The objective of this paper is to establish new variational principles for symmetric boundary value problems. Let
V
V
be a Banach space and
V^*
V
∗
its topological dual. We shall consider problems of the type
\Lambda u=D \Phi(u)
Λ
u
=
D
Φ
(
u
)
where
\Lambda: V \to V^*
Λ
:
V
→
V
∗
is a linear operator and
\Phi: V \to \mathbb{R}
Φ
:
V
→
R
is a Gâteaux differentiable convex function whose derivative is denoted by
D\Phi
D
Φ
. It is established that solutions of the latter equation are associated with critical points of functions of the type
I_{\lambda, \mu}(u):= \mu \Phi^* (\Lambda u)-\lambda \Phi(u)- \frac{\mu-\lambda}{2}\langle \Lambda u, u \rangle,
I
λ
,
μ
(
u
)
:
=
μ
Φ
∗
(
Λ
u
)
−
λ
Φ
(
u
)
−
μ
−
λ
2
⟨
Λ
u
,
u
⟩
,
where
\lambda, \mu
λ
,
μ
are two real numbers,
\Phi^*
Φ
∗
is the Fenchel dual of the function
\Phi
Φ
and
\langle.,.\rangle
⟨
.
,
.
⟩
is the duality pairing between
V
V
and
V^*
V
∗
. By assigning different values to
\lambda
λ
and
\mu
μ
one obtains variety of new and classical variational principles associated to the equation
\Lambda u=D \Phi(u)
Λ
u
=
D
Φ
(
u
)
. Namely, Euler-Lagrange principle (for
\mu=0
μ
=
0
,
\lambda=1
λ
=
1
and symmetric
\Lambda
Λ
), Clarke-Ekeland least action principle (for
\mu=1
μ
=
1
,
\lambda=0
λ
=
0
and symmetric
\Lambda
Λ
), Brezis-Ekeland variational principle (
\mu=1
μ
=
1
,
\lambda=-1
λ
=
−
1
) and of course many new variational principles such as
I_{1,1}(u)= \Phi^* (\Lambda u)- \Phi(u),
I
1
,
1
(
u
)
=
Φ
∗
(
Λ
u
)
−
Φ
(
u
)
,
which corresponds to
\lambda=1
λ
=
1
and
\mu=1
μ
=
1
. These new potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems