If my understanding stands: you are trying to find out all available periods when there are some absent-like period lists
Then there is a complete solution to solve when there are lots of absent-like period lists:
public class Extra_1_interval_merge {
    @Test
    public void testAvailablePeriod() {
        List<MyPeriod> absentPeriods0 = new ArrayList<>();
        absentPeriods0.add(makePeriod(LocalDate.now(), LocalDate.now().plusDays(1), PeriodType.ABSENT));
        absentPeriods0.add(makePeriod(LocalDate.now().plusDays(4), LocalDate.now().plusDays(6), PeriodType.ABSENT));
        absentPeriods0.add(makePeriod(LocalDate.now().plusDays(2), LocalDate.now().plusDays(3), PeriodType.ABSENT));
        List<MyPeriod> absentPeriods1 = new ArrayList<>();
        absentPeriods1.add(makePeriod(LocalDate.now(), LocalDate.now().plusDays(2), PeriodType.UNAVAILABLE));
        absentPeriods1.add(makePeriod(LocalDate.now().plusDays(5), LocalDate.now().plusDays(7), PeriodType.UNAVAILABLE));
        List<List<MyPeriod>> absentListList = new ArrayList<>();
        absentListList.add(absentPeriods0);
        absentListList.add(absentPeriods1);
        System.out.println(getAvailablePeriods(absentListList));
    }
    private List<MyPeriod> getAvailablePeriods(List<List<MyPeriod>> absentListList) {
        // Step - 1: Collect all periods;
        List<MyPeriod> tempList = new ArrayList<>();
        absentListList.stream().forEach(list -> tempList.addAll(list));
        // Step - 2: Sort the periods based on the startDate and then endDate;
        List<MyPeriod> absentList = tempList.stream().sorted((period1, period2) -> {
            if (!period1.startDate.isEqual(period2.startDate)) {
                return period1.startDate.compareTo(period2.startDate);
            } else {
                return period1.endDate.compareTo(period2.endDate);
            }
        }).collect(toList());
        // Step - 3: Merge all overlapped periods to form an one-dimension occupied period list;
        List<MyPeriod> mergedPeriods = new ArrayList<>();
        for (MyPeriod period : absentList) {
            if (mergedPeriods.isEmpty()) {
                mergedPeriods.add(period);
            } else {
                MyPeriod lastPeriod = mergedPeriods.get(mergedPeriods.size() - 1);
                if (!lastPeriod.endDate.isBefore(period.startDate)) {
                    if (lastPeriod.endDate.isBefore(period.endDate)) {
                        lastPeriod.endDate = period.endDate;
                    }
                } else {
                    mergedPeriods.add(period);
                }
            }
        }
        // Step - 4: Pick the periods from the occupied period list;
        List<MyPeriod> availablePeriods = new ArrayList<>();
        for (int i = 0, len = mergedPeriods.size(); i < len - 1; i++) {
            availablePeriods.add(makePeriod(mergedPeriods.get(i).endDate, mergedPeriods.get(i + 1).startDate, PeriodType.AVAILABLE));
        }
        return availablePeriods;
    }
    private MyPeriod makePeriod(LocalDate startDate, LocalDate endDate, PeriodType periodType) {
        MyPeriod thePeriod = null;
        switch (periodType) {
            case ABSENT:
                thePeriod = new AbsentPeriod();
                break;
            case UNAVAILABLE:
                thePeriod = new UnavailablePeriod();
                break;
            case AVAILABLE:
                thePeriod = new AvailablePeriod();
                break;
            default:
                thePeriod = new MyPeriod();
                break;
        }
        thePeriod.startDate = startDate;
        thePeriod.endDate = endDate;
        return thePeriod;
    }
    enum PeriodType {
        ABSENT,
        UNAVAILABLE,
        AVAILABLE;
    }
    class MyPeriod {
        LocalDate startDate;
        LocalDate endDate;
        @Override
        public String toString() {
            return String.format("Start: %s, End: %s", startDate, endDate);
        }
    }
    class AbsentPeriod extends MyPeriod {
    }
    class UnavailablePeriod extends MyPeriod {
    }
    class AvailablePeriod extends MyPeriod {
    }
}
For the input:
- Absent period list - 1: [0, 1], [4, 6], [2, 3]
- Absent period list - 2: [0, 2], [5, 7]
The final result will be:
- Available period list: [3, 4]
You can try more absent period lists if you need. 
Updated
I have no idea why the OP need three different types of Period/Interval here. But to solve the specific problem, I updated the solution based on the OP's needs. 
Just as other comments point out, why three different types? I have no clue...