Is the trash in my trash cans that must stay on my property during Passover still considered under my ownership?
See Shulchan Aruch Orach Chayim 445, 2.
קודם זמן איסורו - יכול להשליכו במקום שהעורבים מצויים שם. ואם מצאו אחר זמן איסורו שלא אכלוהו העורבים - אף על פי שמקום הפקר, לא יניחנו שם אלא יבערנו.
Before the time of prohibition, one can discard the Chamets in a place in wich crows are widespread. If one found it after the starting time of the prohibition, although that this place is Hefker, one must not leave it, but he must destroy it.
במקום שהעורבים וכו': ובהכי מיירי ההיא דחצר א"צ בדיק' (הג"מ) וא"כ מוכח דאיירי בחצר דידיה דאי בחצר של הפקר כגון ברחוב מאי בדיקה שייך דאפי' חמצו אם הפקירו קודם שש שרי כמ"ש הטור סי' תל"ז וב"י סי' תמ"ח בשם הירושל' וד"מ בשם א"ז דמה לי נתנו לעכו"ם או הפקירו אלא ע"כ לא מיירי במקום הפקר ומה שסיים אף על פי שמונח במקום הפקר מ"מ וכו' כלו' כיון שמונח בחצר א"כ הוא מופקר לכל הרוצה ליטול מ"מ כיון שמונח ברשותו חייב לבערו כדינו דהיינו בשריפה לר"י וכ"מ בהג"מ כמ"ש וז"ל הכ"מ ואותן בני אדם המשימין חמץ בצידי ר"ה בשעה ו' שלא כדין הם עושים דמשהגיע שעה ו' צריך לבערו מן העולם וכ"מ בהג"מ עכ"ל משמע דקודם שעה ו' שרי ועמ"ש סי' תל"ג וביורה דעה סי' קמ"ו:
(Loose and summarized translation) When the Shulchan Aruch said an Hefker place, he meant that this is his own courtyard, available for whomever desires take it. But considering that this courtyard is his own property, he must destroy. If the courtyard is itself hefker, Magen Avraham added in name of Hagaot Maymoniot that people who leave Chamets on the edge of the street make mistake, after the sixth hour, they need to destroy it. Magen Avraham inferred from this sentence that if they leave the chamets on the edge of the street before th sixth hour, this is good enough.
Mishna Berura agrees from the halachic point of view with the Magen Avraham.
cans that must stay on my property during Passover still considered under my ownership?
--> Yes, you need to destroy it if you find it when it's already prohibited, because it's still in your property(1).
And Magen Avraham 446 sk1 says that you need to bless before the destruction, despite you already annulled this chamets
נ"ל דיברך על ביעור חמץ אף על פי שכבר ביטל מ"מ מברך על תקנת חכמים
Rashi Pesachim 6a:
ומיהו בבל יראה לא עבר דהא בטליה בליביה מאתמול כדאמרינן לקמן הבודק צריך שיבטל בלבו אלא משום שלא ישכח ויאכלנו צריך כפיית כלי:
Chashash perhaps he will eat it.
The Mishnah Berurah (431:2) says that we don't rely on Bitul (that is, being mafkir the chameitz by saying kol chamira v'chamiya) because it's possible you didn't have the full intention of getting rid of it. When you throw out your chameitz, though, you definitely have the intention of getting rid of it. Thus, it's out of your reshus and there's no issue of bal yeira'eh.
The shulchan aruch harav website can shed some light on this subject:
May one discard his Chametz trash into a garbage bin, if it will remain there after the 6th hour on Erev Pesach?1
As explained in the above Halacha, it is forbidden to discard one’s Chametz into an area that is not public property. This raises a Halachic issue with discarding one’s Chametz garbage into garbage bins before the 6th hour on Erev Pesach, if the garbage will not be collected before that time. The following is the ruling regarding this matter:
Bin owned by city/municipality: It is permitted to throw Chametz garbage into a city owned garbage bin, even if the garbage will not be collected until after the 6th hour on Erev Pesach.
Bin is privately owned: One may not discard his Chametz into a privately-owned garbage bin, whether it is one’s personal property, or the property of one’s building, if it will remain there past the 6th hour on Erev Pesach. Thus, one may not discard Chametz garbage into a buildings garbage shoot if it will not be cleared out before the 6th hour on Erev Pesach. One who does so, possibly transgresses the prohibition against owning Chametz. The following are the available options in such a case:
Mechiras Chametz: If one sold all of his Chametz to a gentile, as is commonly done today through the community Rabbi, then according to the standard sale contracts used today, all one’s Chametz, including Chametz that is dirty and has no price value, is included in the sale. Accordingly, the Chametz that remains in the garbage in truth is sold to the gentile, and one does not need to do anything further to get rid of it. However, some Poskim negate relying on this sale regarding the Chametz garbage, and thus state one is to use one of the next options. Disown garbage bin: Disown the garbage can and its contents in front of three people. If the bin is standing in one’s property, then one is to also disown that area of one’s property. In such a case, one may use the garbage can throughout Pesach. Spoiling the Chametz: Spill bleach, or another spoiling agent, onto the Chametz that is in the garbage before the 6th hour on Erev Pesach, thus making it inedible even for a dog. Breaking to less than Kezayis: To break the Chametz into pieces of less than a Kezayis and then discard it directly into the garbage together with all the other filth. Burn: To save all leftover Chametz starting from the last garbage pickup before the 6th hour of Erev Pesach, until the time of Biur Chametz, in which case one will burn all the Chametz he has. Garbage pickup: To arrange with the municipality to pick up the Chametz on Erev Pesach, up until the 6th hour of the day. [Any Chametz which is discarded after this pickup would need to follow one of the previous mentioned option.]
and the sources:
1 See Chelkas Yaakov 3:165; Minchas Yitzchak 4:56; Shevet Halevi 1:137; Beir Moshe 1:41; 3:74; Betzel Hachochma 3:109; Mishneh Halacha 4:63; Teshuvos Vehanhagos 2:211; Kinyan Torah 2:86; 7:36; Piskeiy Teshuvos 445:7
 Minchas Yitzchak ibid writes that so is the custom, and defends the practice, although concludes that it is best to pour a spoiling agent [i.e. bleach] into the garbage even in such a case; Kinyan Torah ibid
 The reason: By a gentile majority city, being the dumpster is owned by the gentile municipality, it is automatically acquired to them [according to the opinion who rules that Kinyan Chatzer is valid by a gentile]. This applies even if the dumpster is situated in a private area, such as in the building courtyard, as the area of land in which the dumpster is sitting on is considered rented to the gentile municipality. Furthermore, even if one does not hold of Kinyan chatzar by a gentile, the Chametz is considered to be disowned in a public area, as the area belongs to the municipality. [Minchas Yitzchak ibid; See Admur 436:20 that Chametz which is left in the home of a gentile is considered Hefker and does not have to be cleaned before Pesach]
The law by a Jewish municipality: Some write that the above allowance only applies by a gentile municipality, while in a Jewish municipality, it is forbidden to leave Chametz in their garbage bins, as it is forbidden to leave even disowned Chametz in the property of a Jew on Pesach. [Piskeiy Teshuvos ibid based on Admur 436:20] However, Tzaruch Iyun, as although in 436:20 Admur says that when placing Chametz in a Jews property the decree of the sages still apply, nevertheless, perhaps that only refers to a case that the property belongs to one Jew or the like. However, when the property is public, which is the case with the Chametz which belongs to the municipality, then what difference does it make if its public to Jews or not, and thus the same logic used by the Minchas Yitzchak to exempt a dump of a municipality, should apply whether it belongs to Jew or gentile. To note that the Minchas Yitzchak himself does not differentiate between if the municipality is Jewish or not.
 The transgression, and whether its defined as Biblical or Rabbinical, is dependent on various factors, such as: 1) Is there a Kezayis of Chametz; 2) Is the Chametz still edible? 3) Is the Chametz dirty. 4) Was the Chametz sold to a gentile.
 Chelkas Yaakov ibid; Teshuvos Vehanhagos ibid in name of Steipler
 The common sale contracts that are given to the gentile include the following clause: Included in the sale is all pieces of Chametz found in all the mentioned areas, which do not have any sales worth, and are not purchasable, they belong to the gentile. [Clause in Shtar Mechira of Admur, brought in Hilchos Mechiras Chametz in end of Hilchos Pesach in Shulchan Aruch Harav] Thus, anyone who has performed Mechiras Chametz, under standard practice, has also sold his Chametz garbage. [Teshuvos Vehanhagos ibid in name of Steipler]
 Beir Moshe ibid
 The reason: As if one sells the Chametz garbage, it also includes the garbage bin in the sale, and forbids its use during Pesach. If one continues to use it on Pesach he shows that the entire sale was a mere trick, and is stealing the items of the gentile. Furthermore, it is unclear if in truth Chametz garbage is included in the sale, as the gentile certainly has no intent to purchase something that will be removed by the city in one to two days. [Beir Moshe ibid]
 Chelkas Yaakov ibid, in name of Agudas Eizov; Mishneh Halachos ibid; Teshuvos Vehanhagos ibid; Betzel Hachochma
 Minchas Yitzchak ibid; Shevet Halevi ibid
 Minchas Yitzchak ibid in name of Haggadas Agudas Eizov
 The reason: Chametz which is less than a Kezayis and is dirty [as a result of the garbage] does not contain an owning prohibition if disowned. [See Admur 442:28; M”A 442:10; M”B 442:33]
 Orchos Rabbeinu Pesach 34 that so was custom of Chazon Ish
 Minchas Yitzchak ibid that so he arranged, and so is done today