What is the real functional way?

gistfile1.java

private Map<String, List<String>> mapHeaders(Header[] headers) {
    Map<String, List<String>> headerMap = new HashMap<>();
    Arrays.stream(headers)
            .forEach(h -> {
                List<String> values = headerMap.get(h.getName());
                if(values == null) {
                    values = new ArrayList<String>();
                    headerMap.put(h.getName(), values);
                }
                values.add(h.getValue());
            });
    return headerMap;
}

// slightly improved version. But I still have the forEach
private Map<String, List<String>> mapHeaders(Header[] headers) {
    Map<String, List<String>> headerMap = new HashMap<>();
    Arrays.stream(headers)
            .forEach(h -> {
                headerMap.computeIfAbsent(h.getName(), k -> new ArrayList<>()).add(h.getValue());
            });
    return headerMap;
}

// The answer!
private Map<String, List<String>> mapHeaders(Header[] headers) {
    Map<String, List<String>> headerMap = Arrays.stream(headers)
            .collect(Collectors.groupingBy(Header::getName, Collectors.mapping(Header::getValue, Collectors.toList())));
    return headerMap;
}

With static imports and inlining:

// The answer!
private Map<String, List<String>> mapHeaders(Header[] headers) {
    return stream(headers).collect(groupingBy(Header::getName, mapping(Header::getValue, toList())));
}

Could be more readable.