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Dear students, hello, this is the last exercise class and we're going to see as usually the
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solution to the previous exercise sheet, which was again partly on Lax-Milgram theorem, partly
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on Poincare inequality because, I mean, you used it so it's also good that you know how
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to prove it at least one of the true, the one with zero boundary value and in particular
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how to find the somehow sharp constant to some extent and then the last exercise instead
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concerned some general properties of compact operators which then are related to the study
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of existence and uniqueness of weak solutions for elliptic PDEs in that this result it directly
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implies the so-called freedom of the derivative. But let's go in with order. So first of all there
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is the usual definition of sublet space is WKP, WKP0, all standard Poincare inequality. So we
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define the second order linear differential operators on our open set U that we're going
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to use as L of U defined formally as minus the sum of i and j from 1 to n aij uxi all derived
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with respect to xj plus the sum over i from 1 to n bi uxi plus cu where the coefficients
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are generally infinity. Okay, of course this definition is only formal because here I'm
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taking derivatives of functions that are priori are not differentiable. The point is that while
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this would make sense if say aij would be c1 and u would be c2 in the classical setting in the weak
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sense and we can make this meaningful by not considering it pointwise but seeing this as the
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the weak derivative where at least this term in interpreting this term here as a weak derivation
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and therefore considering not LU but the integration of LU against some function v and then by using
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the definition of weak derivatives one can move the derivative in xj over the test function v and get
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an associated bilinear map which associated to this second order differential operator. In addition
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we assume I mean this is the general form of a second order differential operator period then
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we assume that the aij are symmetric that is aij is equal to aji and they are uniform elliptic
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which means that the sum of i and j from 1 to n of aij x psi i xij is larger equal than
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some theta positive norm of psi squared for almost every x in u and xi in Rn. In other words
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if we consider the matrix those components are the functions aij of x for i and j in 1n
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then this is a symmetric matrix which is positive defined that's what it means the
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uniform ellipticity condition and that's why these kind of operators are also called elliptic
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because if you remember of course in geometry the elliptic quadrics the figures described in
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the space by second order polynomials are indeed elliptic if the matrix associated to the to the
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coefficients of the second order terms is a positive defined that is this condition here.
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Well okay now it's as we call also Lax-Meagre theorem because it's fundamental for the first two
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exercises so if h is an Hilbert space and b from h times h into R is a continuous coercive bilinear
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map that is there exists constant c and k such that the absolute value of b of uv is controlled
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by c the h norm of u the h norm of v for u and v in h and b of u u is larger equal than small k
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the h norm of u squared for all u in h if then L from h into R is a continuous linear functional
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then there exists a unique u in h such that b of uv is equal to L of v for all v in h. Exercise one
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you take u to be an open-mounted set b is a bilinear form over h1 zero times h1 zero into R
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associated to the linear to this partial differential operator L given by formula
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so before where the first order terms are all zero the coefficients of the first order terms
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the first order derivatives are all zero so basically there is only aij and c and of course
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it's elliptic prove that there exists a constant mu zero positive that you have to find such that
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b satisfies Lax-Megan theorem if for all mu in zero mu zero we have that basically the function
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c of x has a lower bound for almost every x given by minus mu and so now the continuity
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sorry here is again the text the continuity which we had our form b uv is of course the integral
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over u of the sum over i and j of aij u xi v xj plus c uv now the continuity is trivially implied
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by schwarz inequality we take out we bring the absolute value inside and we take out the maximum
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between the l infinity norm of the aj in the first term and the l infinity norm of c in the
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second term then we have this double sum this product of sums then we use by here in the second
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term we simply use Cauchy-Schwarz to make appear the l term of u times the l term of v and that's
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simple here instead we use the fact the sum of u xj absolute value of u xj is controlled by square
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root of n square root of the sum of the absolute value of u xj squared which is the norm of the u
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and the same for v xj so in the end one gets a factor n integral of absolute value of the u
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against sorry norm of the u against norm of the v Cauchy-Schwarz and we are done take a
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constant which is the sum of the constant that you get the h one the coercivity notice that b of u u
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is the sum of the i and j a i j u x i u xj plus c u squared well in the first term you use the
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ellipticity so you get the theta integral over u of norm of the u squared in the x the second term
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use the fact that c in the worst possible case is minus mu okay for almost every x in u but then
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we have the integral over u of u squared which is the l2 norm of u but u is in h1 zero remember
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and therefore we can use Poincare inequality that tells us that the l2 norm of u is controlled
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for above by a constant times the l2 norm of the gradient so that this becomes larger equal than
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theta minus mu the constant Poincare constant squared the l2 norm of the u squared but then
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one again uses Poincare to basically get that the l2 norm of the gradient squared the l2 norm
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of the gradient controls is controlled from above the h1 norm up to dividing by a constant which is
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c squared plus one where c is Poincare constant and so we get this lower bound but now this lower
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bound will be giving us indeed the coercivity as long as theta minus i mean the constant is
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positive that is theta minus mu c squared is positive which means mu should be less than theta
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over c squared so what is the constant the optimal constant mu zero that you can find theta over c
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squared that's it