EXTREMUM PROBLEMS FOR EIGENVALUES OF ELLIPTIC OPERATORS
CONTENTS EIGENVALUES OF ELLIPTIC OPERATORS NOTATION AND PREREQUISITES NOTATION AND SOBOLEV SPACES PARTIAL DI?ERENTIAL EQUATIONS EIGENVALUES AND EIGENFUNCTIONS ABSTRACT SPECTRAL THEORY APPLICATION TO ELLIPTIC OPERATORS FIRST PROPERTIES OF EIGENVALUES REGULARITY OF EIGENFUNCTIONS SOME EXAMPLES FREDHOLM ALTERNATIVE MIN-MAX PRINCIPLES AND APPLICATIONS MIN-MAX PRINCIPLES MONOTONICITY NODAL DOMAINS PERFORATED DOMAINS TOOLS SCHWARZ REARRANGEMENT STEINER SYMMETRIZATION DE?NITION PROPERTIES CONTINUOUS STEINER SYMMETRIZATION CONTINUITY OF EIGENVALUES INTRODUCTION CONTINUITY WITH VARIABLE COE?CIENTS CONTINUITY WITH VARIABLE DOMAINS (DIRICHLET CASE) THE CASE OF NEUMANN EIGENVALUES TWO GENERAL EXISTENCE THEOREMS DERIVATIVES OF EIGENVALUES DERIVATIVE WITH RESPECT TO THE DOMAIN CASE OF MULTIPLE EIGENVALUES DERIVATIVE WITH RESPECT TO COE?CIENTS THE ?RST EIGENVALUE OF THE LAPLACIAN-DIRICHLET THE FABER-KRAHN INEQUALITY THE CASE OF POLYGONS AN EXISTENCE RESULT THE CASES N = 3, 4 A CHALLENGING OPEN PROBLEM DOMAINS IN A BOX MULTI-CONNECTED DOMAINS THE SECOND EIGENVALUE OF THE LAPLACIAN-DIRICHLET MINIMIZING ?2 THE THEOREM OF KRAHN-SZEG¨O CASE OF A CONNECTEDNESS CONSTRAINT A CONVEXITY CONSTRAINT OPTIMALITY CONDITIONS GEOMETRIC PROPERTIES OF THE OPTIMAL DOMAIN ANOTHER REGULARITY RESULT THE OTHER DIRICHLET EIGENVALUES CONNECTEDNESS OF MINIMIZERS EXISTENCE OF A MINIMIZER FOR ?3 A CONCENTRATION-COMPACTNESS RESULT EXISTENCE OF A MINIMIZER CASE OF HIGHER EIGENVALUES FUNCTIONS OF DIRICHLET EIGENVALUES RATIO OF EIGENVALUES THE ASHBAUGH-BENGURIA THEOREM SOME OTHER RATIOS A COLLECTION OF OPEN PROBLEMS SUMS OF EIGENVALUES SUMS OF INVERSES DESCRIPTION OF THE SET E = (?1, ?2) EXISTENCE OF MINIMIZERS OTHER BOUNDARY CONDITIONS FOR THE LAPLACIAN NEUMANN BOUNDARY CONDITION MAXIMIZATION OF THE SECOND NEUMANN EIGENVALUE SOME OTHER PROBLEMS ROBIN BOUNDARY CONDITION THE BOSSEL-DANERS THEOREM OPTIMAL INSULATION OF CONDUCTORS STEKLO?EIGENVALUE PROBLEM EIGENVALUES OF SCHR¨ODINGER OPERATORS NOTATION A GENERAL EXISTENCE RESULT MAXIMIZATION OR MINIMIZATION OF THE ?RST EIGENVALUE THE MAXIMIZATION PROBLEM THE MINIMIZATION PROBLEM MAXIMIZATION OR MINIMIZATION OF OTHER EIGENVALUES SINGLE-WELL POTENTIALS MINIMIZATION OR MAXIMIZATION WITH AN L?CONSTRAINT MINIMIZATION OR MAXIMIZATION WITH AN LP CONSTRAINT MAXIMIZATION OF RATIOS MAXIMIZATION OF ?2(V )/?1(V ) IN ONE DIMENSION MAXIMIZATION OF ?N(V )/?1(V ) IN ONE DIMENSION NON-HOMOGENEOUS STRINGS AND MEMBRANES EXISTENCE RESULTS A ?RST GENERAL EXISTENCE RESULT A MORE PRECISE EXISTENCE RESULT NONLINEAR CONSTRAINT MINIMIZING OR MAXIMIZING ?K(?) IN DIMENSION 1 MINIMIZING ?K(?) MAXIMIZING ?K(?) MINIMIZING OR MAXIMIZING ?K(?) IN HIGHER DIMENSION CASE OF A BALL GENERAL CASE SOME EXTENSIONS 10 OPTIMAL CONDUCTIVITY 10 1 INTRODUCTION 10 2 THE ONE-DIMENSIONAL CASE 10 2 1 A GENERAL EXISTENCE RESULT 10 2 2 MINIMIZATION OR MAXIMIZATION OF ?K(?) 10 2 3 CASE OF NEUMANN BOUNDARY CONDITIONS 10 3 THE GENERAL CASE 10 3 1 THE MAXIMIZATION PROBLEM 10 3 2 THE MINIMIZATION PROBLEM 11 THE BI-LAPLACIAN OPERATOR 11 1 INTRODUCTION 11 2 THE CLAMPED PLATE 11 2 1 HISTORY 11 2 2 NOTATION AND STATEMENT OF THE THEOREM 11 2 3 PROOF OF THE RAYLEIGH CONJECTURE IN DIMENSION N = 2, 11 3 BUCKLING OF A PLATE 11 3 1 INTRODUCTION 11 3 2 THE CASE OF A POSITIVE EIGENFUNCTION 11 3 3 AN EXISTENCE RESULT 11 3 4 THE LAST STEP IN THE PROOF 11 4 SOME OTHER PROBLEMS 11 4 1 NON-HOMOGENEOUS ROD AND PLATE 11 4 2 THE OPTIMAL SHAPE OF A COLUMN CHAPTER 0(?) FOR SOME NON-LINEAR PROBLEMS, FOR EXAMPLE WHEN WE ARE INTERESTED IN THE P-LAPLACE OPERATOR, IT IS MORE CONVENIENT TO WORK WITH THE SPACES LP, P ?1 INSTEAD OF L2 IN THIS CASE, THE SOBOLEV SPACES, DE?NED EXACTLY IN THE SAME WAY, ARE DENOTED BY W 1,P(?) AND W 1,P N 1/2 DENOTES THE EUCLIDEAN NORM OF THE VECTOR ? WE WILL ALSO ASSUME A SYMMETRY ASSUMPTION FOR THE AIJ NAMELY: ?X ??, ?I, J AIJ(X) = AJI(X) (1 3) 1 1 NOTATION AND PREREQUISITES I (1 5) WHICH WILL BE CONSIDERED IN THE MAIN PART OF THIS BOOK (CHAPTERS 3 TO 7) IN CHAPTER 8, WE CONSIDER THE SCHR¨ODINGER OPERATOR LV U = ??U+V (X)U WHERE V (THE POTEN- TIAL) IS A BOUNDED FUNCTION, WHILE CHAPTERS 9 AND 10 DEAL WITH MORE GENERAL ELLIPTIC OPERATORS IN THAT CASE, WE WILL KEEP THE NOTATION L WHEN WE WANT TO CONSIDER GENERAL OPERATORS GIVEN BY (1 4) AT LAST, IN CHAPTER 11, WE CONSIDER OPERATORS OF FOURTH ORDER REMARK 1 1 3 LET US REMARK THAT, SINCE WE ARE ONLY INTERESTED IN EIGENVALUE PROBLEMS, WE DO NOT PUT ANY SIGN CONDITION ON THE FUNCTION A0(X) WHICH APPEARS IN (1 4) INDEED, SINCE A0(X) IS BOUNDED, WE CAN ALWAYS REPLACE THE OPERATOR L BY L + (?A0??+ 1)ID, I E REPLACE THE FUNCTION A0(X) BY A0(X) + ?A0??+ 1 IF WE NEED A POSITIVE FUNCTION IN THE TERM OF ORDER 0 OF THE OPERATOR L FOR THE EIGENVALUES, THAT WOULD JUST INDUCE A TRANSLATION OF ?A0??+ 1 TO THE RIGHT DIRICHLET BOUNDARY CONDITION LET F BE A FUNCTION IN L2(?) WHEN WE CALL U A SOLUTION OF THE DIRICHLET PROBLEM LU = F IN ?, U = 0(?) ?L2(?), F ? U SOLUTION OF (1 7) (1 8) CHAPTER 1 EIGENVALUES OF ELLIPTIC OPERATORS NEUMANN BOUNDARY CONDITION IN THE SAME WAY, IF F IS A FUNCTION IN L2(?), WE WILL ALSO CONSIDER U A SOLUTION OF THE NEUMANN PROBLEM LU = F IN ?, N I,J=1 AIJ ?U ?XJ NI = ABSTRACT SPECTRAL THEORY LET US NOW GIVE THE ABSTRACT THEOREM WHICH PROVIDES THE EXISTENCE OF A SEQUENCE OF EIGENVALUES AND EIGENFUNCTIONS LET H BE A HILBERT SPACE ENDOWED WITH A SCALAR PRODUCT ( , ) AND RECALL THAT AN OPERATOR T IS A LINEAR CONTINUOUS MAP FROM H INTO H WE SAY THAT: T IS POSITIVE IF, ?X ?H, (TX, X) ?0, T IS SELF-ADJOINT, IF ?X, Y ?H, (TX, Y) = (X, T Y), T IS COMPACT, IF THE IMAGE OF ANY BOUNDED SET IS RELATIVELY COMPACT (I E HAS A COMPACT CLOSURE) IN H 1 2 EIGENVALUES AND EIGENFUNCTIONS ? AIJ(X)?UN ?XI ?V ?XJ DX + ? A0(X)UN(X)V(X) DX ? ?= ? UN(X)V(X) DX WHICH MEANS L UN = ?N UN SETTING ?N = ?N , WE HAVE PROVED CHAPTER 1 EIGENVALUES OF ELLIPTIC OPERATORS THEOREM 1 2 2 LET ?BE A BOUNDED OPEN SET IN RN THERE EXISTS A SEQUENCE OF POSITIVE EIGENVALUES (GOING TO +?) AND A SEQUENCE OF CORRESPONDING EIGENFUNCTIONS (DE?NING A HILBERT BASIS OF L2(?)) THAT WE WILL DENOTE RESPECTIVELY 0 < ?D 1 (L, ?) ? ?D 2 (L, ?) ??D 3 (L, ?) ? AND UD 1 , UD 2 , UD 3 , SATISFYING: LUD N = ?D N (L, ?) UD N IN ?, UD N = ?(?) (ENDOWED WITH THE NORM ?U?2 = ?U2 DX + ??|?U|2 DX) THEN, IT HAPPENS THAT THE PREVIOUS THEORY WORKS AS SOON AS THE FUNCTION 1/? BELONGS TO SOME LP(?) SPACE WITH P > N/2 WE REFER E G TO [155] OR [205] AT LAST, THERE IS AN INTERESTING ALTERNATIVE TO PROVE COMPACTNESS THANKS TO THE GREEN FUNCTION, ONE CAN USUALLY WRITE THE OPERATOR AD L (OR AN L ) WITH AN INTEGRAL REPRESENTATION THEN, IT SU?CES TO PROVE THAT IT IS A HILBERT-SCHMIDT OPERATOR WE WILL SEE AN EXAMPLE OF THIS STRATEGY IN SECTION 10 2 3 WHEN L = ??IS THE LAPLACIAN, WE WILL SIMPLY DENOTE THE EIGENVALUES BY ?N(?) (OR ?N WHEN NO CONFUSION IS POSSIBLE) AND THE CORRESPONDING EIGENFUNCTIONS BY UN SINCE THE EIGENFUNCTIONS ARE DE?NED UP TO A CONSTANT, WE DECIDE TO NORMALIZE THE EIGENFUNCTIONS BY THE CONDITION ? UN(X)2 DX = 1 (1 14) OF COURSE, IT CAN OCCUR THAT SOME EIGENVALUES ARE MULTIPLE (ESPECIALLY WHEN THE DOMAIN HAS SYMMETRIES) IN THIS CASE, THE EIGENVALUES ARE COUNTED WITH THEIR MULTIPLICITY REMARK 1 2 4 WHEN ?IS NON-CONNECTED, FOR EXAMPLE IF ?HAS TWO CONNECTED COM- PONENTS ?= ?1 ??2, WE OBTAIN THE EIGENVALUES OF ?BY COLLECTING AND REORDERING THE EIGENVALUES OF EACH CONNECTED COMPONENTS ?D 1 (L, ?) = MIN(?D 1 (L, ?1), ?D 1 (L, ?2)), ?D 2 (L, ?) = MIN(MAX(?D 1 (L, ?1), ?D 1 (L, ?2)), ?D 2 (L, ?1), ?D 2 (L, ?2)), (1 15) MORE GENERALLY, WE CAN ALWAYS CHOOSE EVERY EIGENFUNCTION OF A DISCONNECTED OPEN SET ?TO VANISH ON ALL BUT ONE OF THE CONNECTED COMPONENTS OF ? IN PARTICU- LAR, WHEN THE TWO CONNECTED COMPONENTS ARE THE SAME, WE WILL HAVE ?D 1 (L, ?) = ?D 2 (L, ?), I E ?1 IS A DOUBLE EIGENVALUE 1 2 EIGENVALUES AND EIGENFUNCTIONS THAT CANNOT HAPPEN WHEN ?IS CONNECTED: THEOREM 1 2 5 LET US ASSUME THAT ?IS A REGULAR CONNECTED OPEN SET THEN THE ?RST EIGENVALUE ?D 1 (L, ?) IS SIMPLE AND THE ?RST EIGENFUNCTION UD 1 HAS A CONSTANT SIGN ON ? USUALLY, WE CHOOSE IT TO BE POSITIVE ON ? ACTUALLY, THIS THEOREM IS A CONSEQUENCE OF THE KREIN-RUTMAN THEOREM WHICH IS AN ABSTRACT RESULT THAT WE RECALL HERE (SEE [181] FOR A PROOF) THEOREM 1 2 6 (KREIN-RUTMAN) LET E BE A BANACH SPACE AND C BE A CLOSED CONVEX CONE IN E WITH VERTEX AT O, NON-EMPTY INTERIOR INT(C) AND SATISFYING C ?(?C) = {O} LET T BE A COMPACT OPERATOR IN E WHICH SATIS?ES T (C \ {O}) ?INT(C); THEN THE GREATEST EIGENVALUE OF T IS SIMPLE, AND THE CORRESPONDING EIGENVECTOR IS IN INT(C) (OR IN ?INT(C)) TO PROVE THEOREM 1 2 5, WE APPLY THE KREIN-RUTMAN THEOREM WITH E = C0(?), T = AD L AND C = {V ?C0(?), SUCH THAT V(X) ?0 } THEN, THE ASSUMPTION T (C \ {O}) ?INT(C) COMES FROM THE STRONG MAXIMUM PRINCIPLE THE FACT THAT T CAN BE DE?NED AS AN OPERATOR FROM E TO E AND THE FACT THAT IT IS COMPACT COMES FROM CLASSICAL REGULARITY RESULTS (IF THE RIGHT-HAND SIDE OF F IS CONTINUOUS, THE SOLUTION U OF (1 6) IS ALSO CONTINUOUS AND EVEN H¨OLDERIAN (DE GIORGI-STAMPACCHIA THEOREM, SEE [36], [94]) REMARK 1 2 7 WE WILL SEE TWO OTHER PROOFS OF THE NON-NEGATIVITY OF THE ?RST EIGENFUNCTION (AND SIMPLICITY OF THE ?RST EIGENVALUE) IN SECTION 1 3 3 IN PARTICULAR, NO REGULARITY ASSUMPTIONS ARE ACTUALLY NEEDED FOR THIS RESULT NEUMANN BOUNDARY CONDITION IN THE SAME WAY, WHEN ?IS A BOUNDED LIPSCHITZIAN OPEN SET IN RN, WE CAN PROVE THE FOLLOWING THEOREM (THE LIPSCHITZ REGULARITY OF ?IS NECESSARY TO HAVE THE COMPACT EMBEDDING H1(?) ?L2(?)): THEOREM 1 2 8 LET ?BE A BOUNDED OPEN LIPSCHITZIAN SET IN RN THERE EXISTS A SE- QUENCE OF NON-NEGATIVE EIGENVALUES (GOING TO +?) AND A SEQUENCE OF CORRESPONDING EIGENFUNCTIONS (DE?NING A HILBERT BASIS OF L2(?)) THAT WE WILL DENOTE RESPECTIVELY 0 ?µN 1 (L, ?) ?µN 2 (L, ?) ?µN 3 (L, ?) ? AND UN 1 , UN 2 , UN 3 , SATISFYING: LUN N = µN N (L, ?) UN N IN ?, ?UN N ?N = ON ?? (1 16) WHEN L = ??IS THE LAPLACIAN, WE WILL SIMPLY DENOTE THE EIGENVALUES BY µN(?) (OR µN WHEN NO CONFUSION IS POSSIBLE) AND THE CORRESPONDING EIGENFUNCTIONS BY UN WE OBSERVE THAT, FOR THE LAPLACIAN AND, MORE GENERALLY IN THE CASE A0(X) = 0, THE ?RST EIGENVALUE IS ALWAYS µ1 = 0 AND A CORRESPONDING EIGENFUNCTION IS A NON- ZERO CONSTANT (ON A CONNECTED COMPONENT OF ?)