07 - SN Partial Differential Equations and Applications Webinars
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Join Yoshikazu Giga of University of Tokyo as he surveys recent progress on flow type equations with special emphasis on a crystalline mean curvature flow whose solvability was left open more than ten years. Don’t want to miss another webinar? Register here: https://go.sn.pub/lXgEaxHave an article on the N-body problems? Submit Your Article: https://go.sn.pub/GT525VLearn more about publishing with SN Partial Differential Equations and Applications: https://go.sn.pub/UFulhi
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Professor Yoshikazu Giga he is the world leading experts in
00:11
mathematical analysis of Piddies.
00:14
As we all know. He made fundamental contributions in analysis
00:19
of blow up phenomena of nonlinear piddies.
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Maybe a stalk equations surface evolution of by the curvature
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and discussed solutions.
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Professor Giggle received his PhD from University of Tokyo and
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in 1986 he joined the Hokkaido University as a faculty
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member and after having various positions at various places including
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Courant Institute. And then from 2004 until now he has
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been a professor at the University of Tokyo.
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Professor Giga received various prices,
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including Mr Science Prize. Autumn,
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prized by the Mathematical Society of Japan and prestigious the
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Medal of Honor with the purple ribbon by Japanese government.
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He is an editor of many journals,
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including.
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Mass, mass, aniline, and the calculus of variation in PDS.
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Let's welcome Professor Professor Giga Yeah please.
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Thank you very much. Thank you very much for your
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kind introduction and invitation.
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I'd like I'd like to share the screen.
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It is my pleasure to be the speaker of this
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meeting.
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So can I share the screen?
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Is it OK? Yes, yes it should be OK.
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You can share. I don't know.
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You have to click on.
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Shea and then annual. Screen and then it should be
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possible.
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Just a second OK.
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Yeah, OK yeah yeah.
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Yes, OK now I. OK,
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today I'm going to talk on an overview,
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give a give an overview of total biasing flow type
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equations.
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Which is we stand to be a typical non local
02:33
problem.
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Although the operator equation does not have any fractional power
02:39
operators.
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The front of my talk,
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so I'll show you next side.
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His other photos after introducing several aspects of total bias
02:51
of row we we explained 22 important approach variational and
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this costly approach.
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And as an application of this costing approach,
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I'll give crystalline analysis on crystalline in carburetor pro,
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which is quite important in the.
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In the materials tires. OK.
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So what is the total variation flow?
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OK, in a strict sense the equation is equation.
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Looks like this type of equation.
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The This is the du.
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And then of course it is interpreted as a gradient
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of the total variation energy.
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Omitted some domain.
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So this is very classical total variation floor.
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But you see, this is another flow which is often
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analyzed P gradient flow or PDC energy.
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So this is something like if you take energy taken
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by this way then this gradient flowing L2 Norm is
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in the divide and so this is P.
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Heat equation.
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OK, and you cause if P is equal to two,
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it is nothing but the Heat equation.
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But if P is equal to 1,
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this is exactly total bias.
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So from the point of PD strategy,
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this corresponds to one.
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So, but there is a lot of difference between PP
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diffusion creation.
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This is a equation is as I wrote,
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yeah. And if P is greater than two.
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Probabilistic diginet, so if P is greater than two,
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this is the equation. At the gradient is equal to
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1 zero and if P is in between one and
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two for Ableist is so strong it singular at great
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gratitude is equal to 0,
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means the diffusion coefficient become infinite in the gradient is
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equal to 0.
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And if P is equal to 1,
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this is called total visor pro.
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But problem is similar at gradient use.
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Equals you know this is similar to the case P
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desktop 2.
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But also we notice probabilistic degenerate integrating the direction of
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gradient year,
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so this is different from above.
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And of course there's a huge literature to study this
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type of problem and this,
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but still still there some many literature.
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OK.
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And that's the. But to go to the father problem
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that consider very simple example is in one dimension example
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1 dimensional example is just du.
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DT is equal to this signature function signature functions this
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function.
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Bye.
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Of course, if the the one dimensional case,
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this is a signature function.
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This is a very innocent look like Unicenter Lookback equation.
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O'Malley, this equation UT is empty it minus two times
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Delta overview XQXX so diffusion equation just stared up but
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you see this is hard,
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too hard to give a meaning of this each time.
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Becaused you cannot pull back major,
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so this is not well defined.
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Quantity and this is the huge effect is so strong
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at gradient is equal to 0 while it is zero
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at the UX not 0.
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So heuristically this looks. This gives some impression.
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Of this situation, however. This this quantity has no meaning
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at all,
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but that this means that definition of a solution is
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not trivial.
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Of course, it may be trivial for most of audience
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here,
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but what I need the definition of solution is not
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is not trivial.
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At least undergraduate people by undergrad student.
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So.
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Let's see how do we understand the situation.
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Feed over the place where you exit code zero.
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What is the speed of the place where gradient is
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equal to 0?
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A question how does it?
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For example, how does the profile in the figure move?
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So suppose that your profile function is something is everything
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1 dimensional?
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Consider such kind of function and this.
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This part is the flat part.
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You started 1st and cost.
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What is the motion if you start with initial data
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like this one,
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how do you think this?
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How should we move this?
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Type of shape. Here's the answers.
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Crock pot which we will say facet.
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Does look split or bent so that the time derivative
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you is constant on the facet?
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So this movie.
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Keeping this shape.
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So we assume. This stuff.
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Then OK, first of all,
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integrated of integrity of this equation from eight 8 minus
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Epsilon to be plus Epsilon you get.
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OK, this is just a simple analysis.
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Under medical analysis says this is signature function UX and
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B proception minus 8 minus episode.
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And here in this particular picture,
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the slope.
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At.
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Lift. OK is plus one while slope.
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Right from B is minus one.
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Is the negative, so this is minus one.
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So totally this is minus 2.
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OK. Then
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if you assume this and at this speed is constant
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on this interval,
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so that the left hand side just use some tee
09:17
times B -
09:18
8 right as Epsilon Tensity.
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So that's our equation. Previous equation is equal to use
09:25
our T -
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B - a = -
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1 - + 1. OK,
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of course this is the best use of T is
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equal to minus two or B minus and of course
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I need sign here.
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But the This Is a B -
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A is a facet press flat part that answer what
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is minus 2 minus minus because this is in some
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sense concave.
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So this should be minus but the number 2 is
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actually the number of endpoints and this quantity is called
09:56
discontented nonlocal contacting.
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So that's the reason I would like to say this
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this equation nonlocal equation because speed is determined by the.
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I said This is faster dances,
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of course nonlocal quantity and the actually disk caution.
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This quantity is often called chigas measure.
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Of course, this looks like trivial,
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but.
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In higher dimensional case, I show you as I show
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you.
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This is very important quantity turned out.
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So so question you know is this answer is OK.
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Some approximation, this means that the unsupported from approximates.
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That means if you consider such kind of equation yourself,
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B is equal to.
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Everybody was seeing each other UX.
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If you if you ask analyst to analyze this problem,
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usually we modify right signature function and process.
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For example you XX so that this becomes uniformly parabolic.
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Then the question you get the approximate solution and the
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question will.
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It is quite natural to guess this limit is a
11:15
solution of our original problem.
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And the situation boring if the space mentioned just one,
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then this. This follows from approximate such approximation.
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But if the higher dimension is in general not.
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So this is you cannot assume such kind of answer
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for and is greater than or equal to two.
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If you assume this solution obtained by Approximations.
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And I want you if the driving point,
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if there's a driving force like this.
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Then the the value BX depends on T.
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This is another very interesting point.
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Usually such value does not depend upon other term.
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But in our problem this does depend upon G,
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otherwise there's no no no solution,
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satisfying comparison principle. This is second order probability problems,
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so it is expected to have some comparison principle.
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But if you assign this value.
12:21
Value.
12:23
Attacchi gas ratio. There may not exist the solution satisfying
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comparison prices.
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OK.
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And then this is example.
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Suppose it kenji is something like positive function negative function
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having support in this interval.
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So G is something like that equation is something like
12:49
UT minus signature UX.
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X class G. Is the question OK?
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We take L. It's equal to 1 Noble maximum value
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of G.
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Then constant zero initial data.
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Then there no solution satisfying comparison principle.
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This is actually expressed written in our recent paper,
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but with no bad pozar.
13:15
And it is just published as Doctor Horan,
13:19
said Springer Nature.
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By 2020 in this.
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I don't know how. How do I understand but number
13:29
starting line paper?
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I don't but I don't this this was printed maybe
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last week.
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But anyhow, why?
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Such a crazy single couples.
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The idea to prove is not so difficult because constant
13:46
that's kind of solution we A and this is slow
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pay and we ate the actually sub solution because speed
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here is if you assign.
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These two L. This is a Cheeger ratio plus you
14:00
use you tease that stunning,
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just stunning. Create their standing solution.
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Standing function time independent function plus G and if there
14:13
is a taken in this way.
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Then this must be always.
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Made it here. So if you this is a.
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This is at least some solution.
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But here's some solution, so so the solution U is
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equal to 0 is equal to 0 is a.
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It's not so difficult to say this is U is
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equal to 0 is super solution compared compared so the
14:42
only choice if the system satisfies compile some principle it
14:48
must be bigger than all sub solutions.
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So the only possibility is the solution must be used.
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It is not difficult to see you identical 0 is
15:00
not supported.
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So you you cannot assign such kind of the.
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Not necessary, this quantity depends on environment environmental dependent.
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This I would like to point out.
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So now let's you see the difficulty of the problem.
15:30
So what is examples? There are,
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of course, various examples of this problem.
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Trustable from the theory of crystal growth,
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there is something like crystalline mean curvature flow location.
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This is not one special example of such totalizer flow,
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but point is this non divergance type you have extra.
15:57
This takes the time.
16:01
Good.
16:04
Grover and another one is this for soda.
16:10
And this is coming from relaxation,
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for in this example you is a height function of
16:15
crystal.
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On the on the on the on the substrate substrate
16:18
is the proposed by spawn along time ago.
16:20
And here is a little bit confusing example similar to
16:23
this one,
16:24
but you have.
16:26
Gradient this looks like total bias or pro,
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but this is not called Koetter Bias for this is
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the never settle in carburetor flow equation because each level
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set moved by its mean cover charge which was studied
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by Chen and long time ago and got myself advances
16:45
book independently.
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Generalization.
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Which will actually explain data and the important thing is
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that the singularity here is weaker than this stuff because
17:01
of this.
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The gradient you tends to 0 this.
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This is the diffusion question tends to Infinity here,
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but this is not because of this multiplier.
17:12
Another example is 1 harmonic map project,
17:15
so this is the total bison whose value is in
17:18
S2 and this such kind of equation is proposed by
17:22
turn and Sapiro and Casillas long time ago.
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This is a color image.
17:29
From chromaticity, and it is a gradient from total variation.
17:33
So with waiting for total with manifold constraint of this
17:37
two?
17:38
OK, this is the image that noisy,
17:40
so there are many example and each each each topic
17:45
studies extensively in recent 10 years I think.
17:49
Listen 20 years so I do not give any references
17:55
at all.
17:55
Because because of that I restriction.
18:00
And they said OK, but I should say that at
18:03
least two approaches to attack this problem and the first
18:06
one if the equation has a diversion structure,
18:09
the theory of maximum monotone operator works well.
18:14
And the advantages it applies to higher order equation,
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including examples to force order problem.
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This or advantages. Of course it needs some dividing structure,
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so does not apply the first problem and solve problems.
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And here's a typical theory and classical theory initiated by
18:36
Chromera and extensively studied by brazils and others,
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and the simple. The theories vary the conclusion.
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The theory is very simple.
18:48
Suppose that age is a real human space that EB
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convex energy,
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which is of course not identical,
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code Infinity. Then there is a unique solution.
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You
19:03
of gradient flow equation.
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OK.
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Oh, but you is right differentiable for all T and
19:12
the right derivative equals to the Canonical restriction of Subdifferential.
19:17
What is strong? OK, this is some device which are
19:20
expanded soon,
19:21
but what is very interesting and it just equation looks
19:25
for bright,
19:26
very ambiguous. However, solution knows how to evolve.
19:30
And the and the initial data we say is you
19:34
0 is a.
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I don't know where this initial data is taken in
19:42
this in this hilbert space.
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Yeah.
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This is a workspace.
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Initial data for any Hilbert space and this is in
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many cases this is equal to the closure of this
19:56
energy domain.
19:59
And what did somebody pressure so differential?
20:03
Is something like some gradient.
20:05
You say each element is upgraded.
20:09
This is a slope of the function touching from below
20:14
at you.
20:17
And this is the totality some device operator.
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It's elementary course upgrade. Yeah,
20:22
if he is differentiable. Of course some differential is a
20:26
single author,
20:27
but if you X is something models in the inner
20:30
product in user kerbal space,
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then this is something like.
20:35
Signature function, but this is a all value here.
20:38
This is multivalued. And this sub differential is actually maximum
20:44
operate if U is convex.
20:46
Andrew Lawson called here in this Canonical restriction minimum section,
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or which is called minimal section.
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This is the smallest F belongs to.
20:58
Come back set and in here about space.
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This is the closest element.
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This is called Canonical restriction or minimal section.
21:09
This is keep calling. And OK,
21:11
what is the definition? How to apply this theory to
21:15
our total variation problem?
21:17
This is not so difficult to avoid boundary condition bounded
21:21
difficulties,
21:22
we only consider the periodic boundary condition.
21:26
OK, that you you is a total waste to energy.
21:29
So this is BB total bounded variation function and H
21:34
Delta Hilbert space.
21:36
Or maybe different sometimes. And if the value is in
21:44
if outside BBUU is.
21:47
The body is Infinity and if is lower Semicontinuous If
21:51
A is IF802,
21:52
this is awesome. Continuously age and it is also lower
21:56
than continuous in another Hilbert space,
22:00
which is less regular. Something like H -
22:03
1 space to simplify the issue we take average zero
22:07
function,
22:08
but this is H on average 0.
22:10
But please do it minus ability to function and this
22:14
is years of course 2 door semicontinuous.
22:18
Of course, convex is trivial,
22:20
but it wasn't in non trivial.
22:22
For example, if you replace B breed by W Space
22:26
W11,
22:26
then this is no longer continuous.
22:30
So we have to be added to be careful,
22:33
not not like this W one,
22:36
but we have to take B with.
22:41
So, but the space of bounded variation,
22:43
which I don't define because we don't use it.
22:47
But it is. It's a bigger space having a discontinuous
22:52
function.
22:57
Space contains jump discontinued. And stick under the property here
23:02
'cause the secondary problem we take Cuba Space L2 and
23:06
is it just me?
23:07
The problem we take 8 miles on other.
23:11
OK, and this applying this camera brings positive we get
23:16
unique solvability.
23:18
There exists unique global in time solution feed you with
23:24
initial data U0H.
23:26
Go to or 8 -
23:28
100.
23:30
And this is a price for both of the program
23:33
by Bob coin.
23:34
Myself long time, long time ago and also fractional total
23:38
bytes of flow is constant,
23:39
but how?
23:41
More recently, but anyhow, we can apply that kind of
23:45
silly,
23:46
but so far it's very classical,
23:48
and in general it's different.
23:50
Calculate subdifferential for total bytes of role is with respect
23:56
to it is very static and I just point out
23:59
a book by Andrew bio.
24:01
Very nice book and.
24:04
Will study for somebody function.
24:07
But if you used this type of result we can
24:10
characterize some differential.
24:13
For example, suppose that you is a set that UX
24:17
is equal to minimum value.
24:19
So suppose you have smooth boundary,
24:22
then the value of summer differential U is a minus
24:27
minus devices.
24:28
So this is as usual.
24:30
But these these looks formally looks like EU of gradients
24:35
du.
24:36
But of course this is may not be defined in
24:40
this place.
24:41
So it turns out this is the best one because
24:45
this is always someone and also you need some boundary
24:49
conditions.
24:50
We we we get some boundary conditions we are new
24:55
is out of normal providers that uses domain of somebody
24:59
financial.
25:00
And so too, this is a graph of you.
25:04
So this is actually you,
25:05
sorry. I think this is a good alcohol use and
25:10
this is a.
25:12
And the value of minimal section.
25:14
So this is the closest element.
25:16
Use overturned by solving over separate property.
25:19
So this is the L2 Norm.
25:21
L2 Norm of. These elements so minimize,
25:24
but there's a constraint with.
25:27
And also boundary conditions. If the The Zero is minimizer
25:32
of this problem,
25:33
then the sub differential BU that you is his value
25:38
deep minus divide divergencies.
25:41
This uniquely done there was zero constant but this is
25:45
if diverted is already taken constant constant.
25:50
This is we we use answer so no fast spring
25:53
more bending forecast.
25:55
If divergencies is constant this is important.
26:01
Select all father.
26:06
And we say I sent you is Calibra.
26:08
If there's a vector field boundary condition.
26:14
And I don't. I don't go to the very detail,
26:17
but I would like to say 1 dimensional intervals all
26:22
on dementia interval,
26:23
Cardio, Grapher. And Albor is considerable,
26:27
but edit soiled may not be calibra if eccentricity is
26:31
fast.
26:31
And then of course various characterization is now available in
26:36
this character ability.
26:37
And.
26:40
OK, and now go back.
26:42
Forget about calibr ability and support the zeros minimizer of
26:48
the problem P.
26:49
We let we set Lambda U at the divide and
26:52
C00 Minimizer,
26:53
and this is called non local curvature of the graph.
26:57
But this looks like secondary but you.
27:00
So we have the comparison place.
27:04
In this way, then we have a such kind of
27:07
compile some place.
27:09
An idea the proof is not so difficult because some
27:14
differential is approximated by.
27:18
So by by solving this kind of resolvent equation.
27:25
So compile this situation is of course elliptic location.
27:29
You approach make this ception to get the user elliptic
27:33
smooth elliptic program and the comparison principle host.
27:37
So the ordering of U is preserved for brief,
27:41
so this is not so difficult.
27:44
So compiles on the Lambda is reduced to the maximum
27:48
principle,
27:48
but important thing is this.
27:50
This curvature like quantity known.
27:53
How much are complicated bigger than this corresponding Omega One
27:57
is bigger than Omega 2.
28:01
And of course if you is colorable then this non
28:05
local curvature is just the Cheeger ratio.
28:09
This is actually Cheeger Asia.
28:11
The proof is very simple,
28:14
you take diverse Z0 and integrate against you,
28:17
then integration by parts or Gauss Formula Gauss Theorem it's
28:23
equal to Z0 not new,
28:24
but since this is equal to 1 by boundary condition.
28:29
This is just to let area in the right hand
28:33
left hand side is of course Lambda Times volume of
28:37
you,
28:37
so you get the gas ratio here if you is
28:40
correct.
28:43
And if the there is,
28:45
if there is extra forcing the term then we have
28:51
to set energy.
28:53
Like this at having extra time,
28:56
everything fine, but the point is.
29:01
Of course, this energy energy as you expected it just
29:06
somebody functional,
29:08
subdifferential, EEG is equal to E trustee.
29:12
However, if you take Canonical restriction,
29:16
this is no longer true.
29:18
This is not may not equal,
29:21
so this is a way we find that.
29:25
We got quotient Tigger ratio cannot be used for under
29:30
knowingly home driving force even if one dimensional case.
29:36
OK, and no local code here is you have to
29:40
minimize this quantity.
29:42
Now, under the boundary condition.
29:46
And note that this is a.
29:50
Complicate this compiler principle for Lambda G is of course
29:55
still horse.
29:56
OK, now we're going to discuss the approach theory based
30:01
on maximum principle.
30:03
Of course, advantages, no diversion structure necessary,
30:07
but problem is applies only to 2nd order program so
30:13
far.
30:13
Code or any use you know there exists a unique,
30:17
so my our theorem is boring or any now it's
30:21
not the L2,
30:21
everything is based on the maximum principle.
30:25
So this is just for C.
30:28
There is a unique global interim solution,
30:31
continuous, satisfying this equation in this course is set here.
30:35
If it is continuous and non increasing the last variable.
30:39
So that problem with President public.
30:41
An example included. Of course we have to clarify what
30:46
does mean solution.
30:48
So we have to use.
30:50
We have to introduce the notion of disco sisters.
30:54
The standard is the definition of elliptic problem is the
30:58
1st for us function.
30:59
You is continuous. The disco situation of service discusses some
31:05
solution of this equation F if.
31:08
Test function. Satisfying
31:13
this type of inequality whenever you and find it takes
31:17
maximum takes up.
31:18
So if you better to write picture is the graph
31:23
of you and if you touch.
31:25
I then the maximum point is obtained.
31:30
In this exact.
31:32
So supposed fire index heart has such kind of property.
31:36
Then you ask difference inequality in this one.
31:40
And superstition is defined similar way.
31:43
You have to define this way and you have to
31:46
be great with me.
31:48
Then you get the Super source.
31:54
This is completely based on maximum principle definition,
31:58
pretty based on magical principle and then you you if
32:02
both sub and super solution then this is called discourses
32:06
of assertion.
32:06
This is fundamental notion.
32:11
And we have to define suitable OK idea of definition
32:15
of this customer solution is falling.
32:18
We have to take suitable class in test functions so
32:22
not nonlocal coverture faceted define.
32:26
But Lambda Phi is equal to something like this quantity.
32:30
So then, uh, function, you is some solution.
32:33
Discuss is because assumption who?
32:35
Our problem? If I exactly had satisfied.
32:38
That's kind of expected property when you in the file
32:42
a file sounds like that.
32:44
And we did not clarify this one,
32:47
but this one is.
32:49
You see from our previous slides everything like I sometimes
32:55
the Cheeger ratio or maybe the minimizer.
33:03
So I do not want to write any details,
33:05
but if you is a sub solution and with the
33:07
Super solution you have a comparison principle.
33:13
And I hit it to show the idea of proof
33:16
given.
33:16
That's the standard way or the doubling,
33:19
valuable, but we have to use some fronting argument.
33:23
It was applied for long time ago for the fundamental
33:27
problem and to compare value value of Lambda quite easy.
33:32
And then we use comparison or non local curvature.
33:35
But we have the approximate very general facet by faster
33:39
is smoother bounded so that more local coverage PY.
33:43
So this is a yeah,
33:44
this is a fundamental idea of data.
33:47
OK.
33:51
OK, and now we are going to our main stage
33:54
Crystal.
33:54
I mean curvature pro.
34:01
So let's gamma TB evolving hypothesis.
34:05
Think about the four equation is Africa vacation.
34:09
Many people knows this is a.
34:13
In case of program, a key in the gamut is
34:16
abreeza normal velocity and normal velocity equals to the main
34:20
coverage on the socket.
34:22
And of course, that's awesome.
34:23
This problem is goes back to material science.
34:26
We do two Marines 1956 and this is proposed by
34:32
to model motional grain boundaries and eating methods.
34:38
And the point is this equation.
34:40
This equation, if you move the surface in this equation
34:44
hypersoft.
34:45
In this case, the solution may develop singularity.
34:49
This is shown by Grace,
34:51
if heuristically it is easy to understand.
34:56
This is the number is more with scenic.
35:00
Without you see the main character here is very big
35:05
in what in coverage.
35:07
This is more than before this both sides collapse.
35:12
This displays collapse very quickly and this actually happens as
35:18
shown by Grayson long time ago.
35:23
OK. And this is a couple Bucks coach anybody is
35:27
in airports is Bibracte.
35:32
To extend the solution after singularity but is not put
35:36
into this by bracket 1972,
35:37
but it was extended by Edelman and recently but underground
35:42
as us and its original brackets articulate very difficult weed.
35:46
But now it is accessible.
35:50
And this is kind of addition approach.
35:53
And then there's another post code level segments listed by
35:59
Chen and go to and myself.
36:02
Facebook long time ago now have both the book and
36:05
the theory encounter with disco situation.
36:09
This is the first time meet me to discuss the
36:13
theory.
36:14
And there's even deterministic game interpretation by corner cell fatty
36:20
2006.
36:21
But anyhow, this is a very important notion to extend
36:26
it to track the evolution after it creates a simulated
36:30
develop similarity.
36:32
So is it possible? Natural question,
36:35
is it possible to extend the level set approach to
36:38
any store pick covers?
36:40
Therefore this is the other is already included in my
36:43
earlier paper we've gotten 10 provides that intimacy,
36:47
energy density, convex and smooth.
36:49
What happens in terms of energy and not C1?
36:53
So in this therapy, but this is the case,
36:55
a crystalline new culture.
36:59
Increase the time in coverage for a typical example of
37:02
Mr.
37:02
Picking carburetor for proof, which for example describe emotional bond
37:08
if its grain boundary.
37:09
And any sense of being covered?
37:12
Theresa change ratio over an interface energy is expected variation
37:16
of volume and crossed by hypothesis,
37:18
So this is a change of ways of not the
37:21
surface surface surface area,
37:23
but interface energy which I explain.
37:28
And into his energy, that,
37:30
zero just given. Depending upon materials negativity and negative continuous
37:35
function over the unit sphere which is called interface energy
37:39
density,
37:40
but the given hypersurface we said I gamma is this
37:44
is the interface and energy which is called into the
37:47
enemies.
37:48
Of course normal gamma.
37:52
I need to be covered.
37:53
The Pro is introduced any sort of mean curvature,
37:57
so this is sort of being covered and it is
38:00
explicitly written by using minus the surface divergent car Marks
38:06
and she is a gamma gradient of Gamma gamma.
38:09
Is homogenization madness, extension of Gamma Zero and divided by
38:15
surface?
38:16
It's awful cold cut off Monday,
38:19
but anyhow, in the case of an estoppel case,
38:23
gamma dealer equal to 1 in the case,
38:26
This is not true, but.
38:30
And then the notion of rule shape is quite important,
38:35
and this is a substitute to Sophia,
38:38
and this is something like blue shape is defined using
38:43
this,
38:43
and M and yeah and you have to outside kind
38:47
of stuff then weighted mean curvature is equal to a
38:52
constant.
38:53
The Converse is also true.
38:55
This is Alexander Paris attack,
38:57
Pfizer, Novartis standard problem. If Gamma is compacted embedded in
39:02
Cambodia to smoothen strictly convex,
39:05
and certifies under ice Alexander Horse.
39:08
So this is substitute of severe.
39:11
What is crystal I mean.
39:14
You got one. This is nothing but area.
39:17
Of Gamma and each, insulin culture.
39:19
But in this case, of course gamma fees must be.
39:23
In general, may not be convex North rumors,
39:26
but we say government zeros crystallize.
39:29
If gamma is convex and piecewise linear,
39:32
unless entropy mean coverage, is Chris.
39:35
Timing covers if gamma is crystal,
39:37
so the key is this is a gamma museums or
39:40
gamma Rays crystallizes convex part,
39:43
piecewise linear.
39:45
OK.
39:47
And so the label set the one level sublevel set.
39:51
The gamma is awful code Frank diagram and W shape
39:55
with you may you may understood something like poor or
40:00
F Gamma.
40:01
So this is a Frank diagram,
40:03
so this is my piece by the linear function.
40:07
It's level set is something here,
40:10
but then the poor look shape is it's too hard.
40:16
So, OK, So what is the flaw?
40:19
Prove is just great, just.
40:23
Gamma depends also independent velocity is equal to something depends
40:29
upon normal at 8.
40:31
So this is a.
40:33
Examples of unstopping covered, therefore crystalline.
40:36
Mine covered the pros. Of course.
40:38
80 gamma is crystalline form,
40:40
so there are few examples I just right.
40:45
Yeah.
40:47
And why it's gonna seem interesting.
40:50
Just physics, um, I take.
40:54
Meet.
40:56
This is the shape of crystal.
40:59
If the temperature of the material is very,
41:03
very positive and this correspond to share.
41:08
And of course, if the temperature OK,
41:12
it is a mixture of covered part and part.
41:16
So this is a question of syllables,
41:19
selenite.
41:22
OK, so this is how are you?
41:25
So anyhow.
41:28
This is a slow quick stop.
41:31
OK. So the fundamental problem analysis is considered crystal.
41:35
I mean covers a floor.
41:37
For example, Breezy contagious, does the initial body program its
41:41
unique solution?
41:44
So far given cross office,
41:46
are there any family matters of the second question?
41:50
Is this solution, if it exists approximately by smooth?
41:57
If interface energy gamma is approximate,
42:00
programmer episode, does the corresponding solution gamati efficient approximate this?
42:06
This is important because otherwise you can say all solutions
42:10
just standing solution or anything you like,
42:13
but this is very good criteria.
42:15
Whether your solution is natural solution or not.
42:20
And Norm is out. Let me say the known result
42:24
is that if this problem is well studied,
42:28
if the printer.
42:31
In the case of capitalism.
42:33
OK and then.
42:36
End of Guardian terroristic secure class of.
42:40
So usual classic combo.
42:46
And but this can be.
42:47
This is a very basic class.
42:50
But we. It would extend level such methods for.
42:56
This kind of two dimensional problem,
42:58
in 2001, how administered case even broker existence or not
43:02
know unless initial data is convex.
43:04
A unique existence of global four provided initial date of
43:09
convex,
43:09
so this is a 2006.
43:12
And of course, nowadays it's as I explain more,
43:16
much more things are available.
43:20
So level set row is a general notion of curvature
43:22
for equation which allowed for spare change over the phone.
43:26
We constantly beset equation for.
43:31
I never said the solution.
43:36
We call them in coverage and so it's never set.
43:39
The pace of the for this type of form.
43:43
We consider this equation are encrusted,
43:45
he not only on Gamma,
43:47
so so this in case of sets,
43:49
each level set moves by two is equal to 8
43:52
each levels,
43:52
so that so Fortunately unique,
43:55
unique the initial value problem uniquely server in the sense
43:59
of this situation with uniformly continuous initial data,
44:03
then the more over this,
44:04
T zero never accepted uniquely determined by zero neighbors set
44:09
of initial data does not depend upon choice of.
44:13
Defining function, this is a introduced by OSHA and essential
44:19
or numeric analysis and this model after that.
44:24
Established.
44:28
OK, the main result in this direction is our paper
44:33
2016,
44:33
- But given initial data is by its boundary Co
44:38
exist global level set for the crystal I mean.
44:45
University Kaseva Bui is something like with this type of
44:48
equation.
44:51
You also notice some.
44:55
Tiger. And this is also stable in Somerset.
44:59
Propose that come up shall be a smooth convex interface.
45:04
Energy density, approximate, unit forward.
45:08
Then the level set for the episode combined the deep
45:11
in the House of distance sense in the space of
45:14
time,
45:14
provided that no fattening or car for I did not
45:17
explain fattening,
45:18
fattening means the OK happening means if you consider even
45:22
being covered up or if you consider such kind of
45:26
shape.
45:26
Cross. Then maybe I should like this.
45:32
May
45:34
I'd be happy enough. Proximation looks like this like this
45:39
one,
45:40
but out approximate evolving.
45:44
Different way, so maybe it's better to start,
45:47
yeah?
45:50
So that's what two way to evolve this one way.
45:55
And this is 2 way and two way and in
46:00
in the in the River.
46:02
Oh even if initially the bearable setting just crossed.
46:09
So in this case, convergence result is problematical because solutions
46:15
may not be unique.
46:16
It's unique in the sense of this defining function never
46:20
said so,
46:21
but if it's happening, if we talk about surface then
46:25
it it may have a problem,
46:26
so we just put such kind of.
46:30
This result is a published in already published 2616 for
46:35
the dimension 3 and whole general dimension is in communication.
46:41
Applied masks in 2018 with North Robot.
46:46
And I don't say recent,
46:49
but a few years ago that are very nice work
46:53
by Samboan,
46:54
Morini and Ponzio in 2017,
46:57
2019 and this time the problem with the other quite
47:02
general,
47:03
but this but the idea what is the result is
47:07
their their initial set is arbitrary.
47:11
So and crystalline stuff is,
47:13
we're talking about. Gamma is just crystalline,
47:17
but here the very general gamma to gamma include.
47:23
Tablet part and.
47:25
In the pizza linear linear.
47:29
No, this is great. No boundaries of the zero notice
47:33
going to space station.
47:35
This is great and very more recent Novara Novara extended
47:40
server that is convex and F is given.
47:46
But December 1 this advantage is the equation can be
47:50
handled in a Dania in Kappa Gamma of Max coefficient.
47:53
The idea is interest. If these superstitions anyone expects that
47:57
and it's sort of a distance.
47:59
D fulfilled such kind of equation.
48:04
This looks like a linear equation.
48:06
So it's great.
48:14
And we have to define the correctness.
48:18
What time is up maybe?
48:21
So.
48:24
Time is up.
48:27
Comparison place so to do that,
48:29
we have to define correct,
48:30
correct notion of this consideration and concludes that if initial
48:35
data.
48:36
This compiles and prints perforce.
48:38
We define.
48:41
Start to compare. We also need approximation by smooth surfaces.
48:47
And of course, this cause this theory has been established
48:51
long time ago for 2 dimensional case,
48:54
so carbonation and minimal divergences constant for carbon pollution but
48:59
may not be constant in self separation.
49:01
This is the difficulty the breakthrough is by.
49:06
With me and we haul giga.
49:09
My wife. And this is for this equation extension for
49:13
E homogeneous driving force with
49:15
G. This is exactly paper which is just published.
49:19
And having jeezer problematical, but we are able to overcome.
49:27
And maybe we skip. So we just mentioned the we
49:32
just mentioned.
49:34
Willfulness of the problem, but of course we know the
49:38
property of solution,
49:39
but this is not one dimensional convolution case.
49:42
It is several works, but.
49:45
Surface evolution only work.
49:49
Is it very similar construction or set of similar thinking?
49:52
Sorry, so higher genius and by know about poser.
49:57
So we give several approaches.
49:59
So total bison tablet case.
50:00
We also provide discussed approach to the crystalline linkable support
50:05
to get global separate innocence of devil step.
50:08
And the open problems if the equations both older has
50:12
no device structure where process is often left open.
50:17
Yeah, example is this is coming from a crystal gross.
50:22
This is actually exponential. Stuff,
50:26
so that's problematic. Also, how many map that many results
50:30
for existence,
50:31
but uniqueness is why do we have to offer?
50:34
So I think it's a good time to stop now.
50:36
Thank you very much for listening my talk.
50:39
Thank you very much.
50:42
Thank you, thank you for a wonderful lecture.
50:46
So it's so let's take few questions of the.
50:51
Anybody have any questions you can directly ask questions to
50:57
Professor Giga.
50:58
Question.
51:03
No mate.
51:05
Question to Professor Giga.
51:09
With the question.
51:12
May I put the giga?
51:15
Oh yeah, yeah do you?
51:16
Can you can directly talk to the Prophet giga yeah.
51:21
Different.
51:24
Screen has screen disappeared. I enjoy very much.
51:30
Thank you very much, yes.
51:34
My question calls.
51:37
The. Yeah, the total pre flow in presence of an
51:41
obstacle.
51:42
If you have called for the total variation flow,
51:46
can you still use? This approach,
51:49
do you have any?
51:52
I think if it's a variation,
51:54
OK, it depends upon equation if the equation has a
51:57
variational structure like user obstacle problem,
52:01
I think it's I think it's possible,
52:03
but what kind of obstacle just the obstacle of value?
52:08
Almost home function. For example,
52:11
the you is greater than some PSI or something.
52:14
That's what you mean.
52:24
Yeah, I think I think about the crystal grows against
52:29
some some planets.
52:35
Yeah, this is another story I think.
52:39
Let's see I roasted him.
52:43
This is another story. My.
52:48
My answer is. My answer is if it's there.
52:52
For example, if you are.
52:54
If you have a problem with something like.
53:00
With you is a size that's kind of Overstock.
53:03
Who is, I think, no problem.
53:05
But what you say is if you hosted some body
53:09
and you have a crystal surface.
53:12
And this is move something like this.
53:14
This is what you mean.
53:20
Professor during this situation. Of course,
53:22
the classical opera.
53:25
Start.
53:27
I mean, the classical obstacle product with you large oraculum
53:31
side,
53:32
but can also have the.
53:34
The level surface approach where,
53:36
of course you must define obstacle in a different way.
53:39
Yep, Yep. But you are talking about the surface simulation
53:44
with obstacles.
53:45
This is a interesting topic,
53:47
so you have some stuff and this is a crystal
53:50
surface and trying to move this way and this is
53:53
another story,
53:54
but if it's also stuck with this is that it
53:58
looks like Tangent.
54:00
So.
54:03
So.
54:06
Any other questions?
54:12
Anybody?
54:16
Then the let me ask a question.
54:18
Do you you mentioned this?
54:20
The computational issues? I mean result by the Usher,
54:24
I mean method by the ocean safety and for the
54:28
mean curvature flow.
54:29
And can you?
54:31
I mean can you add some comment on that?
54:34
I mean some other method or the other than the
54:37
level set method for the computation.
54:40
Thank you very much. I I I skip everything.
54:43
So let's see you are talking about.
54:48
Computations.
54:51
Yes yes yes.
54:53
Yes, OK, yeah. Anyway, anyway to compute?
54:59
OK, if you even if you look this level set
55:03
for equation,
55:05
there's them anyway, like for example Duke.
55:11
Using the using finite element method,
55:14
you can, but of course for forming about the publication
55:19
itself does many way to calculate numerical example using phase
55:25
field or Encan Equation.
55:27
So there's many way to approximate and consecration by approach.
55:35
The question or we have some threshold approximation.
55:39
For example, if you this is also proposed by OSHA,
55:43
Berryman, washer and.
55:46
BMOBM Merrymount Benz Marymount Ocean scheme OK.
55:53
I wish this email something faster,
55:56
so heat equation with the to the initial data is
56:00
a characteristic function of the crystal shape and then solve
56:05
heat equation and then with very small small time we
56:09
just chop.
56:12
Of course there's some level set of this solution with
56:17
never half something like that,
56:19
and then we take this set at the initial data
56:23
to solve something like that kind of threshold method is
56:28
another typical message to solve the problem.
56:32
So yeah, anyway, OK. Image on the ocean session or
56:36
regional level set method,
56:38
but numerical method. Of course,
56:40
if you are interesting, only the evolution of smooth in
56:44
coverage of code you have of course particle tracking method
56:48
and many many methods in America.
56:51
And yes, and sometimes crystalline star can be also another
56:57
way to calculate.
56:58
By the way, is the level set.
57:01
Crystalline floor can be also calculated numerically.
57:07
And this is a. Walk by Takia Nausha and Obama
57:11
and therefore sandpaper,
57:13
which I don't remember exactly by using by using.
57:20
By using it.
57:26
OK, that's good enough.
57:29
Let's take out the just one more question quick question.
57:34
If there is any.
57:37
Audiences. Any other question or.
57:43
OK otherwise.
57:45
Thank you very much for wonderful lecture.
57:48
Professor Giga.
57:51
Listening very happy to see you again.
57:54
OK, thank you. Thanks a lot everybody.
57:57
Thank you.
57:59
So goodnight good evening with good morning.
58:03
Thanks a lot, Goodnight everybody in exactly everybody bye.