丈夫 and 元気: sturdiness and energy
Compare sturdy objects with human health and energy, and distinguish 元気がない from 丈夫ではない.
A sturdy object
このかばんは丈夫で、長く使えそうです。 says that the bag is sturdy and looks likely to last a long time. 丈夫, read じょうぶ, describes resistance to damage.
丈夫な椅子を選びました。 says that someone chose a sturdy chair. It does not mean a cheerful or energetic chair, so 元気な椅子 is not the ordinary equivalent.
Health and energy
祖母は体が丈夫です。 says that the grandmother has a robust constitution. 丈夫 also applies to people but does not merely describe a momentary mood.
子どもたちは元気に走っています。 says that the children are running energetically. 元気, read げんき, concerns health and liveliness.
Observing low spirits
今日は少し元気がありません。 says that the speaker is a little low on energy today. Fatigue, mood, or other reasons are possible; it does not identify a particular illness.
お元気ですか asks how someone is doing. Its subject and purpose differ from asking whether an object is sturdy with 丈夫ですか.
Does 元気がありません establish that someone normally has a weak constitution?
Show the answer and why
No. It may describe a current lack of energy. Usual health and the cause require separate information.