(x) := \frac{2^\nu \Gamma(\nu+1)}{x^\nu} J_\nu(x) = \prod_{i=1}^\infty (1 - \frac{x^2}{j_{\nu,i}^2})</math>

			then we have

			:<math>0 \leq \tilde J_\nu(x) \leq 1</math>

			for <math>0 \leq x \leq j_{\nu,1}</math>. Also, it is known that <math>j_{\nu,i}</math> are non-decreasing in <math>\nu</math> (page 508 of WATSON, G.N., A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944; one can also use here the Sturm comparison theorem). Thus we see that we in fact have

			:<math>0 \leq \tilde J_{\nu_0}(x) \leq \tilde J_\nu(x) \leq 1 </math> (1)

			whenever <math>0 \leq \nu_0 \leq \nu</math> and <math>0 \leq x \leq j_{\nu_0,1}</math>.

			Now, consider a solution to the eigenfunction equation <math>-\Delta u = \lambda u</math> in the sector <math>\{ (r,\theta): 0 \leq r \leq R; 0 \leq \theta \leq \alpha\}</math> for some acute <math>0 < \alpha < \pi/2</math> and some radius <math>R>0</math>, obeying the Neumann conditions <math>\partial_\theta u = 0</math> for <math>\theta=0,\alpha</math>. For technical reasons we will also need the upper bound

			:<math>R \leq j_{\pi/\alpha,1}</math> (2)

			which seems to hold in practice.

			By separation of variables, we can write u in polar coordinates as

			:<math>u(r,\theta) = \sum_{k=0}^\infty c_k J_{\pi k/\alpha}(\sqrt{\lambda} r) \cos(\pi k \theta/\alpha)</math>

			for some coefficients <math>c_k</math>. We may renormalise this as

			:<math>u(r,\theta) = u(0) J_0(\sqrt{\lambda} r) + \sum_{k=1}^\infty \tilde c_k \tilde J_{\pi k/\alpha}(\sqrt{\lambda} r) r^{\pi k/ \alpha} \cos(\pi k \theta/\alpha)</math>

			for some renormalised coefficients <math>\tilde c_k</math>. From (1) and the triangle inequality we thus have the bound

			:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq \sum_{k=1}^\infty \|\tilde c_k\| r_0^{\pi k/\alpha}</math>

			for any <math>0 \leq r_0 \leq R</math>. To bound the coefficients, we assume an L^2 normalisation

			:<math> \int_0^R \int_0^{\alpha} \|u(r,\theta)\|^2\ rdr \leq 1.</math>

			The left-hand side may be expanded as

			:<math> \alpha \|u(0)\|^2 \int_0^R J_0(\sqrt{\lambda} r)^2 r dr + \frac{\alpha}{2} \sum_{k=1}^\infty \|\tilde c_k\|^2 \int_0^R \tilde J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r^{1+2\pi k/\alpha} dr</math>

			and thus

			:<math> \sum_{k=1}^\infty \|\tilde c_k\|^2 \int_0^R \tilde J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r^{1+2\pi k/\alpha} dr \leq A</math>

			where A is the quantity

			:<math> A := \frac{2}{\alpha} - 2 \|u(0)\|^2 \int_0^R J_0(\sqrt{\lambda} r)^2 r dr.</math> (3)

			Thus by the Cauchy-Schwarz inequality, we have

			:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq r_0^{\pi/\alpha} A^{1/2} B(r_0)^{1/2} </math> (4)

			where B(r_0) is the quantity

			:<math> B(r_0) := \sum_{k=1}^\infty \frac{r_0^{2(k-1)\pi/\alpha}}{ \int_0^R \tilde J_{\pi k/\alpha}(\sqrt{\lambda} r)^2 r^{1+2\pi k/\alpha} dr }.</math>

			We may upper bound B(r_0) as follows. By (1) and (2), we may replace <math>\tilde J_{\pi k/\alpha}</math> here by <math>\tilde J_{\pi/\alpha}</math>. We also select an intermediate radius <math>r_0 \leq r_1 \leq R</math> and reduce the range of integration from [0,R] to <math>[r_1,R]</math>. Then one can lower bound <math>r^{1+2\pi k/\alpha}</math> by <math>r^{1+2\pi/\alpha} r_1^{2(k-1)\pi/\alpha}</math>, and so

			:<math> B(r_0) \leq \sum_{k=1}^\infty \frac{(r_0/r_1)^{2(k-1)\pi/\alpha}}{ \int_{r_1}^R \tilde J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r^{1+2\pi/\alpha} dr }.</math>

			One may then sum the geometric series and conclude that

			:<math> \|u(r_0,\theta) - u(0) J_0(\sqrt{\lambda} r_0)\| \leq r_0^{\pi/\alpha} A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}</math> (5)

			for any <math>r_0 \leq r_1 \leq R</math>, where

			:<math>C(r_1) := \int_{r_1}^R \tilde J_{\pi/\alpha}(\sqrt{\lambda} r)^2 r^{1+2\pi/\alpha} dr.</math>

			One can use (5) to exclude extrema for small values of r_0. For instance, (5) shows that <math>\|u(r_0,\theta)\|</math> cannot exceed <math>\|u(0)\|</math> unless

			:<math> \|u(0)\| (1 - \|J_0(\sqrt{\lambda} r_0)\|) \geq r_0^{\pi/\alpha} A^{1/2} ((r_1/r_0)^{2\pi/\alpha}-1)^{-1/2} C(r_1)^{-1/2}.</math>

			The left hand side behaves like r_0^2, and the right-hand side like <math>r_0^{\pi/\alpha}</math>, so for acute <math>\alpha</math> this inequality becomes impossible for small enough <math>r_0</math> (and "small enough" can be made numerically explicit if one is willing to compute with Bessel functions). Things become bad when <math>\alpha</math> is close to a right angle, but it seems in those cases that the extremum is obtained elsewhere.

Teorth: /* An alternate approach to the H^2 theory */

2012-07-06T21:52:27Z

An alternate approach to the H^2 theory

← Older revision		Revision as of 14:52, 6 July 2012
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	By Sobolev embedding, (noting also that we have L^2 and H^1 control on v) we can thus obtain L^infty control on v (or equivalently, u) in terms of the quantity X. So the main task is to understand how to bound X, which in the ABC coordinates is basically the L^2 norm of v weighted by a reasonably explicit logarithmic weight.		By Sobolev embedding, (noting also that we have L^2 and H^1 control on v) we can thus obtain L^infty control on v (or equivalently, u) in terms of the quantity X. So the main task is to understand how to bound X, which in the ABC coordinates is basically the L^2 norm of v weighted by a reasonably explicit logarithmic weight.

	This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/~~perturbation~~.pdf this set of notes].		This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation1.pdf this set of notes].

Teorth: /* An alternate approach to the H^2 theory */

2012-07-06T02:04:48Z

An alternate approach to the H^2 theory

← Older revision		Revision as of 19:04, 5 July 2012
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	By Sobolev embedding, (noting also that we have L^2 and H^1 control on v) we can thus obtain L^infty control on v (or equivalently, u) in terms of the quantity X. So the main task is to understand how to bound X, which in the ABC coordinates is basically the L^2 norm of v weighted by a reasonably explicit logarithmic weight.		By Sobolev embedding, (noting also that we have L^2 and H^1 control on v) we can thus obtain L^infty control on v (or equivalently, u) in terms of the quantity X. So the main task is to understand how to bound X, which in the ABC coordinates is basically the L^2 norm of v weighted by a reasonably explicit logarithmic weight.

			This is done more explicitly at [http://terrytao.files.wordpress.com/2012/07/perturbation.pdf this set of notes].

Teorth: /* H^2 theory */

2012-07-04T19:07:15Z

H^2 theory

← Older revision		Revision as of 12:07, 4 July 2012
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	and so as <math>\alpha < \pi</math>, we can bound <math>\int_{\Omega_A} \frac{\|\nabla u\|^2}{\|x\|^2}</math> as required.		and so as <math>\alpha < \pi</math>, we can bound <math>\int_{\Omega_A} \frac{\|\nabla u\|^2}{\|x\|^2}</math> as required.


			== An alternate approach to the H^2 theory ==

			Yet another approach to the H^2 theory, which seems to give better results, comes from using Schwarz-Christoffel transformations, with the upper half-plane H being the reference domain.

			More precisely, given three angles <math>\alpha+\beta+\gamma=\pi</math> for a triangle, we can find a Schwarz-Christoffel transformation <math>\Phi_{\alpha,\beta}: H \to \Omega(\alpha,\beta)</math> from the half-plane <math>H = \{z: Im(z) > 0\}</math> to a triangle <math>\Omega_{\alpha,\beta}</math> with angles <math>\alpha,\beta,\gamma</math> by solving the differential equation

			:<math> \frac{d}{dz} \Phi_{\alpha,\beta}(z) = \frac{1}{z^{1-\alpha/\pi} (z-1)^{1-\beta/\pi}}</math>

			with initial condition <math>\Phi_{\alpha,\beta}(0)=0</math>, keeping <math>\Phi_{\alpha,\beta}</math> conformal. In particular, <math>\Phi_{\alpha,\beta}(z)</math> behaves like <math>A + \frac{\pi}{\alpha} z^{\alpha/\pi}</math> near the origin, <math>B + \frac{\pi}{\beta} (z-1)^{\beta/\pi}</math> near 1, and <math>C -\frac{\pi}{\gamma} z^{-\gamma/\pi}</math> at infinity, for some complex numbers A,B,C with A=0. The image <math>\Omega_{\alpha,\beta}</math> is a triangle ABC with angles <math>\alpha,\beta,\gamma</math>.

			Write <math>w = \Phi_{\alpha,\beta}(z)</math>, then

			:<math> dw = \frac{dz}{z^{1-\alpha/\pi} (z-1)^{1-\beta/\pi}}.</math>

			The Lebesgue measure dm_w on the triangle is related to the Lebesgue measure dm_z on the half-plane by the change of variables formula

			:<math> \int_{\Omega_{\alpha,\beta}} F(w) dm_w = \int_H \frac{F(\Phi_{\alpha,\beta}(z))}{\|z\|^{2(1-\alpha/\pi)} \|z-1\|^{2(1-\beta/\pi)}} dm_z.</math>

			We write the expression <math>\frac{1}{\|z\|^{2(1-\alpha/\pi)} \|z-1\|^{2(1-\beta/\pi)}}</math> as <math>e^{2\omega}</math>, where

			:<math>\omega = (\frac{\alpha}{\pi}-1) \log \|z\| + (\frac{\beta}{\pi}-1) \log \|z-1\|.</math>

			The Rayleigh quotient, pulled back to the reference half-plane H, is then

			:<math>\int_H \|\nabla u\|^2\ dm_z / \int_H \|u\|^2 e^{2\omega}\ dm_z</math>

			and so the second eigenfunction u_2, in the reference half-plane will obey the equation

			:<math> -\Delta u_2 = \lambda_2 e^{2\omega} u_2</math> (C.1)

			with Neumann boundary condition

			:<math> n \cdot \nabla u_2 = 0</math> (C.2)

			on the boundary of the half-plane and the mean zero condition

			:<math>\int_H u_2 e^{2\omega}\ dm_z = 0</math>. (C.3)

			We can normalise u_2 in L^2 norm,

			:<math>\int_H \|u_2\|^2 e^{2\omega}\ dm_z = 1</math> (C.4)

			which by the Rayleigh quotient immediately gives an H^1 bound

			:<math>\int_H \|\nabla u_2\|^2\ dm_z = \lambda</math> (C.5)

			Now suppose that we vary <math>\alpha,\beta</math> smoothly with respect to a time parameter; this means that <math>\omega</math> will also vary by the formula

			:<math>\dot \omega = \frac{\dot \alpha}{\pi} \log \|z\| + \frac{\dot \beta}{\pi} \log \|z-1\|.</math> (C.6)

			The eigenvalue <math>\dot \lambda_2</math> can also vary at a rate which can be controlled by the previous perturbation theory (e.g. equation (14)). As for the variation of the eigenfunction <math>\dot u_2</math>, we can compute this by differentiating (C.1) to obtain

			:<math> - \Delta \dot u_2 = \dot \lambda_2 e^{2\omega} u_2 + 2 \lambda_2 \dot \omega e^{2 \omega} u_2 + \lambda_2 e^{2\omega} \dot u_2</math>. (C.7)

			Also, on differentating (C.2) we see that <math>\dot u_2</math> conveniently satisfies a homogeneous Neumann condition:

			:<math> n \cdot \nabla \dot u_2 = 0</math> (C.8)

			Using the orthonormal basis <math> u_k e^\omega</math> of <math>L^2(H, dm_z)</math> as in the previous notes on L^2 and H^1 stability, we can obtain an explicit formula for <math> \dot u_2</math>. Indeed, integrating (C.7) against another eigenfunction u_l and integrating by parts using (C.1), (C.8) gives

			:<math> \lambda_l \int_H \dot u_2 u_l e^{2\omega}\ dm_z= 2 \lambda_2 \int_H \dot \omega e^{2 \omega} u_2 u_l\ dm_z+ \lambda_2 \int_H e^{2\omega} \dot u_2 e^{2\omega} u_l\ dm_z</math>

			and thus

			:<math> \int_H \dot u_2 u_l e^{2\omega}\ dm_z= 2 \frac{\lambda_2}{\lambda_l-\lambda_2} \int_H \dot \omega e^{2 \omega} u_2 u_l\ dm_z</math>

			and hence by the eigenfunction expansion

			:<math> \dot u_2 = \sum_{l \neq 2} 2 (\frac{\lambda_2}{\lambda_l-\lambda_2} \int_H \dot \omega e^{2 \omega} u_2 u_l\ dm_z) u_l</math>

			which among other things gives the L^2 identity

			:<math> \int_H (\dot u_2)^2 e^{2\omega}\ dm_z =
			\sum_{l \neq 2} 4 \|\frac{\lambda_2}{\lambda_l-\lambda_2} \int_H \dot \omega e^{2 \omega} u_2 u_l\ dm_z\|^2</math>

			and hence by the Bessel inequality

			:<math> \int_H (\dot u_2)^2 e^{2\omega}\ dm_z \leq \frac{4 \lambda_2^2}{\|\lambda_3-\lambda_2\|^2} X</math> (C.9)

			where X is the quantity

			:<math>X : =\int_H \|\dot\omega\|^2 e^{2\omega} u_2^2\ dm_z</math>. (C.10)

			The quantity cannot quite be bounded directly by (C.4) because of the logarithmic divergence in the weight <math>\|\dot \omega\|^2</math>, but it should be controllable by also employing (C.5) and (C.3), perhaps after transferring back to the triangle ABC.

			The quantity X also controls <math>\dot \lambda_2</math>. Indeed, on integrating (C.7) against u_2 and using (C.1), (C.2), (C.4) we obtain that

			:<math> 0 = \dot \lambda_2 + 2 \lambda_2 \int_H \dot \omega e^{2\omega} u_2\ dm_z</math>

			and hence by Cauchy-Schwarz and (C.4)

			:<math> \|\dot \lambda_2\| \leq 2 \lambda_2 X^{1/2}.</math> (C.11)

			If we now multiply (C.7) by <math>e^{-\omega}</math> and then take L^2 norms using the triangle inequality and the previous estimates (C.9), (C.10), (C.11) we conclude that

			:<math> (\int_H \|\Delta \dot u_2\|^2 e^{-2\omega}\ dm_z)^{1/2} \leq 2\lambda_2 X^{1/2} + 2 \lambda_2 X^{1/2} + \lambda_2 (\frac{4\lambda_2^2}{\|\lambda_3-\lambda_2\|^2} X)^{1/2}</math>

			which simplifies to

			:<math> (\int_H \|\Delta \dot u_2\|^2 e^{-2\omega}\ dm_z)^{1/2} \leq \frac{4 \lambda_2 \lambda_3}{\lambda_3-\lambda_2} X^{1/2}.</math>

			We can transfer this back to the triangle <math>\Omega_{\alpha,\beta}=ABC</math> by the change of variables

			:<math> \dot v(w) := \dot u( \Phi_{\alpha,\beta}^{-1}(w))</math>

			giving

			:<math> (\int_{\Omega_{\alpha,\beta}} \|\Delta v\|^2\ dm_w)^{1/2} \leq \frac{4\lambda_2 \lambda_3}{\lambda_3-\lambda_2} X^{1/2}</math>

			which after two integration by parts (Bochner-Weitzenbock inequality) gives an H^2 bound on v:

			:<math> (\int_{\Omega_{\alpha,\beta}} \|\nabla^2 v\|^2\ dm_w)^{1/2} \leq \frac{4\lambda_2 \lambda_3}{\lambda_3-\lambda_2} X^{1/2}.</math>

			By Sobolev embedding, (noting also that we have L^2 and H^1 control on v) we can thus obtain L^infty control on v (or equivalently, u) in terms of the quantity X. So the main task is to understand how to bound X, which in the ABC coordinates is basically the L^2 norm of v weighted by a reasonably explicit logarithmic weight.

Teorth: /* L^2 and H^1 theory */

2012-07-04T18:33:50Z

L^2 and H^1 theory

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	provided that the denominator is positive. In terms of the condition number <math>\kappa := \sigma_2(B)/\sigma_1(B)</math> of B, this becomes		provided that the denominator is positive. In terms of the condition number <math>\kappa := \sigma_2(B)/\sigma_1(B)</math> of B, this becomes

	:<math> \\|\nabla v \\|_2 \leq \frac{(\kappa^2-1) \lambda_2^{1/2}}{1 - \kappa^2 \frac{\lambda_2}{\lambda_3}} </math>		:<math> \\|\nabla v \\|_2 \leq \frac{(\kappa^2-1) \lambda_2^{1/2}}{1 - \kappa^2 \frac{\lambda_2}{\lambda_3}} </math> (15)

	which is a non-trivial bound when <math>\kappa < (\lambda_3/\lambda_2)^{1/2}</math>, and is of the order of <math>O(\lambda_2^{1/2} (\kappa-1))</math> when <math>\kappa</math> is close to 1. Using (8), we conclude in particular that		which is a non-trivial bound when <math>\kappa < (\lambda_3/\lambda_2)^{1/2}</math>, and is of the order of <math>O(\lambda_2^{1/2} (\kappa-1))</math> when <math>\kappa</math> is close to 1. Using (8), we conclude in particular that

	:<math> \\|v \\|_2 \leq \frac{(\kappa^2-1) (\lambda_2/\lambda_3)^{1/2}}{1 - \kappa^2 \frac{\lambda_2}{\lambda_3}} </math>		:<math> \\|v \\|_2 \leq \frac{(\kappa^2-1) (\lambda_2/\lambda_3)^{1/2}}{1 - \kappa^2 \frac{\lambda_2}{\lambda_3}} </math> (16)

	For these calculations performed on a third reference triangle, http://www.math.sfu.ca/~nigam/polymath-figures/Perturbation.pdf		For these calculations performed on a third reference triangle, http://www.math.sfu.ca/~nigam/polymath-figures/Perturbation.pdf

Teorth: /* Sobolev inequalities in logarithmic coordinates */

2012-07-04T00:26:28Z

Sobolev inequalities in logarithmic coordinates

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	Combining this with (A.16) for various choices of cutoff <math>\eta</math> gives integral control on <math>\partial_s E(s)</math>, and this combined with (A.16) gives pointwise control on E(s) and hence on v(s). This gives L^infty control on the eigenfunction u near the vertex. (The computations are somewhat ugly, though, particularly if one wants to optimise in eta.)		Combining this with (A.16) for various choices of cutoff <math>\eta</math> gives integral control on <math>\partial_s E(s)</math>, and this combined with (A.16) gives pointwise control on E(s) and hence on v(s). This gives L^infty control on the eigenfunction u near the vertex. (The computations are somewhat ugly, though, particularly if one wants to optimise in eta.)

			== H^2 theory ==

			Formally, if on a triangle <math>\Omega</math>, a function v obeys the inhomogeneous Laplace equation

			:<math>-\Delta v = F</math> (B.1)

			with homogeneous Neumann boundary data <math>n \cdot \nabla v = 0</math>, then after two integrations by parts we have the H^2 bound

			:<math>\int_\Omega \|\nabla^2 v\|^2 = \int_\Omega \|F\|^2</math>

			(Bochner-Weitzenbock identity.) The integrations by parts can be justified if v is smooth on Omega, with first derivative uniformly bounded and second derivative blowing up by at most <math>O(r^{-1+\varepsilon})</math> at each vertex for some <math>\varepsilon>0</math>, by the usual trick of inserting a cutoff function to smoothly cut out the integral at the vertices, performing the integration by parts, and then sending the cutoff parameter to zero. I think this sort of regularity for acute-angled triangles can be achieved in practice without difficulty.

			Now suppose that one is solving the inhomogeneous Laplace equation (B.1) but with inhomogeneous Neumann data

			:<math>n \cdot \nabla v = f</math>. (B.2)

			Then the integration by parts becomes much less favorable, as there are now a number of nasty boundary terms to take care of. But suppose that one can find a reasonably smooth function w (of the same regularity needed for v, i.e. first derivative uniformly bounded and second derivative blowing up slower than 1/r) which solves (B.2) but not (B.1), i.e.

			:<math>n \cdot \nabla w = f</math>

			on the boundary. Then the difference v-w solves (B.1) with forcing term <math>F + \Delta w</math> and thus

			:<math> \\| \nabla^2(v-w) \\|_{L^2(\Omega)} = \\| F + \Delta w \\|_{L^2(\Omega)}.</math>

			From the pointwise estimate <math>\|\Delta w\| \leq \sqrt{2} \|\nabla^2 w\|</math> and the triangle inequality we conclude that

			:<math> \\| \nabla^2 v \\|_{L^2(\Omega)} \leq \\| F \\|_{L^2(\Omega)} + (1+\sqrt{2}) \\| w \\|_{L^2(\Omega)}.</math> (B.3)

			We can apply this to the first variation <math>\dot u = \dot u_2</math> of the second eigenfunction at M=I. From the L^2 and H^1 stability theory one has

			:<math>-\Delta \dot u = \lambda_2 \dot u + \nabla \cdot \dot M \nabla u + \dot \lambda_2 u</math>

			with boundary condition

			:<math>n \cdot \nabla \dot u = - n \cdot \dot M \nabla u</math> (B.4)

			and thus we have the H^2 bound

			:<math> \\| \nabla^2 \dot u \\|_{L^2(\Omega)} \leq \lambda_2 \\| \dot u \\|_{L^2(\Omega)} + \\| \nabla \cdot \dot M \nabla u \\|_{L^2(\Omega)} + \|\dot \lambda_2\| \\|u \\|_{L^2(\Omega)} + (1+\sqrt{2}) \\|w\\|_{L^2(\Omega)}</math> (B.5)

			where w is any function with boundary value (B.4). From the L^2 stability theory one has

			:<math> \\| \dot u \\|_{L^2(\Omega)}\leq \frac{1}{\lambda_3 - \lambda_2} \\| \dot M \nabla u \\|_{L^2}</math>

			and so one has bounds on everything on the RHS of (B.5) except for the last term.

			To control this term, we divide the triangle <math>\Omega = ABC</math> into three overlapping regions <math>\Omega_A, \Omega_B, \Omega_C</math>, each one being a region around one of the three vertices A,B,C that avoids the other two vertices, as well as the edge joining those vertices. In each of these regions, we find a function w that solves the boundary condition (B.5) in that region, which obeys good H^2 bounds; combining these functions together by a partition of unity will then give a global w in <math>H^2(\Omega)</math> that obeys (B.5) on the entire boundary.

			Let's look at the region <math>\Omega_A</math> near the vertex A, which we set to be the origin. In polar coordinates <math>(r,\theta)</math>, we know that <math>\partial_\theta u = 0</math> on the edges AB, AC, and so the boundary condition (B.4) becomes

			:<math> n \cdot \nabla \dot u = c_{AB} \partial_r u</math>

			on AB and

			:<math> n \cdot \nabla \dot u = c_{AC} \partial_r u</math>

			on AC for some explicit scalar constants <math>c_{AB}, c_{AC}</math> depending only on geometry of the triangle ABC and on <math>\dot M</math>. On <math>\Omega_A</math>, we can build a function w_A that exhibits this data by the formula

			:<math> w_A(x) := u(\Phi(x))</math>

			where <math>\Phi: \Omega_A \to \Omega</math> is a smooth function that is the identity on the edges AB, BC inside <math>\Omega_A</math>, and whose Hessian maps the outward normal n to a vector with radial component <math>c_{AB}</math> on AB and <math>c_{AC}</math> on AC. It is not difficult to see that one can choose such a function on <math>\Omega_A</math> so that <math>\nabla \Phi, (\nabla \Phi)^{-1} = O(1)</math> and <math>\nabla^2 \Phi = O(1/r)</math> (one can choose a function of the form <math> \Phi: (r,\theta) \mapsto (f(\theta) r, \theta)</math> for a suitable function f). The function w_A defined as above will solve the Neumann conditions, and from the chain rule one has

			:<math> \nabla^2 w_A(x) = O( \|\nabla \Phi(x)\|^2 \|\nabla^2 u(\Phi(x))\| ) + O( \|\nabla^2 \Phi(x)\| \|\nabla u(\Phi(x))\| ) </math>

			and so

			:<math> \\| \nabla^2 w_A \\|_{L^2(\Omega_A)} \ll \\| \nabla^2 u \\|_{L^2(\Omega)} + \\| \frac{\nabla u}{\|x\|} \\|_{L^2(\Omega)}.</math>

			A similar estimate (with some additional lower order terms) obtains when one glues together the three partial solutions w_A, w_B, w_C to create a solution w to the boundary problem with

			:<math> \\| \nabla^2 w \\|_{L^2(\Omega)} \ll \\| u \\|_{H^2(\Omega)} + \sum_{v=A,B,C} \\| \frac{\nabla u}{\|x-v\|} \\|_{L^2(\Omega)}.</math>

			To finish the task of estimating <math>\dot u</math> in H^2, one therefore needs bounds on <math>\\| \frac{\nabla u}{\|x\|} \\|_{L^2(\Omega)}</math> (and similarly for the other two vertices of the triangle). I think such bounds can be obtained as follows. Let's work in <math>\Omega_A</math> with some suitable smooth cutoff and ignore terms coming from the derivatives of the cutoff (which don't involve the singularity 1/\|x\| and so are easy to estimate). If we subtract off <math>u(0) J_0(\sqrt{\lambda} r)</math> then u now has mean zero in angular directions. From integration by parts we have

			:<math> \int_{\Omega_A} \frac{\|\nabla u\|^2}{\|x\|^2} =\int_{\Omega_A} \frac{\lambda \|u\|^2}{\|x\|^2} + \int_{\Omega_A} \frac{u \partial_r u}{\|x\|^3} </math>
			:<math> = \int_{\Omega_A} \frac{\lambda \|u\|^2}{\|x\|^2} + \int_{\Omega_A} \frac{\|u\|^2}{\|x\|^4}.</math>

			But if A has angle <math>\alpha</math>, then the Poincare inequality in the angular direction gives

			:<math> \int_{\Omega_A} \frac{\|u\|^2}{\|x\|^4} \leq \frac{\alpha^2}{\pi^2} \int_{\Omega_A} \frac{\|\nabla u\|^2}{\|x\|^2}</math>

			and so as <math>\alpha < \pi</math>, we can bound <math>\int_{\Omega_A} \frac{\|\nabla u\|^2}{\|x\|^2}</math> as required.

Teorth: /* Sobolev inequalities in logarithmic coordinates */

2012-07-01T18:54:29Z

Sobolev inequalities in logarithmic coordinates

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	and hence on a second integration		and hence on a second integration

	:<math> \|\bar{v}(s)-\bar{v}(0)\| \leq 2^{-1/2} \alpha^{-1/2} \lambda e^s</math>. (A.9)		:<math> \|\bar{v}(s)-\bar{v}(-\infty)\| \leq 2^{-1/2} \alpha^{-1/2} \lambda e^s</math>. (A.9)

	From the triangle inequality, we conclude that		From the triangle inequality, we conclude that

	:<math> (\int_{-\infty}^{\log R} \|\bar v\|^2 e^{2s}\ ds)^{1/2} \geq \|\bar v(0)\| 2^{-1/2} R - 2^{-3/2} \alpha^{-1/2} \lambda R^2</math> (A.10)		:<math> (\int_{-\infty}^{\log R} \|\bar v\|^2 e^{2s}\ ds)^{1/2} \geq \|\bar v(-\infty)\| 2^{-1/2} R - 2^{-3/2} \alpha^{-1/2} \lambda R^2</math> (A.10)

	and hence by (A.6)		and hence by (A.6)

	:<math> \|\bar v(0)\| \leq 2^{1/2} \alpha R^{-1} + 2^{-1} \alpha^{-1/2} \lambda R </math> (A.11)		:<math> \|\bar v(-\infty)\| \leq 2^{1/2} \alpha R^{-1} + 2^{-1} \alpha^{-1/2} \lambda R </math> (A.11)

	(which already bounds the original eigenfunction at the vertex) and hence by (A.9)		(which already bounds the original eigenfunction at the vertex) and hence by (A.9)

Teorth: /* A Sobolev inequality */

2012-06-30T17:05:25Z

A Sobolev inequality

← Older revision		Revision as of 10:05, 30 June 2012
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	Because <math>\Delta(\eta u)</math> has mean zero, one can also replace <math>\log \|x\|</math> here by <math>\log \|x\|-C</math> for any constant C, for instance one can use <math>\log(\|x\|/R)</math>.		Because <math>\Delta(\eta u)</math> has mean zero, one can also replace <math>\log \|x\|</math> here by <math>\log \|x\|-C</math> for any constant C, for instance one can use <math>\log(\|x\|/R)</math>.

			== Sobolev inequalities in logarithmic coordinates ==

			The computations appear to become cleaner if we work in logarithmic coordinates

			:<math>(s,\theta) := (\log r, \theta)</math>.

			Thus, for instance, a sector <math>\{ (r,\theta): 0 < r < R; 0 \leq \theta \leq \alpha \}</math> in polar coordinates, becomes a half-infinite strip <math>\{ (s,\theta): s < \log R; 0 \leq \theta \leq \alpha \}</math> in logarithmic coordinates. A triangle <math>\Omega</math> which contains this sector as one of its corners, then becomes a slightly enlarged version <math>\log \Omega</math> of this half-infinite strip, with a concave boundary connecting the two half-infinite edges.

			Consider an eigenfunction <math>-\Delta u = \lambda u</math> in the original domain <math>\Omega</math>. In polar coordinates, the eigenfunction equation is

			:<math> -\partial_{rr} u - \frac{1}{r} \partial_r u - \frac{1}{r^2} \partial_{\theta\theta} u = \lambda u</math>

			the Rayleigh quotient is

			:<math> \int_\Omega (\partial_r u)^2 + \frac{1}{r^2} (\partial_\theta u)^2\ r dr d\theta / \int_\Omega u^2\ r dr d\theta</math>

			and the Neumann boundary condition on the two radial edges are

			:<math> \partial_\theta u = 0</math>.

			If we convert to logarithmic coordinates <math>v: \log \Omega \to \R</math> by the change of variables

			:<math> v(s,\theta) := u(e^s, \theta)</math>

			then one can compute that the eigenfunction equation now becomes

			:<math> - \partial_{ss} v - \partial_{\theta\theta} v = \lambda e^{2s} v</math> (A.1)

			the Rayleigh quotient becomes

			:<math> \int_{\log \Omega} (\partial_s v)^2 + (\partial_\theta v)^2\ ds d\theta / \int_{\log \Omega} \|v\|^2 e^{2s}\ ds d\theta</math>

			and the Neumann boundary conditions on the two half-infinite edges are

			:<math> \partial_\theta v = 0</math>. (A.2)

			We can reflect <math>\log \Omega</math> infinitely often across the half-infinite edges, extending v to a half-infinite domain that contains the half-space <math>\{ (s,\theta): s < \log R, \theta \in \R \}</math>. Thanks to the Neumann condition (A.2), v will remain smooth and obey the eigenfunction equation (A.1) in the entirety of this half-space.

			Let's suppose that we have normalised our original eigenfunction u to have an L^2 norm of 1, then after transforming into logarithmic coordinates we have

			:<math> \int_{\log \Omega} \|v\|^2 e^{2s}\ ds = 1</math> (A.3)

			and hence by the Rayleigh quotient we have H^1 control:

			:<math> \int_{\log \Omega} (\partial_s v)^2 + (\partial_\theta v)^2\ ds d\theta = \lambda</math>. (A.4)

			Now we can start moving towards L^infty control of v. We first look at the mean

			:<math> \bar{v}(s) := \frac{1}{\alpha} \int_0^\alpha v(s,\theta)\ d\theta</math>.

			From the eigenfunction equation and the Neumann boundary condition, the mean obeys the equation

			:<math> \partial_{ss} \bar{v} = \lambda e^{2s} \bar{v} </math>. (A.5)

			On the other hand, the L^2 normalisation of v and Cauchy-Schwarz gives

			:<math> \int_{-\infty}^{\log R} \|\bar v\|^2 e^{2s}\ ds \leq 1/\alpha</math> (A.6)

			and the H^1 bounds similarly gives

			:<math> \int_{-\infty}^{\log R} \|\partial_s \bar v\|^2\ ds \leq \lambda/\alpha</math>. (A.7)

			Among other things, (A.7) tells us that <math>\partial_s \bar v</math> must go to zero along some subsequence as <math>s \to -\infty</math>. (Indeed, a Bessel expansion of the original eigenfunction shows that it goes to zero uniformly, but we won't need this fact here.) Integrating (A.5) and using (A.6) and Cauchy-Schwarz then gives us the bound

			:<math> \|\partial_s \bar{v}(s)\| \leq 2^{-1/2} \alpha^{-1/2} \lambda e^s</math> (A.8)

			and hence on a second integration

			:<math> \|\bar{v}(s)-\bar{v}(0)\| \leq 2^{-1/2} \alpha^{-1/2} \lambda e^s</math>. (A.9)

			From the triangle inequality, we conclude that

			:<math> (\int_{-\infty}^{\log R} \|\bar v\|^2 e^{2s}\ ds)^{1/2} \geq \|\bar v(0)\| 2^{-1/2} R - 2^{-3/2} \alpha^{-1/2} \lambda R^2</math> (A.10)

			and hence by (A.6)

			:<math> \|\bar v(0)\| \leq 2^{1/2} \alpha R^{-1} + 2^{-1} \alpha^{-1/2} \lambda R </math> (A.11)

			(which already bounds the original eigenfunction at the vertex) and hence by (A.9)

			:<math> \|\bar v(s)\| \leq 2^{1/2} \alpha R^{-1} + 2^{-1} \alpha^{-1/2} \lambda R + 2^{-1/2} \alpha^{-1/2} \lambda e^s</math>. (A.12)

			This gives L^infty bounds on the mean <math>\bar v</math>. Now we move on to bounding v itself. We have the Sobolev inequality

			:<math> \sup_\theta \|v(s,\theta)-\bar v(s)\| \leq \int_0^\alpha \|\partial_\theta v(s,\theta)\|\ d\theta \leq \alpha^{1/2}
			(\int_0^\alpha \|\partial_\theta v(s,\theta)\|^2\ d\theta)^{1/2}</math> (A.13)

			so it will suffice to get pointwise bounds on the angular energy

			:<math>E(s) := (\int_0^\alpha \|\partial_\theta v(s,\theta)\|^2\ d\theta)^{1/2}</math> (A.14)

			since we then have

			:<math> \|v(s,\theta)\| \leq \|\bar v(s)\| + \alpha^{1/2} E(s)</math> (A.15)

			from the triangle inequality.

			Note that (A.4) gives an integrated bound

			:<math> \int_{-\infty}^{\log R} E(s)^2\ ds \leq \lambda</math>. (A.16)

			On the other hand, differentiating (A.1) in <math>\theta</math> gives

			:<math> - \Delta \partial_\theta v = \lambda e^{2s} \partial_\theta v</math>

			and hence by integration by parts

			:<math> \int_{\log \Omega} \|\nabla \partial_\theta v\|^2 \eta^2 = \int_{\log \Omega} \lambda e^{2s} \|\partial_\theta v\|^2 \eta^2 - 2 \int_{\log \Omega} \partial_\theta v \nabla \eta \cdot \nabla \partial_\theta v \eta</math>

			for any test function <math>\eta</math>. Bounding

			:<math>-2 \int_{\log \Omega} \partial_\theta v \nabla \eta \cdot \nabla \partial_\theta v \eta
			\leq \frac{1}{2} \int_{\log \Omega} \|\nabla \partial_\theta v\|^2 \eta^2 + 2 \int_{\log \Omega} \|\partial_\theta v\|^2 \|\nabla \eta\|^2</math>

			we conclude that

			:<math> \int_{\log \Omega} \|\nabla \partial_\theta v\|^2 \eta^2 \leq \int_{\log \Omega} \|\partial_\theta v\|^2 (2 \eta^2 \lambda e^{2s} + 4 \|\nabla \eta\|^2)</math>.

			In particular, if <math>\eta</math> is a function just of s and is supported in <math>(-\infty,\log R]</math>, we conclude that

			:<math> \int_{-\infty}^{\log R} \|\partial_s E(s)\|^2 \eta^2(s)\ ds \leq \int_{-\infty}^{\log R} E(s)^2 (2 \eta^2(s) \lambda e^{2s} + 4 \eta'(s)^2)\ ds.</math>. (A.17)

			Combining this with (A.16) for various choices of cutoff <math>\eta</math> gives integral control on <math>\partial_s E(s)</math>, and this combined with (A.16) gives pointwise control on E(s) and hence on v(s). This gives L^infty control on the eigenfunction u near the vertex. (The computations are somewhat ugly, though, particularly if one wants to optimise in eta.)

Teorth: /* A Sobolev inequality */

2012-06-27T06:18:27Z

A Sobolev inequality

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	In principle, the above inequality, when combined with the previous L^2 and H^1 estimates, gives L^infty control on <math>\dot u</math> in the interior of the triangle. One should also be able to get control near the interior of an edge by reflection. One needs to modify the argument a bit though to handle what is going on at vertices.		In principle, the above inequality, when combined with the previous L^2 and H^1 estimates, gives L^infty control on <math>\dot u</math> in the interior of the triangle. One should also be able to get control near the interior of an edge by reflection. One needs to modify the argument a bit though to handle what is going on at vertices.

			Here is an alternate approach that avoids the use of Bessel functions of the first and second kinds. Suppose one has a solution to the inhomogeneous eigenfunction equation <math>-\Delta u = \lambda u + F</math> on a ball B(0,R). Let <math>\eta</math> be a smooth (or at least C^2) cutoff on this ball which equals 1 at the origin. Then <math>u\eta</math> is compactly supported with Laplacian

			:<math>-\Delta(\eta u) = \lambda \eta u + \eta F - 2 \nabla \eta \cdot \nabla u -\Delta \eta u</math>

			and hence by the fundamental solution to the Laplacian

			:<math>u(0) = \eta u(0) = \frac{1}{2\pi} \int_{B(0,R)} \log \|x\| (\lambda \eta u + \eta F - 2 \nabla \eta \cdot \nabla u -\Delta \eta u)(x)\ dx</math>

			and hence by Cauchy-Schwarz

			:<math>\|u(0)\| \leq \frac{1}{2\pi} (\\|u\\|_{L^2(B(0,R))} \\| \log\|x\| (\lambda \eta - \Delta \eta) \\|_{L^2(B(0,R)} + 2 \\|\nabla u \\|_{L^2(B(0,R)} \\| \nabla \eta \\|_{L^2(B(0,R))} + \\|F\\|_{L^2(B(0,R)} \\| \log\|x\| \eta \\|_{L^2(B(0,R))}.</math>

			Because <math>\Delta(\eta u)</math> has mean zero, one can also replace <math>\log \|x\|</math> here by <math>\log \|x\|-C</math> for any constant C, for instance one can use <math>\log(\|x\|/R)</math>.