To elaborate on what taejo said, in Danish personal names do not change spelling, even if they mean the same. It is not uncommon to see people's given names or surnames being small variation of similar names. But in these terms, you should think of »Kierkegård« and »Kierkegaard« in the same manner as 'Philip' or « Philippe ». They are the same name, pronounced the same way and mean the same, but spelt differently.
Of course, some do make mistakes and spell it in a more common way. My surname, for instance, is on occasion spelt »Utzon« because of Jørn Utzon (who designed the Sydney Opera House), who's surname sound like mine.
But the problem in Danish isn't just limited to personal names and proper nouns. In English, words like 'subtle', 'segue' and 'colonel' are not pronounced as you'd think (from their spelling), because they are loanwords, and you just need to know this. This problem is more unfortunate in Danish, as it is far more common, even amongst Germanic words.
It can be hard to explain to foreigners, why the o's in »mormor« is pronounced differently ([ɒ] and [o], respectively).
I don't know about Danish, but in German, ü "decomposes" into "ue" and is sorted that way in phonebooks. Herr Müller wouldn't be surprised to see his name spelled Mueller in ASCII-only contexts. OTOH, in some families it is traditional to write the name without the umlaut, and in these cases it's wrong to write it with, so Frau Mueller wouldn't like to see her surname written Müller (though it has the same pronunciation and origin). Famous examples are Goebbels and Goethe (not Göbbels and Göthe). There's a similar situation with ß vs. ss.
So if Danish is like German, you can't call Kierkegaard Kierkegård, but they're still the "same" surname and appear next to each other in the phone book.
That's generally the same in Danish, although one historical tidbit to add to it is that the aa->å respelling in Danish happened only in 1948. With that spelling reform, all normal words using 'aa' as a digraph were respelled to use 'å', so for example "cemetery" went from "kirkegaard" to "kirkegård". The latter is now the standard spelling, and "kirkegaard" would only be accepted in situations where it's a technical necessity (e.g. using 7-bit ASCII).
If that had been done across the board, Danish would be easy to sort, since the only digraph would have been eliminated, replaced with an atomic character. However the reform was not made mandatory for personal names; people could choose to retain the 'aa' or switch to 'å', and many retained the traditional spelling (the traditional spelling was also retained for historical figures). There was some discussion about officially reforming the names of cities to use the new 'å' spellings, but both Aarhus and Aalborg (the 2nd- and 3rd-largest cities) strongly objected. So placenames were not reformed (unless a specific locality opted in). You do occasionally see the spellings Århus and Ålborg, especially in writing closer to the time of the reform, but those spellings didn't catch on.
Actually, Århus changed its name immediately after the reform, but Aalborg did not. It was not until 2012 that Århus decided to change its name back to »Aarhus« to become 'easier to spell internationally' (which was the argument).
I still spell it Århus to taunt the idea that it was the »å« that prevented Århus from the same business opportunities as Copenhagen -- I'm sorry, København.
