HTML 5 form which send infos in a PHP e-mail - php

Question

I would use some help because I don't know what to do at all...
I created an HTML form (which isn't a problem) :
<form method="post" action="form-page.php" class="cta">
<div class="row gtr-uniform gtr-50">
<div class="col-8 col-12-xsmall"><input type="text" name="user_name" id="name" required placeholder="Votre nom"></div>
<div class="col-8 col-12-xsmall"><input type="email" name="email" id="email" placeholder="Votre adresse e-mail" /></div>
<div class="col-8 col-12-xsmall"><textarea name="user_message" id="msg" required placeholder="Votre message"></textarea></div>
<div class="col-4 col-12-xsmall"><input type="submit" value="Envoyer." class="fit primary" /></div>
</div>
</form>
I created the file "form-page.php" which treat the form with this code :
<?php
// We get the infos of the form
$user_name = $_POST['user_name']; // Nom / Pseudo.
$email = $_POST['email']; // e-mail.
$user_message = $_POST['user_message']; // Message.
//=========
// We put the adress where the mail is going to be send.
//P.S : "myemail#gmail.com" isn't MY email, I just put it here to hide my email :)
$mail = 'myemail#gmail.com';
//=========
// We filtrate servers to avoid bugs.
if (!preg_match("#^[a-z0-9._-]+#(hotmail|live|msn).[a-z]{2,4}$#", $mail))
{
$passage_ligne = "\r\n";
}
else
{
$passage_ligne = "\n";
}
//=========
// We define the subject of the mail.
$sujet = "Suggestion ou bug de ton site, Pokemon Reality";
//=========
// We create the header of the email.
$header = "From: \"$user_name\" $email".$passage_ligne;
$header.= "MIME-Version: 1.0".$passage_ligne;
//==========
// We create the message.
$message = $user_message;
//==========
// Finally, we send the email.
mail($mail,$sujet,$message,$header);
//==========
?>
Buuuut... This doesn't work! After I press the button "Envoyer." of the form, I wait while the page is loading and I see an error :
ERROR
ERROR2
I didn't touch to anything, and before that, I test it 3 times, and it send the mail, but there was a blank page.
So I want to fix the error, and then, I want to redirect the user to the form and say "Thank You !".
Thanks for your help!