I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\nabla u|^2 + \int_\Omega \chi_{\{u > 0\}}$$ (the constraint $u \geq 0$ could also be omitted if convenient). Here $\chi_A$ means the indicator function of the set $A$; so it is $1$ on $A$ and $0$ elsewhere.

This is (I believe) the Alt-Caffarelli functional analyised in their paper [1], which cites an old (and hard to decipher) German text by Friedrichs [2] for the derivation of the E-L equations. Is there a more modern textbook/paper or other that gives the proof of this functional (or related ones, involving the indicator function)?

References

[1] H. W. Alt; L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325, 105-144 (1981). MR618549, Zbl 0449.35105.

[2] K. O. Friedrichs, Über ein Minimumproblem für Potentialströmungen mit freiem Rande., Math. Ann. 109, 60-82 (1933). JFM 59.1447.01, Zbl 0008.16605.

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\nabla u|^2 + \int_\Omega \chi_{\{u > 0\}}\label{1}\tag{$J$}$$ (the constraint $u \geq 0$ could also be omitted if convenient). Here $\chi_A$ means the indicator function of the set $A$; so it is $1$ on $A$ and $0$ elsewhere.

This is (I believe) the Alt-Caffarelli functional analyised in their paper [1], which cites an old (and hard to decipher) German text by Friedrichs [2] for the derivation of the E-L equations. Is there a more modern textbook/paper or other that gives the proof of this functional (or related ones, involving the indicator function)?

References

[1] H. W. Alt; L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325, 105-144 (1981). MR618549, Zbl 0449.35105.

[2] K. O. Friedrichs, Über ein Minimumproblem für Potentialströmungen mit freiem Rande., Math. Ann. 109, 60-82 (1933). JFM 59.1447.01, Zbl 0008.16605.

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\nabla u|^2 + \int_\Omega \chi_{\{u > 0\}}$$ (the constraint $u \geq 0$ could also be omitted if convenient). Here $\chi_A$ means the indicator function of the set $A$; so it is $1$ on $A$ and $0$ elsewhere.

This is (I believe) the Alt-Caffarelli functional analyised in their paper, which cites an old (and hard to decipher) German text by Friedrichs for the derivation of the E-L equations. Is there a more modern textbook/paper or other that gives the proof of this functional (or related ones, involving the indicator function)?

I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for $$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\nabla u|^2 + \int_\Omega \chi_{\{u > 0\}}$$ (the constraint $u \geq 0$ could also be omitted if convenient). Here $\chi_A$ means the indicator function of the set $A$; so it is $1$ on $A$ and $0$ elsewhere.

This is (I believe) the Alt-Caffarelli functional analyised in their paper [1], which cites an old (and hard to decipher) German text by Friedrichs [2] for the derivation of the E-L equations. Is there a more modern textbook/paper or other that gives the proof of this functional (or related ones, involving the indicator function)?

References

[1] H. W. Alt; L. A. Caffarelli, Existence and regularity for a minimum problem with free boundary, J. Reine Angew. Math. 325, 105-144 (1981). MR618549, Zbl 0449.35105.

[2] K. O. Friedrichs, Über ein Minimumproblem für Potentialströmungen mit freiem Rande., Math. Ann. 109, 60-82 (1933). JFM 59.1447.01, Zbl 0008.16605.