Ah well...

An interjection expressing resignation or mild disappointment. Almost just a verbalised sigh meaning "too bad"; "pity", or "so that's it". In short, for situations it is a softer form of "too bad"
The phrase is a variation of "Oh well":
-Oh well, it'll come back to me in a few minutes.
But
-"Ah well... I gave it my best shot.”

"Ah well" sounds a little more disappointed, while "Oh well" sounds more optimistic.
What is a Greek equivalent?

No idea, but I would say δε βαριέσαι in the second example. Δε βαριέσαι, έκανα ό,τι μπορούσα.
Before Daeman adds some music to that, here's my contribution (lyrics displayed on the clip, so not hard to follow)

Theseus said:

"Ah well" sounds a little more disappointed, while "Oh well" sounds more optimistic.

Click to expand...

Is there really any difference between "Ah..." and "Oh..."?

I would add that "τέλος πάντων" tends nowadays to cover various meanings, ranging from optimism to disappointment through boredom, resignation and/or indifference.

Ε, τι να κάνουμε;

Thanks for all the help here. To SBE thanks for the idiom & the song. Δεν βαριέσαι proves to be a very useful idiom. At https://forum.wordreference.com/threads/δεν-βαριέσαι.2418125/
The second meaning is the more relevant::-
"In the more colloquial version that you probably seek, imagine that two people are talking, someone mentions a problem or something unpleasant and the other responds "Δεν βαριέσαι..." with the meaning "what can you do...", "(I know it's rough) but..." or "take it easy..." or "let it slide...". Sometimes of course someone may talk about his own problems and may also use it to conclude like
"Ναι, το ξερώ ότι δεν είναι σωστό, αλλά δεν βαριέσαι..."
To rogne I think when I use 'ah well' I am expressing mere resignation. The syllable 'ah' uses more breathing out & a feeling of giving up. In 'oh well' I think there is a shade more annoyance & perhaps a slight shrug of the shoulders. Perhaps, perhaps not.
Τέλος πάντων & Ε, τι να κάνουμε are also useful phrases in my armoury. Thanks. :)