Use request dispatching instead of redirection.
RequestDispatcher view = Request.getRequestDispatcher("index.jsp");
view.forward(request, response);
This would forward the same request object to index.jsp. If the username is not already a request parameter pass it as an attribute
request.setAttribute("username", username); // before doing the forward
and retrieve it in your index.jsp as
<%= "Welcome " + request.getAttribute("username")%>
Alternatively, since the user has logged-in (through backend authentication via JDBC as you state) you can save the username (and other things pertaining to him) in the HttpSession
.
HttpSession session = request.getSession();
session.setAttribute("username", username);
Now, you can forward (recommended) or choose to redirect as before but index.jsp would change to
<%= "Welcome " + session.getAttribute("username")%>