“The solutions that we need to find must take into consideration each person’s reality — it’s almost like tailored to fit,” he said. “The anger that is being expressed is sometimes very different from one area to another.”

The Yellow Vests take their name from the fluorescent hazard vests adopted by the protesters as a sign of their economic distress.

The challenges Mr. Macron will face in speaking to his fellow citizens are considerable.

He must connect with the many who feel abandoned and make clear not only that he understands their anger but also that he is prepared to listen to their ideas and take action, analysts said.

At the same time, he must project strong resolve against the destruction of private property to preserve the allegiance of the small-business owners who have seen their shops vandalized or who have been forced to close on successive Saturdays. Mr. Macron must also stand with the many French who have been outraged by the desecration of national monuments, or who have seen their cars burned or other property destroyed.