VERFEINERTE PARTIELLE INTEGRATION: AUSWIRKUNGEN AUF DIE KONSTANTEN

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1 VERFEINERTE PARTIELLE INTEGRATION: AUSWIRKUNGEN AUF DIE KONSTANTEN IN MAXWELL- UND KORN-UNGLEICHUNGEN CARL VON OSSIETZKY UNIVERSITÄT OLDENBURG 37. NORDWESTDEUTSCHES FUNKTIONALANALYSIS-KOLLOQUIUM GASTGEBER: Andreas Defant, Peter Harmand, Michael Langenbruch Dirk Pauly Universität Duisburg-Essen 14. November 015

2 OVERVIEW MAXWELL INEQUALITIES TWO MAXWELL INEQUALITIES PROOFS KORN S FIRST INEQUALITIES STANDARD HOMOGENEOUS SCALAR BOUNDARY CONDITIONS NON-STANDARD HOMOGENEOUS TANGENTIAL OR NORMAL BOUNDARY CONDITIONS REFERENCES DISTURBING CONSEQUENCES FOR VILLANI S WORK(FIELDS MEDAL) CITATIONS SOME FUN...

3 TWO MAXWELL INEQUALITIES R 3 bounded, weak Lipschitz (even weaker possible) ) R( ) \ rot R( ),! L ( ), R( ) \ rot R( ),! L ( ) ) Maxwell estimates: 9 c m > 0 8 E R( ) \ rot R( ) E L ( ) apple cm rot E L ( ) 9 c m > 0 8 H R( ) \ rot R( ) H L ( ) apple cm rot H L ( ) note: best constants 1 rot E L = inf ( ), c E m 06=ER( )\rot R( ) L ( ) 1 c m = inf 06=HR( )\rot R( ) rot H L ( ) H L ( ) Theorem (i) c m = c m (ii) convex ) c m apple c p Poincaré estimate: 9 c p > 0 8 u H 1 ( ) \ R? u L ( ) apple cp ru L ( ) best constant: 1 = c p ru L inf ( ) 06=uH 1 ( )\R? u L ( )

4 PROOF OF MAXWELL INEQUALITIES step one: two lin., cl., dens. def. op. and their reduced op. A : D(A) X! Y, A : D(A) :=D(A) \ R(A ) R(A )! R(A), A : D(A ) Y! X, A : D(A ):=D(A ) \ R(A) R(A)! R(A ) crucial assumption: D(A),! X, D(A ),! Y + gen. Poincaré estimates: note: best constants Theorem c A = c A 9 c A > 0 8 x D(A) x applec A Ax 9 c A > 0 8 y D(A ) y applec A A y 1 = inf c A 06=xD(A) Ax x, 1 c A A y = inf 06=yD(A ) y

5 PROOF OF MAXWELL INEQUALITIES step two: two lin., cl., den. def. op. and their reduced op. choose A : D(A) X! Y, A : D(A) :=D(A) \ R(A ) R(A )! R(A), A : D(A ) Y! X, A : D(A ):=D(A ) \ R(A) R(A)! R(A ) A := rot : R( ) L ( )! L ( ), rot : R( ) \ rot R( ) rot R( )! rot R( ), rot = rot : R( ) L ( )! L ( ), rot = rot : R( ) \ rot R( ) rot R( )! rot R( ) crucial assumption: R( ) \ rot R( ),! L ( ), R( ) \ rot R( ),! L ( ) + gen. Poincaré estimates (Maxwell estimates): 9 c m > 0 8 E R( ) \ rot R( ) E L ( ) apple cm rot E L ( ) 9 c m > 0 8 H R( ) \ rot R( ) H L ( ) apple cm rot H L ( ) Theorem c m = c m

6 PROOF OF MAXWELL INEQUALITIES step three: Proposition (integration by parts (Grisvard s book and older...)) Let R 3 be piecewise C. Then for all E C 1 ( ) div E + rot L ( ) E re L ( ) L ( ) LX Z LX Z = div E n +((r ) E t ) E t + {z } `=1 ` `=1 curvature, sign!... 0, if convex. ` (E n div E t E t r E n). {z } boundary conditions, no sign! approx. convex from inside by convex and smooth ( k ) k ) Corollary (Gaffney s inequality) Let R 3 be convex and E R( ) \ D( ) or E R( ) \ D( ). Then E H 1 ( ) and rot E L ( ) + div E L ( ) re L ( ) 0.

7 PROOF OF MAXWELL INEQUALITIES step four: (Poincaré) 9 c p > 0 8 u H 1 ( ) \ R? u L ( ) apple cp ru L ( ) Let be convex and E R( ) \ D 0 ( ). Note D 0 ( ) = rot R( ). Cor. (Gaffney) ) E H 1 ( ) and E = rot H with H R( ). ) E H 1 ( ) \ (R 3 )? \ D 0 ( ), since he, ai L ( ) = hrot H, ai L = 0 for a R3 ( ) + + Theorem convex ) c p apple c m = c m apple c p Here: E L ( ) apple cp re L ( ) apple cp rot E L ( ) c m apple c p (Poincaré/Friedrichs) 9 c p > 0 8 u H 1 ( ) u L ( ) apple cp ru L ( )

8 MATRICES Let A R N N. sym skw A := 1 (A ± A> ), (pointwise orthogonality) ) id A := tr A N id, tr A := A id, dev A := A id A A = dev A + 1 N tr A, A = sym A + skw A, sym A = dev sym A + 1 tr A N ) dev A, N 1/ tr A, sym A, skw A apple A R N and A := rv := J > v for v H 1 ( ) ) (pointwise) skw rv = 1 rot v, tr rv = div v, Moreover rv = dev sym rv + 1 N div v + 1 rot v (1) rv = rot v + hrv, (rv) > i since skw rv = 1 rv (rv)> = rv hrv, (rv) > i.

9 KORN S FIRST INEQUALITY: STANDARD BOUNDARY CONDITIONS Lemma (Korn s first inequality: H 1 -version) Let be an open subset of R N with apple N N. Then for all v H 1 ( ) rv L ( ) = dev sym rv L ( ) + and equality holds if and only if div v = 0 or N =. Proof. note: =rot rot r div (vector Laplacian) N N div v apple dev sym L ( ) rv L ( ) ) 8v C 1 ( ) rv L ( ) = rot v L ( ) + div v L ( ) (Gaffney s equality) () () extends to all v H 1 ( ) by continuity. Then rv = dev sym L ( ) rv + 1 L ( ) rv + N L ( ) N div v L ( ) follows by (1), i.e., rv = dev sym rv + 1 N div v + 1 rot v, and ().

10 KORN S FIRST INEQUALITY: TANGENTIAL/NORMAL BOUNDARY CONDITIONS main result: Theorem (Korn s first inequality: tangential/normal version) Let R N be piecewise C -concave and v H 1 t,n ( ). Then Korn s first inequality holds. If is a polyhedron, even rv L ( ) = dev sym rv L ( ) + rv L ( ) apple p dev sym rv L ( ) is true and equality holds if and only if div v = 0 or N =. N N div v apple dev sym L ( ) rv L ( )

11 KORN S FIRST INEQUALITY: TANGENTIAL/NORMAL BOUNDARY CONDITIONS tools: Proposition (integration by parts (Grisvard s book and older...)) Let R N be piecewise C. Then LX Z div v + rot L ( ) v rv = div v L ( ) L ( ) n +((r ) v t ) v t {z } `=1 ` curvature, sign! LX Z + (v n div v t v t r v n), {z } `=1 ` boundary conditions, no sign! LX Z div v + rot L ( ) v rv = div v L ( ) L ( ) n +((r ) v t ) v t. `=1 ` holds for all v C 1 ( ) resp. v C 1 t,n ( ). Corollary (Gaffney s inequalities) Let R N be piecewise C and v H 1 t,n ( ). Then 8 >< apple 0, if is piecewise C -concave, rot v + div L ( ) v rv = 0, if is a polyhedron, L ( ) L ( ) >: 0, if is piecewise C -convex.

12 KORN S FIRST INEQUALITY: TANGENTIAL/NORMAL BOUNDARY CONDITIONS Proof. (1), i.e., rv = dev sym rv + 1 N div v + 1 rot v, and the corollary ) rv apple dev sym L ( ) rv + 1 L ( ) rv + N L ( ) N div v L ( ) ) first estimate polyhedron ) equality holds

13 REFERENCES I Pauly, D.: Zapiski POMI, (014) On Constants in Maxwell Inequalities for Bounded and Convex Domains I Pauly, D.: Discrete Contin. Dyn. Syst. Ser. S, (015) On Maxwell s and Poincaré s Constants I Pauly, D.: Math. Methods Appl. Sci., (015) On the Maxwell Constants in 3D I Bauer, S. and Pauly, D.: submitted, (015) On Korn s First Inequality for Tangential or Normal Boundary Conditions with Explicit Constants I Bauer, S. and Pauly, D.: submitted, (015) On Korn s First Inequality for Tangential or Normal Boundary Conditions I Desvillettes, L. and Villani, C.: ESAIM Control Optim. Calc. Var., (00) On a variant of Korn s inequality arising in statistical mechanics. A tribute to J.L. Lions. I Desvillettes, L. and Villani, C.: Invent. Math., (005) On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation

14 CITATIONS I Desvillettes, L. and Villani, C.: ESAIM Control Optim. Calc. Var., (00) On a variant of Korn s inequality arising in statistical mechanics. A tribute to J.L. Lions. page 607 page 608 page 609 Proposition 5 (end of) Theorem 3 (continued) page 609 (closed graph theorem) I Desvillettes, L. and Villani, C.: Invent. Math., (005) On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation page 306

15 HOW ONE CANNOT APPLY THE CLOSED GRAPH THEOREM! generally: compact embedding or regularity + closed graph theorem (hard analysis to do!) ) Poincaré type estimate surprisingly: 9 people closed graph / open mapping / bounded inverse theorem (example on next slide) ) Poincaré type estimate!!! THIS IS WRONG!!!

16 to understand that this number may be useful in the context of the Boltzmann equation. Even more, to our M AXWELL INEQUALITIES KORN S FIRST INEQUALITIES REFERENCES DISTURBING CONSEQUENCES FOR V ILLANI S WORK ( FIELDS MEDAL ) knowledge his paper is the only one to mention this fact. This justifies our terminology of Grad s number. The present work drew a lot of inspiration from Grad s paper [9], which is at the same time quite obscure, definitely false and really illuminating in certain respects as we will discuss in [4]. 3. If is simply connected, THE which is CLOSED presumably the most natural case for applications, then H OW ONE CANNOT APPLY GRAPH THEOREM! V contains a unique element (we shall show in a moment that V is never empty). 0 = 0 and V 4. Our primary goal was to obtain fully explicit lower bounds for K( ) in terms of simple geometrical information about ; to achieve this completely with our method, we would have to give quantitative estimates on CH. Unfortunately, we have been unable to find explicit estimates about CH in the literature, although it seems 608unlikely that nobody has been interested L. DESVILLETTES ANDOf C. course, VILLANIwhen N = 3 and in this problem. is simply connected, estimate (10) is equivalent to sym view of fluid dynamics, our context of application may seem rather strange, because u L looks like an CHin ( the ) theory u Lof + Navier Stokes u L ( ), equations; but the tangency (13) energy dissipation term, of the kind uencountered ( )the L ( ) boundary condition on u is typical of inviscid models, like the Euler equation. There is no contradiction at the sym which is well-known to many people, but for CH + 1.the This is an estimate possible replacement u L levelupoftothe modelling, becauseofinch ourbymethod term is not obtained as a dissipation term, but which it seems very to find an accurate reference. ofinequality can be seen as a consequence of the as the leading order, in difficult some sense, of the second derivative a certain(10) functional. closed graphpoint theorem; for instance, in the of a simply one just which needs to (i) The second on which we attract thecase attention of theconnected reader is domain, the importance we note give that to the a a u + u is bounded by u, (ii) the identities u = 0, u =the 0, value u n =of0 the (onconstant the boundary), value of the K( ) in (8). In K( ) L positive constant L L our study of trend to equilibrium, together imply uthe = deviation 0; so in fact norms appearing on left and of ongreat the right-hand have to be is used to quantify of thefrom axisymmetry. It the is therefore interest toside haveofas(10) much insight equivalent. The proof value of point (ii) is follows:of from Poincare s in a the simply connected domain, there as possible in the explicit of K( ), as in terms the geometry of lemma. In fact, main interest of the present exists real-valued function estimates such thaton K(= ). u; then is a harmonic function with homogeneous Neumann work is toaprovide the following boundary condition, so it has to be a constant, and u = 0. Of course this argument no insight on how to estimate As pointed out to us independently Theorem 3 (continued). Thegives largest admissible constant K( )the inconstants. (8) satisfies by Druet and by Serre, one can choose CH ( ) = 1 if is convex, but the general case seems to be much harder. Anyway this is a separate nothing Nhas 1+ CH ( )to 1do+with K( axisymmetry; ) G( )all1the, relevant information about (9) K(issue ) 1which axisymmetry lies in our estimates on G( ) 1. organization of the paper as follows: after a short proof of Theorem 3 in Section 3, we shall give where The the various constants above areisdefined as follows: some quantitative estimates on the positivity of G( ) in Section 4, and finally give a brief discussion of the ) isina Section constant to the homology of and the of Hodge decomposition, estimate defined byof the axisymmetric CH = CH ( case 5. related In an Appendix, we reproduce a proof the abovementioned CH inequality when is convex, which was communicated to us by Druet. sym v L ( )/V0 ( ) CH v L ( ) + a v L ( ), (10) or (almost) equivalently by inequality (13) below. Here v stands for the divergence of the vector field v, v = i vi / xi, and V0 ( ) is the space of all vector fields v0 H 1 ( ; RN ) such that v0 = 0, a v0 = 0. We recall that V0 is a finite-dimensional vector space whose dimension depends only on the topology of K( ) is the constant in (1); (copies from and original finally, G =paper...) G( ) is what we shall call Grad s number: ;