FUNCTIONAL METHODS IN DIFFERENTIAL EQUATIONS

By J. JOLIA

1 PRELIMINARIES 2 ELLIPTIC BOUNDARY VALUE PROBLEMS 3 PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS 4 PARABOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS 5 HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC BOUNDARY CONDITIONS 6 HYPERBOLIC BOUNDARY VALUE PROBLEMS WITH ALGEBRAIC-DIFFERENTIAL BOUNDARY CONDITIONS 7 THE FOURIER METHOD FOR ABSTRACT DIFFERENTIAL EQUATIONS 8 THE SEMIGROUP APPROACH FOR ABSTRACT DIFFERENTIAL EQUATIONS 9 NONLINEAR NONAUTONOMOUS ABSTRACT DIFFERENTIAL EQUATIONS 10 IMPLICIT NONLINEAR ABSTRACT DIFFERENTIAL EQUATIONS CONTENTS CHAPTER FUNCTION AND DISTRIBUTION SPACES THE LP-SPACES WE DENOTE IR = (??, ?), IN = {0, 1, 2, }, AND IN?= {1, 2, } LET X BE A REAL BANACH SPACE WITH NORM ? ?X IF ??IRN, N ?IN?, IS A LEBESGUE MEASURABLE SET, WE DENOTE, AS USUAL, BY LP(?; X), 1 ?P < ?, THE SPACE OF ALL EQUIVALENCE CLASSES (WITH RESPECT TO THE EQUALITY A E IN ?) OF (STRONGLY) MEASURABLE FUNCTIONS F: ??X SUCH THAT X 7??F(X)?P X IS LEBESGUE INTEGRABLE OVER ? IN GENERAL, EVERY CLASS OF LP(?; X) IS IDENTI?ED WITH ONE OF ITS REPRESENTATIVES LP(?; X) IS A REAL BANACH SPACE WITH THE NORM ?U?LP(?;X) = Z ? ?U(X)?P X DX P WE SHALL DENOTE BY L?(?; X) THE SPACE OF ALL EQUIVALENCE CLASSES OF MEASURABLE FUNCTIONS F: ??X SUCH THAT X 7??F(X)?X ARE ESSENTIALLY BOUNDED IN ? AGAIN, EVERY CLASS OF L?(?; X) IS IDENTI?ED WITH ONE OF ITS REPRESENTATIVES L?(?; X) IS A REAL BANACH SPACE WITH THE NORM ?U?L?(?;X) = ESS SUP X?? ?U(X)?X IN THE CASE X = IR WE SHALL SIMPLY WRITE LP(?) INSTEAD OF LP(?; IR), FOR EVERY 1 ?P ?? ON THE OTHER HAND, IF ?IS AN INTERVAL OF REAL NUMBERS, SAY ?= (A, B) WHERE ???A < B ??, THEN WE SHALL WRITE LP(A, B; X) INSTEAD LOC(?) (I E , U IS LEBESGUE INTEGRABLE ON EVERY COMPACT SUBSET OF ?), THEN THE FUNCTIONAL DE?NED BY D(?) ?? 7? Z ? U(X)?(X) DX IS A DISTRIBUTION ON ? SUCH A DISTRIBUTION WILL ALWAYS BE IDENTI?ED WITH THE CORRESPONDING FUNCTION U AND SO IT WILL BEDENOTED BY U NOW, RECALL THAT THE PARTIAL DERIVATIVE OF A DISTRIBUTION U ?D?(?) WITH RESPECT TO XJ IS DE?NED BY ?U ?XJ (?) = ?U( ?? ?XJ ) FOR ALL ? ?D(?), AND THE HIGHER ORDER PARTIAL DERIVATIVES OF U ARE DE?NED ITERATIVELY, I E , D?U(?) = (?1)|?|U(D??) FOR ALL ? ?D(?), LET 1 ?P ??AND LET K ?IN?BE ?XED THEN, THE SET W K,P(?) = {U: ??IR | D?U ?LP(?) FOR ALL ? ?INN WITH |?| ?K} (WHERE D?U ARE THE DERIVATIVES OF U IN THE SENSE OF DISTRIBUTIONS) IS SAID TO BE A SOBOLEV SPACE OF ORDER K RECALL THAT, FOR EACH 1 ?P < ?AND K ?IN?, W K,P(?) IS A REAL BANACH SPACE WITH THE NORM ?U?W K,P(?) = X |?|?K ?D?U?P LP(?) P W K,?(?) IS ALSO A REAL BANACH SPACE WITH THE NORM ?U?W K,?(?) = MAX |?|?K ?D?U?L?(?) THE COMPLETION OF D(?) WITH RESPECT TO THE NORM OF W K,P(?) IS DENOTED BY W K,P (?) IN GENERAL, W K,P 0(?) = {U ?H1(?) | THE TRACE OF U ON ??VANISHES } IF, IN PARTICULAR, ?IS AN INTERVAL OF REAL NUMBERS, SAY ?= (A, B) WITH A < B, THEN WE SHALL WRITE C? 0 (A, B), W K,P(A, B), HK(A, B), AND W K,P (A, B) INSTEAD OF C? (A, B), W K,P(A, B), HK((A, B)), AND W K,P ((A, B)), RESPECTIVELY IF A, B ARE ?NITE NUMBERS, THEN EVERY ELEMENT OF W K,P(A, B), K ?IN?, 1 ?P ??, CAN BE IDENTI?ED WITH AN ABSOLUTELY CONTINUOUS FUNCTION F: [A, B] ?IR SUCH THAT DJF/DTJ, 1 ?J ?K ?1, EXIST AND ARE ALL ABSOLUTELY CONTINUOUS ON [A, B], AND DKF/DTK (THAT OBVIOUSLY EXISTS A E IN (A, B)) BELONGS TO LP(A, B) (MORE PRECISELY, THE EQUIVALENCE CLASS OF DKF/DTK, WITH RESPECT TO THE EQUALITY A E FUNCTIONAL METHODS IN DIFFERENTIAL EQUATIONS ON (A, B) BELONGS TO LP(A, B)) MOREOVER, EVERY ELEMENT OF W K,P ?U(J)?P LP(A,B;X) P ALSO, FOR EACH K ?IN?, W K,?(A, B; X) IS A BANACH SPACE WITH THE NORM ?U?W K,?(A,B;X) = MAX 0?J?K ?U(J)?L?(A,B;X) LET X BE A REAL RE?EXIVE BANACH SPACE AND LET U ?LP(A, B; X) WITH ??< A < B < ?AND 1 < P < ? THEN, THE FOLLOWING TWO CONDITIONS ARE EQUIVALENT: (I) U ?W 1,P(A, B; X); EVERY U ?W(A, B) HAS A REPRESENTATIVE U1 ?C([A, B]; H) AND SO U CAN BE IDENTI?ED WITH SUCH A FUNCTION FURTHERMORE, IF U, ˜U ?W(A, B), THEN THE FUNCTION T 7?(U(T), ˜U(T))H IS ABSOLUTELY CONTINUOUS ON [A, B] AND D DT U(T), ˜U(T) H = ?U(T), ˜U?(T)?+ ?˜U(T), U?(T)?FOR A A T ?(A, B) HENCE, IN PARTICULAR, D DT U(T) LET F BE A FAMILY OF MAPPINGS FROM A SEPARABLE TOPOLOGICAL SPACE X TO A COMPACT METRIC SPACE M IF F IS EQUI-CONTINUOUS, THEN EVERY SEQUENCE OF F HAS A SUBSEQUENCE, WHICH CONVERGES UNIFORMLY ON COMPACT SETS OF X